\begin{document}

\title{Convergence of the mirror to a rational elliptic surface}
\author{Lawrence Jack Barrott}
\date{}

\maketitle

\begin{abstract}

The construction introduced by Gross, Hacking and Keel in~\cite{MirrorSymmetryforLogCY1} allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In that paper they proved two convergence results. Firstly that if the intersection matrix of $D$ is not negative semi-definite then the family they construct lifts to an algebraic family. Secondly they prove that if the intersection matrix is negative definite then their construction lifts along certain analytic strata on the base, and then over a formal neighbourhood of this. In the original version of that paper they claimed that if the intersection matrix were negative semi-definite then family in fact extends over an analytic neighbourhood of the origin but gave an incorrect proof.

In this paper we correct this error. We explain how the general Gross-Siebert program can be used to reduce construction of the mirror to such a surface to calculating certain relative Gromov-Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an $I_4$ fibre.

\end{abstract}

\section*{Introduction}

The geometry of surfaces often is a rich playground in which to test new conjectures and constructions. In~\cite{MirrorSymmetryforLogCY1} the authors introduced a construction of a mirror family to certain pairs $(S,D)$, called Looijenga pairs. This built upon work of~\cite{AurouxKatzarkovOrlov} who had made predictions of the mirror families to various del Pezzo surfaces. These del Pezzo surfaces are the smooth Fano surfaces, they have ample anti-canonical class. However both constructions make sense when $-K_S$ is not ample, but merely nef. This corresponds to the case where $S$ is a rational elliptic surface and $D=F$ a rational fibre.

There is another direction from which to approach this question. Gross, Hacking and Keel were trying to understand mirror symmetry over the moduli space of $K3$ surfaces. Recall that there is a classification of the degenerations of a $K3$ surface.

\begin{theorem}[Structure of $K3$ degenerations]

Let $\pi:X \rightarrow \AA^1$ be a semi-stable family such that the fibre $X_t$ for $t \neq 0$ is a smooth $K3$ surface and the canonical bundle of the family is trivial. Then the central fibre $X_0$ is isomorphic to one of the following types

  \begin{itemize}
  \item[I] A smooth $K3$ surface.
  \item[II] A chain $S_1, \ldots S_n$ of surfaces such that $S_1$ and $S_n$ are rational, the other components are ruled surfaces over elliptic
curves and $S_i \cap S_{i+1}$ are smooth elliptic curves.
    \item[III] A complex of rational surfaces $S_1 \ldots S_n$ such that $V_i \cap V_j$ are smooth rational curves. The dual complex gives a triangulation of $S^2$.
  \end{itemize}

In the second case one can perform a birational modification of the family to produce a type II degeneration where the central fibre is the union of only two components. In this case the class $S_1 \cap S_2$ is an anti-canonical curve on each component.

\begin{proof}

  See~\cite{DegenerationsOfK3SurfacesAndEnriquesSurfaces}.
  
\end{proof}  

\end{theorem}

So one could hope to apply their construction to a point on this boundary, a type II degeneration, and then extend out the family constructed over the entire moduli space. This paper therefore answers first half to constructing the mirror to a type II degeneration. There is much more work to be done before one can extend this example to the type II fibre.

This paper begins by recalling the general philosophy of the Gross-Siebert program, explaining the appearance of scattering diagrams and broken lines. This should be seen as a continuation of my work in~\cite{ExplicitEquationsForMirrorFamiliesToLogCalabiYauSurfaces} which features a slightly introduction to this philosophy. We explore how the combinatorial nature of these objects limits the data we must calculate. In our case this reduces the problem to calculating the number of rational curves meeting the fixed fibre $F$ with tangency orders one and two.

Having done that we discuss work of Bryan and Leung who proved that the number of curves in a class $E + nF$ for $E$ a section appear as coefficients of a known modular form. These include all the curves tangent to $F$ to order one. We extend their results to our case, where we are interested in relative invariants rather than absolute invariants.

With this warm up done we move to the main challenge of this paper, counting the curves tangent to the boundary at a single point with tangency order two. To do this we construct a threefold whose geometry allows us to lift curves from $S$. This threefold is a nef complete intersection in a toric variety, hence we can construct the Gromov-Witten theory via Givental's $I$ and $J$ functions. Proving relations between the invariants on $S$ and on the threefold requires that we delve into the obstruction theories on both, applying techniques from~\cite{VirtualPullbacks} and~\cite{VirtualPushforward}. 

Having done this we then return to the original construction and deduce that it converges in a neighbourhood of the origin, exactly as desired.

Throughout this $S$ will denote a rational elliptic surface, which by Lemma IV.1.2 of~\cite{GeometryofEllipticSurfaces} is isomorphic to the blowup in the nine points lying on the intersection of two cubics in $\PP^2$. The Chow group $A_1 (S)$ is spanned by $H$, the pullback of a hyperplane, and $E_1, \ldots, E_9$, the exceptional curves of the blowup. The Chow groups $A_0(S)$ and $A_2(S)$ are both one dimensional, spanned by a point and a fundamental class respectively. $F$ will denote a general fibre, whilst $F_0$ will denote an $I_4$ fibre, a cycle of four $-2$-curves. We will write $S^\dagger$ for $S$ together with the divisorial log structure coming from $F_0$. By an inclusion of components $i:X \rightarrow Y$ we mean an isomorphism from $X$ to a union of connected components of $Y$.

This work was the bulk of my thesis project, and I must thank Mark Gross for suggesting this problem to me. Paul Hacking and Sean Keel also contributed both in writing their paper with Mark Gross and through discussions and motivation for the various different parts of the construction. Tom Coates gave a wonderfully clean description of the work of Givental which greatly improved my understanding of how those techniques work. My thesis was funded by an Internal Graduate Studentship of Trinity College Cambridge, an extension grant from the Department of Pure Maths and Mathematical Statistics in Cambridge, and a research studentship from the Cambridge Philosophical Society. 

\section{The Gross-Siebert program}

The Gross-Siebert program began as an attempt to make rigorous geometric constructions from the SYZ conjecture. 

\begin{conjecture}[The SYZ conjecture]

Let $X, \check{X}$ be a mirror pair of \emph{Calabi-Yau varieties}. Then there is an affine manifold with singularities $B$ and maps $\phi: X \rightarrow B$, $\check{\phi}: \check{X} \rightarrow B$ which are dual special Lagrangian fibrations. This is shown schematically on the next page.

\end{conjecture}

In the case of toric varieties this conjecture is tractible, the base is a polytope for the variety and the fibration the momentum map. This map degenerates over the boundary in a very controlled manner, showing that this fibration is certainly not trivial. Gross and Siebert suggested that even without knowing the exact form of the fibration one could still recover enough information to reconstruct the mirror family. To do this one must reincorporate information about the singularities via a scattering diagram.

The construction of~\cite{MirrorSymmetryforLogCY1} studies pairs $(S,D)$ where $S$ is a smooth rational surface and $D$ a cycle of rational curves (otherwise known as a \emph{Looijenga pair}). It states that the base for the fibration of should be the \emph{dual intersection complex of $D$} which we construct below. This is inspired by the toric case where this complex is the base of the momentum map. Of course this base need not embed into $\RR^2$, rather it has a singularity at the origin. This is precisely as expected from the SYZ picture and is where interesting geometry can enter the picture.

\begin{construction}

Let $D = D_1 \cup D_2 \cup \ldots \cup D_n$ be a cycle of rational curves on a smooth surface $S$ such that $D_i \cap D_j$ is a single point just when $i$ and $j$ differ by 1 mod $n$ and otherwise is empty (in the case $n=2$ we relax this to saying that there are two points in the unique intersection $D_1 \cap D_2$). Then $\Delta_{(S,D)}$ contains precisely one zero-dimensional cell 
$\{0\}$ corresponding to the interior $S \setminus D$. For each component $D_i$, $\Delta_{(S,D)}$
contains a one-dimensional cone with $v_i$ its primitive generator.
Attach the zero-dimensional cone as 0 inside each of these rays. Now introduce a two-dimensional cone in $\Delta_{(S,D)}$ for each intersection point of $D_i \cap D_j$, spanned by $v_i$ and $v_j$. This produces the \emph{dual intersection complex} $\Delta_{(S,D)}$ as a cone complex but it carries more structure. 
We write $B$ to be the underlying topological space of the cone complex
$\Delta_{(S,D)}$. It is homeomorphic to $\RR^2$. We give $B\setminus\{0\}$
an affine manifold structure by defining an affine coordinate chart by embedding
the union of the cones $\RR^{\geq 0} v_{i+1} \oplus \RR^{\geq 0} v_i $ and $\RR^{\geq 0} v_{i} \oplus \RR^{\geq 0} v_{i-1}$ into $\RR^2$ via the relations $v_{i-1} \mapsto (1,0)$, $v_i \mapsto (0,1)$ and $v_{i+1} \mapsto (-1,-D_i ^2)$. This expresses $\Delta_{(S,D)}$ not just as a complex of sets but an affine manifold with singularities (indeed a single singularity at the origin).

\end{construction}

Let us draw out in Figure 1 this complex, or rather its universal cover, in the case where $S$ is a rational elliptic surface and $D = D_1 \cup \ldots \cup D_4$ is a cycle of $-2$-curves, i.e. an $I_4$ fibre.

\begin{figure}
\caption{The universal cover of the dual intersection complex for $S^\dagger$.}
\label{fig:DualIntersectionRES}
\end{figure}

In the toric case one can construct the mirror family via the Mumford degeneration. We give the general construction first, then apply it to our setting and then explain how to apply it to the toric case.

\begin{example}

  Let $B \subset N_\RR$ be a lattice polyhedron, $\mathcal{P}$ a lattice polyhedral decomposition of $B$ and $\phi: B \rightarrow \RR$ a strictly convex piecewise linear integral function, i.e. it is strictly convex between different cells of $\mathcal{P}$. One takes the graph over $\phi$ to produce a new polyhedron

\[\Gamma (B, \phi) := \{ (n, r) \in N_\RR \oplus \RR \mid n \in B, r \geq \phi (n)\}\]

This produces a lattice polyhedron unbounded in the positive direction on $\RR$. To construct a family from this we perform a cone construction, let $C (\Gamma)$ be the closure of the cone over $\Gamma (B, \phi)$:
\[
C(\gamma) = \overline{\{(n,r_1,r_2)\in N_{\RR}\oplus\RR\oplus\RR\,|\,
(n,r_1)\in r_2 \Gamma(B,\phi)\}}.
\]
 This carries an action of $\RR^+$ given by translating the second component of $\Gamma(B, \phi)$. The integral points of $C (\Gamma)$ form a graded monoid and we can take $Proj$ of this to produce a variety projective over $\AA^1$. We write this $\PP_{\Gamma (B,\phi)}$. The general fibre is a toric variety, 
whilst the fibre over the origin is a union of toric varieties.

\end{example}

Gross, Hacking and Keel modify this construction by incorporating singularities into the base. Outside the toric case we lack global coordinates for the Mumford construction. Instead they introduce a bundle with local coordinates.

\begin{definition}
\label{def:sheaf-of-monoids}
  Let $(S,D)$ be a Looijenga pair with associated dual intersection complex $\Delta_{(S,D)}$ and let $\eta: NE (S) \rightarrow M$ be a homomorphism of monoids. We want to construct a multi-valued piecewise linear function on $\Delta_{(S,D)}$ which will bend only along the one-cells. This will be a collection of
piecewise linear functions defined on open subsets of $B$ which differ by linear
functions on overlaps. Such functions are determined by their bends at
one-cells, which are encoded as follows. For a
one-cell $\tau=\RR^{\geq 0} v_i$, choose an orientation $\sigma_+$ and 
$\sigma_-$ of the two two-cells separated by $\tau$ and let $n_\tau$ be the unique primitive linear function positive on $\sigma_+$ and annihilating $\tau$. We want
to construct a representative $\phi_i$ for $\phi$ on $\sigma_+\cup \sigma_-$.  Writing $\phi_+$ and $\phi_-$ for the linear function defined by $\phi_i$ on $\sigma_+$ and $\sigma_-$, the function $\phi_i$ is then defined up to a linear
function by the requirement that
\[ \phi_+ - \phi _- = n_\tau \otimes \eta ([D_i])\]
Such a function is convex in the sense of~\cite{MirrorSymmetryforLogCY1} Definition 1.11, and one says that it is \emph{strictly convex} if $\eta([D_i])$ is not invertible for any $i$. 

This function determines a $M^{gp}\otimes\RR$-torsor $\PP$ as defined 
in~\cite{GrossSiebert}, Construction 1.14, on $B\setminus \{0\}$
which is trivial on each $(\sigma_+\cup\sigma_-)\setminus\{0\}$, i.e.,
is given by $((\sigma_+\cup\sigma_-)\setminus \{0\})\times (M^{gp}\otimes\RR)$.
These trivial torsors are glued on the overlap of two adjacent such sets,
namely on $\RR^{\ge 0} v_{i} \oplus\RR^{\ge 0}v_{i+1}$, via the map
\[ (x,p) \mapsto (x, p + \phi_{i+1} (x) - \phi_i (x))\]
which induce isomorphisms on the monoid of points lying above $\phi$.
This construction is designed to allow us to run the Mumford construction locally, even if we cannot run it globally. 
Write
$\pi: \PP \rightarrow B\setminus\{0\}$, and we write $\mathcal{M}$ for the 
sheaf $\pi_* (\Lambda _{\PP})$, bearing in mind that $\PP$ also has the
structure of an affine manifold. Then there is a canonical exact sequence
  \[ 0 \rightarrow \underline{M^{gp}} \rightarrow \mathcal{M} \xrightarrow{r} \Lambda_{B} \rightarrow 0 \]
  We write $r$ for the second map in this sequence. This will not be mentioned again until we define the canonical scattering diagram so keep this in mind until then.
Furthermore, \cite{GrossHackingSiebert}, Definition 1.16 gives a subsheaf
of monoids $\mathcal{M}^+\subset \mathcal{M}$.
 
\end{definition}

If one performs this with a toric variety relative to its boundary and pairs the function $\phi$ with an ample class then one obtains a height function for the Mumford degeneration. In particular the general fibre of this family will be isomorphic to the original toric variety. At the moment this lacks any data about the singularities of the SYZ fibration. Without this data we cannot hope to correctly recreate the mirror. Let us recall how the singularities appear.

The authors in~\cite{MirrorSymmetryforLogCY1} claim that the mirror should be affine, the spectrum of the symplectic cohomology of $(S,D)$. This is not explicated there, and indeed is not true in general. Daniel Pomerleano has written down three dimensional examples whose ``mirror family'' is generically not smooth, one must pass to a resolution of this family to obtain the actual mirror family. In any case we accept this and ask how to describe the symplectic cohomology.

By definition this ring counts Maslov index two disks with boundaries lying on Lagrangians. Furthermore given a Maslov index zero disk we may glue it to a standard index two disk. Now the image of one of these index two disks under the fibration is a topological space retracting onto a piecewise linear skeleton. This skeleton is a balanced piecewise linear graph on $B$ with three legs, one ending on the central singularity and two more passing off to infinity. The linear structure breaks down near points where a Maslov index zero disk attaches. This process goes by the name \emph{tropicalisation}.

Gross, Hacking and Keel take this definition and study all possible trivalent tropical curves, piecewise linear balanced graphs, on $B$. They attempt to reformulate the Maslov index zero disks algebraically as a scattering diagram on $B$.

\begin{definition}

  Let $B$ be an affine manifold with a single singularity, with 
$B$ homeomorphic to $\RR^2$ with singularity at the origin, so that $B^*:=
B\setminus \{0\}$ is an affine manifold.
Let $\mathcal{M}$ be a locally constant sheaf of abelian groups
on $B^*$ with a subsheaf of monoids $\mathcal{M}^+\subset\mathcal{M}$
and equipped with a map $r:\mathcal{M} \rightarrow \Lambda_B$.
Let $\mathcal{J}$ be a sheaf of ideals in $\mathcal{M}^+$ with stalk $\mathcal{J}_x$ maximal in $\mathcal{M}^+_x$ for all $x \in B^*$.
Let
$\mathcal{R}$ denote the sheaf of rings locally given by the completion of 
$k[\mathcal{M}^+]$ at $\mathcal{J}$. 
 A \emph{scattering diagram with values in the pair $(\mathcal{M}, \mathcal{J})$ on $B$} is a function $f$ which assigns to each rational ray from the origin an section of the restriction of $\mathcal{R}$ to the ray.
We require the following properties of this function:

  \begin{itemize}
    \item For each $\mathfrak{d}$ one has $f(\mathfrak{d}) = 1$ mod 
$\mathcal{J}|_{\mathfrak{d}}$.
    \item For each $n$ there are only finitely many $\mathfrak{d}$ for which $f(\mathfrak{d})$ is not congruent to $1$ mod $\mathcal{J}|_{\mathfrak{d}}^n$. These $\mathfrak{d}$ are called \emph{walls}.
    \item For each ray $\mathfrak{d}$ and for each monomial $z^{p}$ appearing in $f(\mathfrak{d})$ one has $r(p)$ tangent to $\mathfrak{d}$. A line for which $r(p)$ is a positive generator of $\mathfrak{d}$ for all $p$ with $c_p\not=0$
 is called an \emph{incoming ray}. If instead
$r(p)$ is a negative generator of $\mathfrak{d}$ for all $p$ with $c_p\not=0$,
it is called an \emph{outgoing ray}.
  \end{itemize}
  
  We denote such an object by the tuple $(B, f, \mathcal{M}, \mathcal{J})$. We say that $(B,f,\mathcal{M},\mathcal{J})$ is obtained from $(B,f',\mathcal{M},\mathcal{J})$ by \emph{adding outgoing rays} if for each ray $\mathfrak{d}$ one can write $f(\mathfrak{d}) = f' (\mathfrak{d}) (1 + \sum (c_p z^p))$ where for each monoid element $p$ with $c_p\not=0$ the vector $-r(p)$ is a generator of $\mathfrak{d}$.

\end{definition}

In our case the sheaf of monoids $\mathcal{M}^+$ will be as given in
Definition \ref{def:sheaf-of-monoids}, with the monoid $M$ being
 a finitely generated sharp submonoid of $H_2(S,\ZZ)$
containing $NE(S)$, the monoid generated by effective curves on $S$. Being sharp means the only invertible element of $M$ is the identity element, and so the maximal ideal is just the complement of the identity. We introduce the choice of scattering diagram, the \Lawrence{\emph{canonical scattering diagram},} and then motivate and define each term appearing. A ray $\mathfrak{d} = (a v_i + b v_{i+1}) \RR^{\geq 0}$ specifies a blow up of $S$ given by refining the fan until $\mathfrak{d}$ is a one-cell and all the two-cells are integrally isomorphic to $(\RR^{\geq 0})^2$. The ray $\mathfrak{d}$ then corresponds to a component $C$ in the inverse image of $D$. To this ray then we assign the power series
\[f(\mathfrak{d}) := \exp \left[ \sum_\beta k_{\beta} N_\beta z^{\eta (\pi_* (\beta)) - k_\beta m_{\mathfrak{d}}'}\right],\]
where $m_{\mathfrak{d}}'$ is the unique lift of a primitive outward
pointing tangent vector $m_{\mathfrak{d}}$ to $\mathfrak{d}$ not contained
in the relative interior of $\mathcal{M}|_{\mathfrak{d}}$.
The number $N_{\beta}$ counts the number of relative curves mapping to $(S,D)$ tangent to $C$ to maximal order $k_\beta$ at a single point as outlined in~\cite{MirrorSymmetryforLogCY1} at the beginning of section 3.
 By the Riemann-Roch formula this is of dimension
\[ dim \: S - 3 - K_S. \beta - k_\beta + 1 = 0.\]
and applying~\cite{GromovWittenInvariantsInAlgebraicGeometry} this produces a Gromov-Witten invariant. This construction may also be run using logarithmic Gromov-Witten invariants rather than these blow ups, see~\cite{LogarithmicGromovWittenInvariants} for the definition, and we take this approach later, defining the relevant theory.

The tropicalisation of a Maslov index two disk is trivalent and therefore we could study each leg separately. A broken line formalises one leg of this picture, the full picture arises once we glue two of these together

\begin{definition}

  A \emph{broken line from $v\in B(\ZZ)$ to $P \in B$} on a scattering diagram $(B, f, R, J)$ is a choice of piecewise linear function $l: \RR^{\leq 0} \rightarrow B$ and a map $m: \RR^{\leq 0} \rightarrow \prod_{t\in\RR^{\leq 0}}
\mathcal{R}_{l(t)}$ such that the following hold:

  \begin{itemize}
  \item $l$ has only finitely many points where it is non-linear and these only occur where $l$ maps into a rational ray of $B$.
  \item $l (0) = P$.
  \item $l(t)$ lies in the same cone as $v$ and is parallel to $v$ for all $t$ sufficiently negative.
  \item $l$ does not map an interval to a ray through the origin.
    \item $m(t)\in \mathcal{R}_{l(t)}$ for all $t$ and is a monomial in
this ring, written as $c_t z^{m_t}$. Further, on each domain of linearity
of $l$, $m(t)$ is given by a section of $\mathcal{R}$ pulled back to 
this domain of linearity.
  \item For $t$ very negative, $m_t\in \mathcal{M}^+_{l(t)}$ is the
unique element not lying in the interior of this monoid satisfying
$r(m_t) = v$.
  \item $r(m(t_0)) = -{ \partial l \over \partial t}\big|_{t=t_0}$ wherever $l$ is linear.
  \item $m$ only changes at those points where $l$ is non-linear.
  \item Let $t \in \RR_{\leq 0}$ be a point where $l$ is non-linear. We write $\partial(l_+)$, $\partial(l_-)$, $m_+$ and $m_-$ for the values of $\partial l \Lawrence{/} \partial t$ and $m$ on either side of $t$. Suppose that $l(t)$ lies on a ray $\mathfrak{d}$ with primitive normal vector $n_{\mathfrak{d}}$, negative on $\partial (l_-)$. Then $m_+$ is a monomial term of $m_- f_{\mathfrak{d}}^{\langle n_{\mathfrak{d}}, \partial (l_-)\rangle}$.
\end{itemize}

\end{definition}

The key data we will need is the count of pairs of pants, which are expressed in terms of broken lines as pairs of broken lines $(l_P, m_P)$  and $(l_Q, m_Q)$ from $P$ and $Q$ respectively to an irrational point near to $R$ such that 
$(\partial l_P/\partial t)|_{t=0} + (\partial l_Q/\partial t)_{t=0} + R=0$. Let $T_{P,Q \rightarrow R}$ denote the set of such pairs.

Now to construct an algebraic version of the symplectic cohomology we follow~\cite{CanonicalBasesForClusterAlgebras} and for each integral point $P$ in $B$ introduce a symbol $\vartheta_P$. As a $k$-vector space the ring $QH (\check{W})$ is freely generated by the $\vartheta_P$. We take the content of~\cite{MirrorSymmetryforLogCY1} Theorem 2.34 as the definition of the product of $\vartheta_P$ and $\vartheta_Q$, so this product is equal to

\[ \sum_{R} \sum_{((l_P, m_P), (l_Q, m_Q)) \in T_{P,Q \rightarrow R}} m_P (0) m_Q (0) \vartheta_R\]

This is analogous to the approach of ~\cite{GrossHackingSiebert}. A key result of ~\cite{MirrorSymmetryforLogCY1} is that for fixed order $J^n$ and for a consistent scattering diagram in the sense of
~\cite{MirrorSymmetryforLogCY1}, Definition 2.26, this does not depend on the choice of irrational point near $R$. This consistency property will hold
in particular for the canonical scattering diagram.
This sum need not terminate and indeed will in general only produce a power series. However, in the case that $D$ supports an ample divisor, this power series will in fact be a polynomial.

Now let us forget everything about the scattering diagram and suppose that it is as bad as possible, so it has rays in every direction. We can use the integral structure to limit the possible terms appearing in the product. Let us explain how to apply this idea in the case of a rational elliptic surface.

\begin{example}

  We apply the balancing condition to limit the bends which can occur. Consider the $y$-coordinate of a point of $B$, $y(P)$. This extends to the tangent bundle of $B$ and we write $y(v)$ for this extension. Importantly for us $y(-)$ is positive on points of $B$, and given vectors $v_i$ with $\sum v_i = 0$ we have $\sum y(v_i) = 0$. Let $(l,m)$ be a broken line, we claim that $E(\partial l / \partial t)$ is an increasing function on the linear components of $l$. At points where $l$ is non-linear by definition the tangent vector changes by a positive generator of the ray $\mathfrak{d}$, and $E$ is positive on such a generator. Similarly if $l$ crosses from one maximal cell to another then the convexity of $y$ shows that $y(\partial l / \partial t)$ increases.

  Now suppose that $(l_1, m_1)$ and $(l_2, m_2)$ are two broken lines from $v_1$ and $v_2$ respectively combining to form a pair of pants at $P$. By definition we have an equality:

  \[ y(\partial l_1/\partial t)|_{t=0} + y(\partial l_2/\partial t)|_{t=0} + y(P) = 0 \]

  Since $y(P)$ is non-negative and $-y(v_i)\le y((\partial l_i/\partial t)|_{t=0})$ we see that $y(v_1)+y(v_2)$ determines how many times such broken lines can bend. By restricting to the case where $F_0$ is an $I_4$ fibre we need only consider products with $y(v_i) = 1$, and so the only possible pairs of pants which can appear are the following:

  \begin{enumerate}

  \item Pairs where neither broken line bends.
  \item Pairs where one broken line bends along a monomial $m$ with $y(r(m)) = 1$.
  \item Pairs where one broken line bends along a monomial $m$ with $y(r(m)) = 2$.
  \item Pairs where both broken line bends along monomials $m_i$ with $y(r(m_i)) = 1$.
  \item Pairs where one broken line bends twice along monomials $m_i$ with $y(r(m_i)) = 1$.
    
  \end{enumerate}

Now $y(r(m))$ describes the order to which the corresponding curve meets the boundary, so we see that we must the calculate the number of curves tangent to the boundary at a single point to order one or two.
  
\end{example}

This also tells us the general form of the equations defining the mirror:

\begin{align}
  \vartheta_{D_1} \vartheta_{D_3} & = f_{(2,2)} \vartheta_{2D_2} + f_{(6,2)} \vartheta_{2D_4} + \sum f_{(i,1)} \vartheta_{D_i} + f_0   \label{eqn:MirrorToRES1} \\
  \vartheta_{D_2} \vartheta_{D_4} & = g_{(0,2)} \vartheta_{2D_1} + g_{(4,2)} \vartheta_{2D_3} + \sum g_{(i,1)} \vartheta_{D_i} + g_0 \label{eqn:MirrorToRES2}\\
  \vartheta_{D_i}^2 & = (1+r^{i}_{(i,2)}) \vartheta_{2D_i} + r^{i}_{(i+2,2)} \vartheta_{2D_{i+2}} + \sum r^{i}_{(i,1)} \vartheta_{D_i} + r^i_0\label{eqn:MirrorToRES3}
\end{align}
with $r^{i}_{(i,2)}$ and $r^{i}_{(i+2,2)}$ vanishing on the central fibre. The functions $f_{(2,2)}$, $f_{(6,2)}$, $g_{(0,2)}$, $g_{(4,2)}$, $r^{i}_{(i,2)}$ and $r^{i}_{(i+2,2)}$ correspond to pairs of broken lines not bending. The terms $f_{(i,1)}$, $g_{(i,1)}$ and $r_{i,1}^i$ come from pairs of broken lines where one of the lines bends once off of a ray with primitive $(k,1)$. The remaining terms correspond to pairs of broken lines bending in three different ways, either one broken line bends off of a ray with primitive $(k,2)$, both broken lines bends off of rays with primitives $(k,1)$ and $(k',1)$ or one of the broken lines bends twice, off of rays with primitives $(k,1)$ and $(k',1)$. Therefore from these we can extract two equations for the primitive $\vartheta$ functions and thus an embedding of the mirror family into affine space.

Whilst the original paper~\cite{MirrorSymmetryforLogCY1} used relative invariants we here use log Gromov-Witten invariants. The compatibility results of~\cite{ComparisonTheoremsForGromovWittenInvariantsOfSmoothPairsAndOfDegenerations} show that these two approaches agree. We recall the basic definitions of the subject.

  \begin{definition}

  A \emph{pre-log scheme} $X$ is a pair $(\underline{X}, \alpha_X: \mathcal{M}_X \rightarrow \mathcal{O}_{\underline{X}})$ of a scheme $\underline{X}$ (the \emph{underlying scheme}), a sheaf of monoids on $\underline{X}$ in the \'etale topology $\mathcal{M}_X$ and a homomorphism $\alpha_X$ of sheaves of monoids from $\mathcal{M}_X$ to the multiplicative monoid $\mathcal{O}_{\underline{X}}$. Saying that this is a \emph{log scheme} means that the restriction of $\alpha_X$ to the inverse image of $\mathcal{O}_{\underline{X}}^\times$ is an isomorphism.

  To any pre-log scheme there is an associated log structure given by taking the amalgamated sum

  \[ \mathcal{M}_X \oplus_{\alpha^{-1} \mathcal{O}_{\underline{X}}^{\times}} \mathcal{O}_{\underline{X}}^\times\]

  A morphism of log schemes $\phi : X \rightarrow Y$ is a pair $(\underline{\phi}, \phi^\#)$ with $\underline{\phi}: \underline{X} \rightarrow \underline{Y}$ a morphism of schemes and $\phi^\#$ fitting into a commutative diagram
\[    \begin{tikzcd}
      \underline{\phi} ^{-1} \mathcal{M}_Y \arrow{r}{\phi^\#} \arrow{d} & \mathcal{M}_X \arrow{d}\\
      \underline{\phi}^{-1} \mathcal{O}_{\underline{Y}} \arrow{r}{\phi^*} & \mathcal{O}_{\underline{X}}
    \end{tikzcd} \]

  Given a morphism of schemes $\underline{\phi}: \underline {X} \rightarrow \underline{Y}$ and the structure of a log scheme on $\underline{Y}$ the inverse image sheaf $\underline{\phi}^{-1} (\mathcal{M}_Y)$ on $X$ is naturally a pre-log structure. We call the associated log structure the \emph{pull-back log structure},
written $\phi^*\mathcal{M}_Y$. A morphism of log schemes $\phi: X \rightarrow Y$ is \emph{strict} if the natural
map $\phi^\#:\phi^*{\mathcal{M}_Y} \rightarrow \mathcal{M}_X$ is an isomorphism.

There is an interesting invariant of a log scheme which contains the combinatorial data not seen by classical geometry. The \emph{ghost} or \emph{characteristic sheaf} is the quotient $\mathcal{M}_X/\mathcal{O}_X^\times$.
  
\end{definition}

We introduced log structures to study pairs $(X,D)$, so we now explain how to construct a log structure from such a pair.

\begin{example}

  Let $\underline{X}$ be a smooth scheme and $\underline{D}$ be a simple normal crossings divisor. Write $\underline{U}$ for the complement of $\underline{D}$ in $\underline{X}$ and let $j$ be the inclusion of $\underline{U}$ into $\underline{X}$. Let $\mathcal{M}$ be the sheaf $j_* \mathcal{O}_{\underline{U}}^* \cap \mathcal{O}_{\underline{X}}$, and $\alpha_X$ the canonical inclusion. This defines a log scheme $X$, the \emph{divisorial log structure} on $\underline{X}$.

\end{example}

Importantly there is a moduli space of basic log stable maps to a target $X$ with a number of marked points and tangency data. The relative obstruction theory is given by the log cotangent bundle in the same way the usual obstruction theory is induced by the usual cotangent bundle.

\begin{definition}

  Let $f: X \rightarrow Y$ be a morphism of log schemes. A log derivation is a pair of maps $D, Dlog$ from $\mathcal{O}_{\underline{X}}$ or $\mathcal{M}_X$ respectively to a sheaf $\mathcal{E}$ satisfying:

  \begin{enumerate}
  \item $D$ is a derivation in the classical sense.
  \item $Dlog$ vanishes on sections of $f^{-1} (\mathcal{\mathcal{M}}_Y)$.
  \item For every element $m \in \mathcal{M}_X (U)$ one has an equality $D (\alpha_X(m)) = \alpha_X (m) Dlog(m)$.
  \end{enumerate}

\end{definition}

There is a universal sheaf classifying these, the logarithmic cotangent bundle $\Omega^{1, \dagger}_{X/Y}$. There is a natural pullback map, given $f:X \rightarrow Y$ and $g: Y \rightarrow Z$ there is a canonical morphism $f^* \Omega^{1, \dagger}_{Y/Z} \rightarrow \Omega^{1,\dagger}_{X/Z}$. We write $\omega^\dagger_{X/Y}$ for the top tensor power in the case where $\Omega^{1, \dagger}_{X/Y}$ is a vector bundle.

\section{Sections of a rational elliptic surface}

Bryan and Leung in~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} studied the Gromov-Witten theory of an elliptically fibred $K3$ surface with a section $E$ and fibre $F$. They considered genus $g$ stable maps with $g$ marked points in the class $E + nF$ and proved a formula for the Gromov-Witten invariants. For such a surface one cannot use the Gromov-Witten theory of $K$, but rather the reduced Gromov-Witten theory. This is defined to be the virtual class defined by $T^\circ$ in the following exact triangle
  \[ \tau_{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K) \rightarrow R \pi_* f^* \Theta_{K/k} \rightarrow T^\circ \rightarrow\]
  In the paper Bryan and Leung constructed explicitly the moduli space of genus zero stable maps and reinterpreted it as the moduli space of maps on certain blow ups of $\PP^2$. In particular in section 5 they showed that any stable map in this moduli space is a union of a component mapping to $E$ and some combinatorial data describing a cover of different rational fibres. From this description it is clear that even infinitesimally there are no deformations smoothing the nodes between the component mapping to $E$ and those mapping to rational fibres.

  By explicit calculation they show that the two induced deformation theories on this moduli space of stable maps are quasi-isomorphic. Then in Lemma 5.7 the authors apply well known formulae for the Gromov-Witten theory of blow ups of $\PP^2$ to show that each component of the moduli space contributes either a one or a zero to the total count.

  They also performed the same analysis, deducing the same geometric results for the moduli space of stable maps to a rational elliptic surface $S$. In particular there is no infinitesimal smoothing of any of the nodes lying on the intersection of the component mapping to the section and those mapping to the fibres. In this case there was no need to work with reduced invariants either and in Theorem 6.2 they provide the following formula for $I_{0,0,E+nF}$, the number of unmarked rational curves in class $E+nF$:
\begin{equation}
\sum_{n=0}^\infty I_{0,0,E + mF} z^m = \prod _{m=1}^\infty (1- z^m)^{-12}
\end{equation}

But we also need to know that there are not too many classes of sections of a given degree on $S$. Once we combine these two results we will produce a bound on the total number of sections. Thankfully bounding the number of curves is an easy exercise in classical algebraic geometry:

\begin{lemma}[Bounds on the number of sections]

Let $d \in \mathbb{N}$, define the \emph{Goldilocks zone} 
\[
GZ(S, d) = \{ [C] \in A_1 (S) \mid [C] \sim dH - \sum a_i E_i, \:  [C].[F] = 1, \: p_a([C]) \geq 0 \}.
\]
\Lawrence{where $p_a([C])$ is the arithmetic genus of a generic member of this family.} Then $| GZ (S,d) | \leq (N\sqrt{d} + N)^{9}$ for some $N$ independent of $d$.

\label{thm:BoundedlyManySections}

\end{lemma}

\begin{proof}
Let $f: C \rightarrow S$ be a stable map. There is an inequality $g(C) \leq p_a (f_* [C])$ and so this counts the number of possible curve classes for stable maps of genus zero. To prove \Lawrence{this lemma} we apply the genus formula together with basic intersection theory. Let $[C] = d[H] - \sum a_i [E_i]$ be a curve class with $[C].[K_S] = -1$, so
$3d-\sum a_i=1$. Then the genus formula then states that $p_a(C) = 1 + 1/2([C].[C]-[C].[F])$ and
$p_g(C)\le p_a(C)$, so
\begin{align*}
0 \leq \; & 1 + 1/2([C].[C]-[C].[F])\\
\leq \; & 3/2 + 1/2 [C].[C]
\end{align*}
hence
\begin{align}
\begin{split}
0 \leq \; & 3 + [C].[C]\\
\leq \; & 3 + d^2 - \sum a_i^2
\end{split}
\label{inequality}
\end{align} 
By the arithmetic-geometric inequality $\sum a_i^2$ is minimised when all the $a_i$ are equal. We need to find a bound for $|a_i-a_j|$ in terms of $d$ given
the inequality above. Thus suppose
that $a_1 + k = 1/8 \sum_2^9 a_i$. We will bound $k$. Again $\sum a_i^2$ is minimised when $a_2 = a_3 = \cdots = a_9$. Using $3d = \sum a_i + 1$ we can write out everything explicitly
\begin{align*}
a_1 & = \frac{3d-1-8k}{9},\\
a_2 = a_3 = \ldots & = \frac{3d - 1 + k}{9}.
\end{align*}
Thus \Lawrence{\eqref{inequality}} implies
\begin{align*}
3 + d^2 & \geq \sum a_i^2\\
& \geq \left(\frac{3d - 1 -8k}{9}\right)^2 + 8 \left(\frac{3d - 1 + k}{9}\right)^2\\
& \geq d^2 - \frac{2}{3} d + \frac{8}{9} k^2 + \frac{1}{9},
\end{align*}
and hence
\[
\frac{26}{9} + \frac{2}{3}d  \geq \frac{8}{9}k^2
\]
or
\[
\frac{13}{4} + \frac{3d}{4}  \geq k^2
\]
Thus we see that for $|k| > \sqrt d + 2$ there can be no classes satisfying
the above inequality. By symmetry we may assume that $a_1$ is the smallest of the $a_i$. This then shows that none of the $a_i$ can be more than $\sqrt{d} + 2$ from $(3d-1)/9$, thus the total possible combinations is bounded by a bound of the above form. 

\end{proof}

In particular the number of rational curves of degree at most $d$ grows slower than $2^d$.

\section{The unravelled threefold}

The idea for this section is to make use of the fibration $\rho$. We will take a family of double covers of $\PP^1$ and construct a threefold $X$ fibred by $K3$ surfaces with known singularities. Each fibre will be a double cover of $S$ and so possesses an elliptic fibration. We will show that this induces a strong relation between the moduli spaces of stable maps to $S$ and the moduli space of stable maps to $X$. Let us begin though by thinking about the geometric content properties of the curves we wish to count.

Let $C$ be an irreducible rational curve on $S$ with $C.F = 2$. Restricting the fibration $\rho$ to $C$ we therefore obtain a double cover $\PP^1 \rightarrow \PP^1$. By the Riemann-Hurwitz formula this map must ramify at two points and up to a choice of involution these two points uniquely specify the double cover. This suggests that there should be a map from the moduli space of stable maps in the class $[C]$ to the Hilbert scheme of two points on $\PP^1$.

Let $[C]$ be an effective curve class in $A_1 (S, \mathbb{Z})$ with $[C].[F] = 2$. We will denote the moduli space of stable rational unmarked maps to $S$ with image in class $[C]$ by $\StableModuli_{0,0} (S, [C])$. This moduli space carries a universal curve, $\pi: \mathcal{C}_{0,0} (S,[C]) \rightarrow \StableModuli_{0,0} (S, [C])$, and a universal stable map, $f: \mathcal{C}_{0,0} (S,[C]) \rightarrow S$. Taking the composition $\rho\circ f$ we obtain a family of branched double covers of $\mathbb{P}^1$. Passing to the stabilisation we therefore obtain a morphism $\StableModuli_{0,0} (S, [C]) \rightarrow \MTwoOne$. The space $\StableModuli_{0,0} (\mathbb{P}^1, 2)$ meanwhile is naturally isomorphic to $[\Hilb / C_2]$. Here the $C_2$ action on $\Hilb$ is trivial, reflecting the involution on a double cover of $\mathbb{P}^1$. Any stable map in $\MTwoOne$ is a double cover of $\mathbb{P}^1$ by a rational curve hence ramified over two points. The comparison map to $\Hilb$ sends such a curve to the two ramified points. Therefore via composition we obtain a morphism $ram: \StableModuli_{0,0} (S, [C]) \rightarrow [\Hilb/C_2]$.

Our goal will be to understand the images of different irreducible components of the moduli space $\StableModuli_{0,0} (S, [C])$ under this map $ram$. To do this we wish to understand how the fibres of $\pi$ vary as double covers of $\PP^1$. This requires that we be able to identify which components contribute to the double cover for which we label the different components as follows. Let $f:C \rightarrow S$ be a stable map corresponding to a closed point of $\StableModuli_{0,0} (S, [C])$. There are three types of components of $C$: contracted components, components covering fibres of $\rho$ and components which cover the base $\mathbb{P}^1$ of the fibration. We will assign one of the two labels $\fib$ and $\tr$ to each component. Components covering fibres will be labelled with $\fib$ whilst components covering the base will be labelled $\tr$. Any contracted component which is part of a chain linking two components covering the base will be labelled $\tr$ and all the remaining contracted components will be labelled $\fib$. This divides $C$ into two types of components based on these labels, which we will denote by $C_{\tr}$ and $C_{\fib}$. The component $C_{\tr}$ consists either of a single component mapping as a double cover to the base $\mathbb{P}^1$ or as a chain of contracted components joining two components mapping isomorphically to the base or is the union of two disconnected components. Note that $C_{\fib}$ may very well be disconnected. We write $C_{\contr}$ for the curve obtained from $C_{\tr}$ by contracting out the contracted components, noting that if $C_{tr}$ is connected then $C_{contr}$ admits a map to $S$. The following lemma shows that in a fibre of $ram \circ \pi$ the component $C_{\tr}$ is rigid, only the component $C_{\fib}$ deforms.

\begin{theorem}[Fibres of $ram\circ \pi$ vary]
\label{thm:varying-fibres}
The component $f(C_{\tr})$ is locally constant as a subscheme of
$S$ along fibres of $ram\circ\pi$. Hence 
the image under $ram\circ \pi$ of a family of curves in $\StableModuli_{0,0} (S, [C])$ with $f(C_{\tr})$ varying as a subscheme of $S$ is positive dimensional.

\end{theorem}

\begin{proof}

Let $B \xleftarrow{\pi} C \xrightarrow{f} S$ be a family of genus zero unmarked stable maps contained inside a fibre of $ram$. First we treat the case where $C_{\tr}$ is irreducible.

Fix a choice of section $E$ for the rational elliptic surface which we take to be the identity for the group law. Choose an \'etale $U\rightarrow B$ with dense
image such that $(C|_U)_{\tr}$ is isomorphic to $U \times \mathbb{P}^1$. 
Thus we may identify fibres over each $u \in U$ as follows, using the assumption
that $C_{tr}$ is irreducible. After passing to another \'etale cover we may identify $(C|U)_{tr}$ with the pull-back of the universal curve over $\MTwoOne$. By assumption the image in $\MTwoOne$ is a point $P$. Therefore for each $u \in U$ we may identify the curve $C_{u,tr}$ with the same double cover of $\PP^1$.

\begin{figure}\label{identificationoffibres}
\end{figure}

Fix a choice of closed point $u \in U$ and so a stable map to $S$ and for each $v \in U$ we identify the two covers via the above isomorphism as described in Figure~\ref{identificationoffibres} and consider $f(C_{tr,v}) - f(C_{tr, u})$. For $u = v$ this defines a double cover of the section $E$, while if $f(C_{\tr})$ is varying then for $u \neq v$ this is not a double cover of that section. By rigidity of sections this does not occur.

Now if $C_{\tr}$ is reducible, $C_{\contr} = C_1 \cup C_2$ then both $C_1$ and $C_2$ are sections and admit no deformations. In particular this shows that $f$ is constant on these fibres.

\end{proof}

Let us now study the deformation theory of the different components of $C$. The virtual dimension of the moduli space of genus $g$, $n$-marked stable maps to $S$ in a class $\beta$ is given by
\[(1-g)(dim \: S - 3) +n - K_S.\beta\]
applying this to our situation we see that $C$ moves in at least one dimensional family, and if $C_{tr}$ is connected then $C_{contr}$ also moves in a one dimensional family. We will work to relate deformations of these two curves. In any case there are three types of irreducible components of $\StableModuli_{0,0} (S, \beta)$, \emph{point components} whose image is dimension zero, \emph{curve components} whose image is dimension one and \emph{bubble components} which surject onto $\Hilb \cong \mathbb{P}^2$. Schematically the types of curves which can occur in each component is contained in Figure~\ref{fig:animals}. We can draw representative curves of each type in the class 
$E_1 + E_2 + 3F$ for sections $E_1, E_2$ satisfying $E_1=E_2$, $E_1\cdot E_2=1$
or $E_1\cdot E_2=0$ in the three cases depicted. 
The dashed line in the first figure shows that this is a double cover. We will later prove that these are representative of the general case.

\begin{figure}
\caption{Bubble}
\label{fig:gull}

\caption{Curve}
\label{fig:tiger}

\caption{Point}
\label{fig:mouse}
\end{figure}

Let us begin by proving that if $C$ is in a point component then $C_{tr}$ is not connected.

\begin{lemma}[The structure curves over point components]

Let $\mathcal{M}$ be a point component. Then the restriction of the universal curve to this component decomposes as two constant components each mapping to potentially distinct sections joined by a tree of curves containing at least one component mapping to a fibre.

\end{lemma}

\begin{proof}

  If the transverse component $C_{\tr}$ is smooth and connected then \Lawrence{we have seen that we can deform it as} a double cover of $\mathbb{P}^1$. This leaves how to glue the component $C_{\fib}$ to this deformation. Suppose that $C_{\fib}$ meets $C_{\tr}$ in $P_1, \ldots , P_n$. \Lawrence{If none of these are ramification points of the map $C_{\tr} \rightarrow \PP^1$ then this deformation gives unique deformations of the $P_i$. If not then take the double cover $\AA^1 \rightarrow \AA^1$ given by $z \mapsto z^2$. Over this family there is a unique deformation given by choosing a branch to move along}. This allows us to lift the glueing data to the entire deformation contradicting Lemma \ref{thm:varying-fibres} and the definition of a point component. Therefore the universal curve must be generically reducible containing two distinct sections. The same argument applies if $C_{\tr}$ is the union of two sections meeting at a point.

Now suppose that $C_{\tr}$ is connected but not smooth hence of the form two sections joined by a chain of contracted components. Then by stability each contracted component in the chain must have an attached component mapping to a rational fibre.

\end{proof}

Now these point components are not relevant to the rest of our work, they cannot carry the correct log structure to be tangent to $F_0$ at a single point, nor are they generic enough to appear in our main construction. However the bubble components for sure will contribute, and controlling these will be key to our construction. In particular we can show that curve and bubble components do not meet.

\begin{theorem}[Curve and bubble components are disjoint]

Let $\mathcal{M}$ be a bubble component. Then $\mathcal{M}$ does not meet any curve components of the moduli space. 

\end{theorem}

\begin{proof}

We will begin by showing $f(C_{\tr})$ does not vary for any stable map in this space, a double cover of some section $\sigma$. We do this by first studying the restriction of the family to the boundary of the moduli space. Let $\partial \Hilb$ denote the conic of degenerate double covers inside $\mathcal{H}ilb^2 (\mathbb{P}^1) \cong \mathbb{P}^2/C_2$. Take a point $P \in \partial \Hilb$ such that the ramification point of the corresponding double cover $C_{\tr}\rightarrow \mathbb{P}^1$ is distinct from the images of any of the rational fibres of $S$. In this case, $C_{\tr}$ is reducible, the union of two sections meeting at a point which does not lie on one of the singular fibres and admits deformations which remain reducible. Sections however are rigid and so this can only happen if one has a double cover of a single section, $\sigma$.

Now suppose a curve component intersects $\mathcal{M}$. In this case, one
can find the spectrum of a DVR $B = \Spec R$ with closed point $b_{0}$ and generic point $b_{gen}$ and
a morphism $cl:B\rightarrow \StableModuli_{0,0}(S,[C])$ such that
$cl(b_0)\in \mathcal{M}$, $cl(b_{gen})\not\in\mathcal{M}$ and $cl(b_{gen})$
lies in a curve
component. Let
$\pi: C \rightarrow B$ be the pullback of the universal curve and $f: C \rightarrow S$ the corresponding stable 
map.

In fact, it is enough to show the following.
First, replace $C$ by removing the closures of irreducible components
of $C_{gen}$ mapping into the rational fibres of $S$ and taking the closure inside $C$. Thus we may assume
that $(C_{gen})_{\tr}=C_{gen}$. If we can show that $(C_0)_{\tr}=C_0$, then
as necessarily $f_*([C_0])=2[\sigma]$ and $\sigma^2=-1$, we see
that $f(C_{gen})\subseteq \sigma$ also. Thus $f$ yields a double cover
of $\sigma$, and hence $cl(b_{gen})$ in fact lies in a bubble component.

Our goal will be to decorate $C_{gen}$ with the structure of a log stable map. Then since the moduli space of log curves is proper the family $B$ admits a unique completion, and this completion must be a base-change of the classical stable map. But we will now see that results of~\cite{TheTropicalVertex} lead to
a contradiction. To this end we take the divisorial log structure on $S$ coming from the union 
$\cup F_i$ of all singular fibres.

To give a log structure on $C_{gen}$ we need to control the location of the singularities. Therefore we first exclude the case
that $C_{gen}$ has any nodes on the transverse component. Indeed, if $C_{gen}$ possesses a node then it is
reducible and the two components map to sections of $S\rightarrow \mathbb{P}^1$.
But then the family is constant since sections admit no deformations.
Thus we assume $f_{gen}^{-1}(\cup F_i)$ is a set of smooth points of $C_{gen}$
and we give $C_{gen}$ the divisorial log structure given by this divisor.
We then have a log morphism $C_{gen}\rightarrow S$.

Now we know that the generic fibre has a log structure and hence there is a unique choice of limit of stable log maps, possibly after passing to a branched
cover of $B$, see \cite{LogarithmicGromovWittenInvariants}, Theorem 4.1. We now claim that there are no log stable curves with components mapping into the fibre $F_0$. This is a consequence of the proof of Proposition 4.3 of~\cite{TheTropicalVertex}. This shows that $C_{gen}$ cannot degenerate to one of these curves. Thus we conclude all curve components are disjoint from $\mathcal{M}$.

\end{proof}

Now if we take a general point of $\Hilb$ there is a canonical way to produce a $K3$ surface by taking the fibre product over $\PP^1$. By the work of Bryan and Leung the moduli space of stable maps to an elliptically fibred $K3$ surface in a class $E + nF$ is independent of the choice of surface. Let us use this idea to describe the fibres of $\pi$ over a bubble component.

\begin{theorem}[A fibration of bubble components]

Let $\mathcal{M}$ be a bubble component. The map $\mathcal{M} \rightarrow \Hilb$ fibres $\mathcal{M}$ by moduli spaces of stable maps to (potentially singular) $K3$ surfaces. This fibration is trivial away from a codimension one set.
  
\end{theorem}

\begin{proof}

One might think that the fibration should be trivial as they correspond to different parametrisations of the rational fibres. This is only true generically but fortunately we can describe the locus where it does not hold. Let $R$ denote the locus inside $\Hilb$ of stable maps ramifying over the images of the $F_i$. Away from $R$ the fibration will be trivial since the only deformations are deformations of the rational tails.

Let $P \in \Hilb \setminus R \cup \partial \Hilb$ be a smooth double cover not ramifying over one of the rational fibres. Let $f: C \rightarrow S$ and $\pi: C \rightarrow B$ be a curve in $ram^{-1} P$ with $B$ an infinitesimal extension over $k[t]/t^2$. Let $B_0, C_0$ denote the reduced structure. We wish to show that this deformation of $C_0$ does not smooth any of the nodes between $C_{0,tr}$ and the other components. To do this we construct a moduli space of stable maps to a $K3$ surface where none of these smoothings exists and compare the obstruction theories. Take the following diagram
\[
\xymatrix@C=30pt
{
K\ar[r]^d\ar[d]&S\ar[d]^{\rho}\\
C_{0,tr}\ar[r]_{\rho\circ f}&\mathbb{P}^1
}
\]

This defines a smooth elliptically fibred $K3$ surface $K$ (here we are using the assumption that the curve does not ramify over the images of any of the rational fibres). $C_0$ maps to both $C_{0, tr}$ and $S$ and hence maps to $K$ as a section together with a collection of covers of the rational fibres. This shows that the fibres of $ram$ are isomorphic to moduli spaces of stable maps to a $K3$ surface. In particular by the beginning of section 5 ~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} there are no infinitesimal deformations smoothing the nodes lying on $C_{0,tr}$ since there are none on $K$. In particular this shows that over this set the fibration is trivial.

We now turn our attention to the boundary $\partial \Hilb$. We will show
that if $P\in\partial\Hilb$, then no map $\Spec k[\epsilon]/(\epsilon^2)
\rightarrow ram^{-1}(P)$ produces an infinitesimal deformation of stable
maps which
smooths any nodes between $C_{\tr}$ and $C_{\fib}$. Since $\partial\Hilb$ meets every codimension one locus in $\Hilb$ this would show that the set of points on which at least one of the deformations smooths these nodes is codimension at least two. Take $P \in \partial \Hilb$ and $f: C \rightarrow S$, $\pi:C \rightarrow B$ an infinitesimal deformation of a point $\pi: C_0 \rightarrow B_0$, $f_0: C_0 \rightarrow S$ in $ram^{-1}P$. Then $C_{0}$ contains a chain of rational curves connecting two components $D_1$ and $D_2$ which are mapped to sections of $S$. Since the deformation remains within the fibre it fails to smooth at least one node in this chain. Removing this node disconnects the domain curve $C_0$ into two components $C_a$ and $C_b$ and the deformation restricts to each of these. But $C_a$ and $C_b$ are sections together with rational tails, hence deformations of them fail to smooth any of the nodes connecting the rational tails to the section by the discussion of Section 5 of ~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms}. Pulling back to the curve $C_0$ we obtain the desired statement.

Finally we claim that the fibres over any point in $\Hilb \setminus R$ are isomorphic. This is trivial from the above description. All these spaces are just moduli spaces of covers of the rational tails. It is worth saying that it is not true that this fibration is globally trivial. We can construct examples where the fibres of $ram$ over $R \cap \partial \Hilb$ are much higher dimension than the surrounding fibres since there could be more moduli in the way we connect the two transverse components together and such examples exist for three sheeted covers of one of the rational tails.

\end{proof}

Given an irreducible such curve, $C$ it is an easy calculation that the product $C \times_{\PP^1} S$ is a $K3$ surface. But $C$ maps both to itself via the identity and to $S$, compatibly over $\PP^1$. Therefore it also maps to this $K3$ surface. The moduli space we are considering provides us with a collection of components mapping to $\Hilb$. Of these we will be interested in those whose image is at least dimension one so we should search for a curve inside $\Hilb$. So let us perform this construction in a family setting.

\begin{construction}

   Let $D$ be a generic tri-degree $(1,1,2)$ surface inside $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ with projection maps $\pi_L$, $\pi _M$ and $\pi_R$ to the respective factors. We view this as a family of curves over $\mathbb{P}^1$ via the projection $\pi_L$. Taken this way each fibre is a bi-degree $(1,2)$ hypersurface inside $\mathbb{P}^1 \times\mathbb{P}^1$, so via $\pi_L$ the graph of a double cover of $\mathbb{P}^1$ by $\mathbb{P}^1$. This produces a morphism $\PP^1 \rightarrow \MTwoOne$ such that the family $\pi_L : D \rightarrow \PP^1$ is the pullback of the universal family of double covers. We wish to know that the trivial $C_2$ action on the Hilbert scheme $\Hilb$ is intertwined with the involution of $D$ swapping the two sheets of the double cover.

\end{construction}

  \begin{lemma}

    With the above notation the family $[D/C_2] \rightarrow [\PP^1/C_2]$ is the pull-back of the universal family of double covers of $\PP^1$ over the base.

  \end{lemma}

  \begin{proof}

    By construction the family $D$ is the pull-back of the universal family of
double covers to $\PP^1$. Therefore we have a diagram

    \[\begin{tikzpicture}
    \node (P1) at (0,0) {$\PP^1$};
    \node (M21) at (4,0) {$\MTwoOne$};
    \node (P1/2) at (2,1) {$[\PP^1/C_2]$};
    \node (D) at (0,2) {$D$};
    \node (C) at (4,2) {$\mathcal{C}$};
    \node (D/2) at (2,3) {$[D/C_2]$};
    \draw [->] (P1) -- (M21);
    \draw [->] (P1) -- (P1/2);
    \draw [->] (P1/2) -- (M21);
    \draw [->] (D) -- (P1);
    \draw [->] (C) -- (M21);
    \draw [->] (D/2) -- (P1/2);
    \draw [->] (D) -- (C);
    \draw [->] (D) -- (D/2);
    \draw [->] (D/2) -- (C);
    \end{tikzpicture}\]
    where the front face is a pullback square. Now $\PP^1$ is simply connected, hence there is only the trivial $C_2$ torsor over $\PP^1$. This shows that $\PP^1 \rightarrow [\PP^1/C_2]$ has a chart given by $\PP^1 \coprod \PP^1 \rightarrow \PP^1$. Taking the pull-back of $[D/C_2]$ to this and using that $D$ is the universal curve over $\PP^1$ we obtain the desired result.

\end{proof}

Let us explore some general features of this via an example.

\begin{example}

  Let us study the family with defining equation
  \[ x_1x_2x_3^2 + x_1y_2x_3y_3 + 2y_1x_2x_3y_3 + y_1y_2y_3^2 \]
where the $i^{th}$ factor has coordinates $x_i$ and $y_i$. The fibre over $(x_2, y_2)$ is the graph of a double cover of $\mathbb{P}^1$, ramifying where the discriminant $(x_1y_2+2y_1x_2)^2 - 4 x_1x_2y_1y_2 = x_1^2y_2^2 + 4y_1^2x_2^2$ vanishes. This gives two values for $(x_1:y_1)$ for each value of $(x_2:y_2)$. The fibre is singular just when these two values coincide, so when the discriminant of this quadratic vanishes, $-16 x_2^2y_2^2$. Therefore there are two singular fibres each with multiplicity two.
  
\end{example}

In general this shows that there will be four singular fibres and it is easy to construct examples where these occur as distinct fibres. We have constructed here a collection of double covers of $\mathbb{P}^1$, so the base $\mathbb{P}^1$ should map to $\Hilb$. The degenerate fibres inside $\Hilb$ form a conic inside $\mathbb{P}^2$, so by counting intersection points we know that the image of this family is also a conic. Therefore to search for curve and bubble components of the moduli space we are lead naturally to study the following fibre product:
\[\centerline{\xymatrix{X \ar[r]^{d} \ar[d] & S \ar[d]^{\rho} \\ D \ar[r]^{\pi_L} & \mathbb{P}^1}}\]
We call the constructed threefold $X$ an \emph{unravelled threefold}. Such a threefold is the intersection of a $(3,1,0,0)$ hypersurface and a $(0,1,1,2)$ hypersurface inside $\mathbb{P}^2 \times (\mathbb{P}^1)^3$. The $(0,1,1,2)$ hypersurface we may take to be generic whilst the $(3,1,0,0)$ hypersurface depends on the choice of rational elliptic surface $S$. The first hypersurface gives rise to a choice of line $\iota: [\PP^1/C_2] \rightarrow \MTwoOne \cong [\PP^2/C_2]$. We will restrict our range of choices later in the discussion to reflect the surfaces we choose to work with. Let $\mathcal{M}$ be a component of the moduli space $\StableModuli_{0,0} (S, f_*[C])$ and consider the image of $\mathcal{M}$ under the morphism $ram:\StableModuli_{0,0}(S,f_*[C])\rightarrow \mathcal{H}ilb^2 (\mathbb{P}^1) \cong \mathbb{P}^2/C_2$. Whenever the image intersects the image of the family of double covers of $\PP^1$ one can lift the corresponding stable map to $X$. The existence of a lift suggests that $\StableModuli_{0,n} (X, \tilde{f}_* ([C])) \cong \StableModuli_{0,n} (S, f_*[C]) \times_{\mathcal{H}ilb ^2 \mathbb{P}^1} \mathbb{P}^1$ along the map $ram$. Soon we will see that this description induces a relation between the Gromov-Witten invariants of these two moduli spaces.
  
\section{Calculating Gromov-Witten invariants}

We can describe this space $X$ as a nef complete intersection inside $\PP^2 \times \PP^1 \times \PP^1 \times \PP^1$. This allows us to apply techniques of Givental from~\cite{AMirrorTheoremforToricCompleteIntersection} to calculate the Gromov-Witten theory of $X$. We will begin by giving an explanation of this in the case of $S$ itself, and then explain why this is not enough to calculate the relative invariants we want. Instead we will prove that the $J$ function of $X$ is holomorphic, which provides strong bounds on the growth of the coefficients.

\begin{example}

  From the description above the threefold $X$ is the intersection of a $(3,1,0,0)$ hypersurface and a $(0,1,1,2)$ hypersurface in $\PP^2 \times \PP^1 \times \PP^1 \times \PP^1$.

\end{example}

Such hypersurfaces were studied in~\cite{AMirrorTheoremforToricCompleteIntersection}. The Gromov-Witten theory of a toric variety is very well understood and by using these line bundles we can localise the invariants to the intersection. Givental constructed solutions to two different quantum differential operators, and then proved that the solution spaces to these two were equal. Therefore there is a change of basis relating the constructed solutions. Let us recall the theory, starting with a toric variety $\mathbb{T}$ with codimension one strata $D_\rho$. Take a basis for the rational Chow ring $A^* (\mathbb{T})$, $\{H_i\}_1^n$, and a dual basis $\{H^i\}_1^n$ under cup product.

\begin{definition}

The $I_X$ and $J_X$ functions are defined for a smooth complete intersection $X \subset \mathbb{T}$. Suppose that $X$ is the common vanishing of sections of line bundles $\mathcal{L}_i$ with $-K_\mathbb{T} - \sum c_1(\mathcal{L}_i)$ nef. Then the $I_X$ function is defined by

\begin{align*}I_X (t_0, \ldots t_n) &= \\ e^{(t_0+\sum t_i H_i) / \hbar} Eul& (\oplus \mathcal{L}_i) \sum_\beta q^\beta \frac{\prod_i (\prod_{m=1} ^{\mathcal{L}_i (\beta)} (c_1 (\mathcal{L}_i) + m\hbar) \prod_\rho \prod_{m=-\infty}^{0} (D_\rho + m\hbar))}{\prod_\rho \prod_{m=-\infty}^{D_\rho \cap \beta} (D_\rho + m\hbar)} \end{align*}
where $q^{\beta} = e^{\sum t_i H_i \cap \beta}$ and the sum is over $\beta$ effective. This function is valued in $A^* (\mathbb{T})[\![t_0, \ldots t_n, \hbar^{-1}]\!]$. The function $J_X$ is a generating function for the gravitational correlators of $X$ and is valued in $A^* (X)[ \! [t_0, \ldots t_n, \hbar^{-1} ] \! ]$. It is defined by the formula
\[J_X (t_0 \ldots t_n) = e^{(t_0+\sum t_i H_i)/ \hbar} \left(1 + \sum_{\beta, i, k} \hbar^{-(k+1)} q^\beta \langle \tau_k H_a, 1 \rangle_{0,\beta} H^a \right)\]
where $\langle \tau_k H_a, 1 \rangle_{0,\beta}$ are the gravitational descendants, the sum is over positive $i, k$ and effective curve classes $\beta$.

\end{definition}

These two functions are not valued in the same ring so cannot be equal. Let $\iota: X \rightarrow \mathbb{T}$ be the inclusion. The main result of~\cite{AMirrorTheoremforToricCompleteIntersection} is an equality of generating functions $I_X(t_i) = \iota_* J_X (s_i)$  after some change of basis $t_i \mapsto s_i \in A^* (\mathbb{T})[\! [t_0, \ldots t_n ]\!]$.

\begin{theorem}[Mirror symmetry for Givental's $I$ and $J$ functions]

The functions $I_X$ and $\iota _* J_X$ are equal up to a homogeneous change of variables of the form $t_0 \mapsto t_0 + f_0 (z^{\beta}) \hbar + h (z^\beta)$, $t_i \mapsto t_i + f_i (z^\beta)$ where the $f_i$ and $h$ are homogeneous power series of weights $\deg f_i = 0$ and $\deg h = 1$ with degrees of
the variables being given by $c_1 (\mathbb{T}) - c_1 (\sum \mathcal{L}_i) = \sum deg (z^{H_i}) H_i$, $\deg \hbar = 1$, $\deg t_0=1$ and $\deg t_i=0$ for
$i>0$.

\begin{proof}

  See~\cite{AMirrorTheoremforToricCompleteIntersection} Theorem 0.1.
  
\end{proof}

\end{theorem}

A well written explanation of how to perform these calculations can be found in~\cite{MirrorSymmetryAndAlgebraicGeometry}. The rational elliptic surface $S$ is itself a $(3,1)$ hypersurface in $\PP^2 \times \PP^1$. Unfortunately this is not powerful enough to reconstruct the relative or open invariants and so it cannot answer our questions by directly calculating with $S$. Instead let us calculate the $J$ function of $X$ and attempt to interpret the answer geometrically. Since we are practising numerology we will not attempt to prove our claims. Firstly we take a basis for the cohomology $H_1, \ldots, H_4$ given by the duals of pullbacks of hyperplanes from each factor.

The invariants we wish to calculate intersect $H_2$, the class which restricts on each $K3$ fibre to a fibre of the fibration of that $K3$, at a single point. Therefore the number of such curves appear as the coefficient of $H_2$ in the corresponding term in the $J_X$ function. We must double the invariant to account for the tangency condition and double it again since we are really intersecting with a conic inside $\Hilb$. This suggests that each curve on $S$ should be counted with multiplicity four compared to the corresponding curve on $X$.

When we expand out the function $J_X$ the first three coefficients we calculate, the coefficients of $z^{(0,2,0,1)}, z^{(1,2,0,1)}$ and $z^{(2,2,0,1)}$, are $-9, 144$ and $1980$. The first of these we can easily explain, there are nine sections each admitting a bubble component, by the double cover formula the relative invariant should be $-9/4$. The 144 reflect the fact that for each family class $H - E_i$ there are four lines which are tangent to the boundary to order two, the coefficient therefore is $9 \times 4 \times 2 \times 2 = 144$.

For the term 1980 we begin to see an interaction between the bubble components and the curve components. Each of the lines $H - E_i - E_j$ admits a bubble component, there are 36 choices of such classes so we expect a contribution of $-36 = -36/4 \times 2 \times 2$. The classes $2H - E_i - E_j - E_k - E_l$ give a curve component of which four elements are tangent to the boundary. This gives a contribution of $126 \times 4 \times 2 \times 2 = 2016$, combining these we obtain the predicted contribution of $1980$. Unfortunately as the degree grows the presumed correspondence with the log invariants becomes intractable. To find the next term we would have to answer the question ``given a rational cubic $E$ passing through seven points $P_i$ what is the virtual contribution from rational cubics pass through those points meeting $E$ in one other point of tangency order two?''. This is why we must pass to the threefold.

Our main result for this section is that $J_X$ is actually a holomorphic function. We believe that this should be part of the general yoga of hypergeometric equations, but do not know of a reference. Let us recall the definition and some key facts about holomorphic functions. We will prove that $I_X$ is holomorphic and then deduce the holomorphicity of $J_x$.

Given a vector $\underline{v} = (v_1, \ldots v_n) \in \NN^n$ we write $x^{\underline{v}}$ for the monomial $\prod x_i^{v_i}$. Recall that a holomorphic function in variables $x_1, \ldots x_n$ is given locally by an absolutely convergent power series $\sum a_{\underline{v}} x^{v}$ converging on some disc and that if such a power series exists then operations such as differentiation and so on may be performed term by term. Let $diag (\underline{v})$ be the function sending $v$ to $\sum (v_i + 1)$. Suppose now that $a_{\underline{v}}$ are real numbers satisfying $|a_{underline{v}}| < c r^{diag (\underline{v})}$ for some fixed choice of real constants $c$ and $r$. Then by construction the power series
\[ f (x_1, \ldots, x_n) = \sum a_{\underline{v}} x^{\underline{v}}\]
converges on the polydisc around the origin in $\CC^n$ of radius $1/r$. We call such a family of $a_{\underline{v}}$ \emph{exponentially bounded}. We provide a converse to this statement.

\begin{proposition}

  Let $f: D^n (0,r) \rightarrow \CC$ be a holomorphic function on a polydisc around $0$ of radius $r$. Then $f (x_1, \ldots x_n) = \sum a_{\underline{v}} x^{\underline{v}}$ with the $a_{\underline{a}}$ exponentially bounded.
  
\end{proposition}

\begin{proof}

  The existence of the power series expansion is standard. We apply the Cauchy Integral formula for many variables to find $a_{\underline{v}}$.
  \[a_{\underline{v}} = \frac{1}{2\pi i}^n\int_{\partial D^n(0,r/2)} \frac{f(x_1, \ldots x_n)}{x^{\underline{v}} x_1 \ldots x_n} dx_1 \ldots dx_n.\]
  Therefore we have
  \[|a_{\underline{v}}| \leq \frac{\max_{z \in D^n (0,r/2)}|f(x_1, \ldots x_n)|}{(r/2)^{diag(\underline{v})}} \]
  where $f(x_1, \ldots x_n)$ is bounded on $D^n (0,r/2)$ . Therefore we have the result.
\end{proof}

\begin{definition}

  We work in slightly greater generality than is standard. Let $R = \CC[t_1, \ldots t_m] / \langle t_1^{k_1}, \ldots t_m^{k_m} \rangle$ be a finite dimensional $\CC$ algebra. We say that an element of $R[\! x_1, \ldots x_n \! ]$ is holomorphic if the coefficients of each monomial in the $t_{i}$'s are all holomorphic in the $x_i$.
  \label{def:exp-bounded}
\end{definition}
  
  The class of holomorphic functions is closed under many operations. This is a list of properties we need of them.

\begin{lemma}[A holomorphic toolkit]

The set of germs of holomorphic functions around $0$ is closed under the following operations:

\begin{enumerate}
  \item If an element $f \in \mathbb{C} [ \! [ t_1, \ldots t_n, x_1, \ldots x_m ] \! ]$  is holomorphic then its image in the quotient ring $R [ \! [ x_1, \ldots x_m ] \! ]$ obtained by dividing out by the ideal $\langle t_1^{m_1},\ldots,t_n^{m_n}\rangle$ is holomorphic.
\item If $f = \sum a_{\underline{i}, \underline{j}} t^{\underline{i}} x^{\underline{j}}$ is holomorphic and $g = \sum b_{\underline{i}, \underline{j}} t^{\underline{i}} x^{\underline{j}}$ is such that $| a_{\underline{i}, \underline{j}} | \geq | b_{\underline{i}, \underline{j}} |$ then $g$ is holomorphic.
\item Taking products of elements.
\item Taking inverses of invertible elements.
\item Taking exponentials of elements.
\item Substitutions of the form $x_i \mapsto x_i f_i$ where the $f_i$ are also holomorphic around $0$.
\item If $s_i = x_i f_i (x_0,  \ldots , x_m)$ is a change of basis with $f_i$ holomorphic and non-vanishing at zero then this change of basis locally has an inverse where the inverse map is given by $x_i = s_i g_i (s_0, \ldots, s_m)$ with the $g_i$ holomorphic.
\end{enumerate}

\end{lemma}

\begin{proof}

  The proof of i) is trivial.
  
  The proofs of ii) through vi) are standard for genuinely holomorphic functions of multiple variables. However the first result allows us to lift holomorphic elements of $R [ \! [ x_1, \ldots x_m ] \! ]$ to genuinely holomorphic functions of more variables by treating the $t_i$ as being free under the natural inclusion and then take the quotient.

  The second result says that if a collection $a_{\underline{v}}$ is exponentially bounded then any subcollection of the $a_{\underline{v}}$ are exponentially bounded too.
  
  To see the final result note that the derivative $\partial s_i / \partial x_j = \delta_{ij}$. Therefore the change of basis $\CC^n \rightarrow \CC^n$ is locally invertible by the inverse function theorem with holomorphic inverses. Since this maps $(0, \ldots 0)$ to $(0, \ldots 0)$ it has the required form.
  
\end{proof}

We now show that $I_X$ is exponentially bounded. Then the function $J_X$ was constructed using the tools coming from the above toolkit, and hence is also holomorphic.

\begin{theorem}[The $I$ function is holomorphic]

The function $I_X$ is holomorphic for $X$ an unravelled threefold,
where the coordinates $x_i$ in Definition \ref{def:exp-bounded} coincide
with $e^{t_i}$ in $I_X$, and the coordinates $t_i$ in
Definition \ref{def:exp-bounded} coincide with the $H_i$ in $I_X$.

\end{theorem}

\begin{proof}

To do this we prove the the coefficients of $I_X$ form an exponentially bounded set. We write out the definition of the $I_X$ function

\begin{equation}
I_X (t_0, \ldots t_n) = e^{\sum t_i H_i / \hbar} Eul (\oplus \mathcal{L}_i) \sum_\beta q^\beta \frac{\prod_i (\prod_{m=1} ^{\mathcal{L}_i (\beta)} (c_1 (\mathcal{L}_i) + m\hbar) \prod_\rho \prod_{m=-\infty}^{0} (D_\rho + m\hbar))}{\prod_\rho \prod_{m=-\infty}^{D_\rho \cap \beta} (D_\rho + m\hbar)}
\end{equation}

To prove boundedness, with $x_i=e^{t_i}$ and $x^{(a_1,\ldots,a_4)}
=\prod_i x_i^{a_i}$, we only need to prove boundedness of the sum

\begin{equation}
\sum x^{(a,b,c,d)} \frac{\prod_{m=1} ^{3a + b} (3H_1 + H_2 + m\hbar) \prod_{m=1} ^{b+c+2d} (H_2 + H_3 + 2H_4 + m\hbar)}{\prod_{m=1}^a (H_1 + m\hbar)^3\prod_{m=1}^b (H_2 + m\hbar)^2\prod_{m=1}^c (H_3 + m\hbar)^2\prod_{m=1}^d (H_4 + m\hbar)^2}
\end{equation}
Fix a coefficient $H_{\underline{i}} = \prod H_i ^{\alpha_i}$ and consider the coefficient of $H_{\underline{i}} x^{(a,b,c,d)}\hbar^{-c + \sum \alpha_i}$. Expanding out the above product we obtain the expression
\begin{align*}
  \sum x^{(a,b,c,d)} \hbar^{-c} & \frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2} \\
  & \frac{\prod_{m=1} ^{3a + b} (\frac{3H_1 + H_2}{m\hbar} + 1) \prod_{m=1} ^{b+c+2d} (\frac{H_2 + H_3 + 2H_4}{m\hbar} + 1)}{\prod_{m=1}^a (\frac{H_1}{m\hbar} + 1)^3\prod_{m=1}^b (\frac{H_2}{m\hbar} + 1)^2\prod_{m=1}^c (\frac{H_3}{m\hbar} + 1)^2\prod_{m=1}^d (\frac{H_4}{m\hbar} + 1)^2} \end{align*}
any monomial appearing in this consists of two parts, a term $\frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2}$ with no $\hbar$ term times a correction factor involving a sum of products of terms $1/i$. The correction factor is at most a polynomial in $a,b,c,d$, hence exponentially bounded since there are only finitely many choices of $\underline{i}$. We now bound the term $\frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2}$ using Stirling's formula:
\[
\frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2} 

< k e^c \cdot \frac{(3a + b)^{3a+b} (b+c+2d)^{b+c+2d}}{a^{3a} b^{2b} c^{2c} d^{2d}} \cdot \frac{2 \pi \sqrt{(3a + b) (b + c + 2d)}}{\sqrt{512 \pi ^ 9 a^3 b^2 c^2 d^2 }}
\]
The middle term is the only one not obviously exponentially bounded. We rewrite it as

\begin{equation}
\frac{(3a + b)^{3a}}{a^{3a}} \cdot \frac{(3a+b)^b}{b^b} \cdot \frac{(b + c + 2d)^b}{b^{b}} \cdot \frac{(b + c+ 2d)^{c}}{c^{2c}} \cdot \frac{(b + c +2d) ^{2d}}{d^{2d}}
\end{equation}

Rearranging the first term we get $(3+b/a)^{3a}$ and we wish to compare this to $r^{a + b}$. Let $r = 27$, then $27^{a+b} = 3^{3a+3b}$ and we can rewrite the quotient $(3+b/a)^{3a} / r^{a + b}$ as
\[ \frac{(1+\frac{b}{3a})^{3a}}{3^b}.\]
Taking log and using that $\log (1+x) < x$ we find that
\[ 3a \log \left(1 + \frac{b}{3a} \right) - b \log (3) < b - b \log (3) < 0\]
Hence for $r > 27$ we see that $(3 + b/a)^{3a} < r^{a + b}$. There are similar choices of $r$ for each other term. Our choice of $r$  is the product of all of these minimal choices. Then since $r^{a+b+c+d}$ dominates all terms of the form $r^{a+b}$ etc. we see that the coefficients of $I_X$ are exponentially bounded. Thus $I_X$ defines a holomorphic function in a neighbourhood of the origin.

\end{proof}

\begin{corollary}

  The function $J_X$ is also holomorphic.
  
\end{corollary}

\section{Relating Gromov-Witten invariants}

This is the real technical heart of the paper. In this section we will prove that the Gromov-Witten count of curves on $X$ and on $S$ are related by a Gysin map induced by a regular embedding $\PP^1 \hookrightarrow \Hilb$. The invariants we actually wish to count are then related by the composition of two Gysin maps, one induced by an inclusion of components and the other by a regular embedding $\PP^1 \hookrightarrow \Hilb$. This inclusion of components causes some problems for us, since we do not know if we have thrown away components with large virtual degree.

Let us begin by proving our philosophy, that the moduli spaces of stable maps to $X$ and to $S$ are highly related.

\begin{lemma}[The unravelled threefold lifts curves]
Let $X \subset \mathbb{P}^2 \times (\mathbb{P}^1)^3$ be an unravelled three-fold, $\pi_M$ the projection to the third factor, a copy of $\mathbb{P}^1$,
and let
\[
d^{-1} \beta = \{ \alpha \in A_1 (X) \mid \pi _{M, *} \alpha = 0, d_* \alpha = \beta\}
\]
be the set of effective classes on $X$ contained inside a fibre of $\pi_M$  
and such that $d_* \alpha = \beta$. \Lawrence{We write $\StableModuli(X,d^{-1}(\beta))$ for the union of moduli spaces $\bigcup _{\alpha \in d^{-1} \beta} \StableModuli_{0,0} (X, \alpha)$}. Then consider the following diagram
\[
\xymatrix{\StableModuli(X,d^{-1}(\beta)) \ar@/^2pc/[rrrd]^{d_*} \ar@/_2pc/[rdd]^{ pr _X} \ar[rd]^{\psi} & & & \\
 & \mathbb{P}^1 \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)  \ar[rr]^{\iota} \ar[d] ^{pr _S } & & \StableModuli_{0,0} (S, \beta) \ar[d]^{ram} \\
 & \mathbb{P}^1 \ar[rr]^{\iota} & & \MTwoOne}
\]
where $pr_X$ associates to a stable map $f:C/B\rightarrow X$ the morphism
$B\rightarrow \mathbb{P}^1$ given by the ramification of the composed
map $\rho\circ d\circ f$, where $\rho:S\rightarrow \mathbb{P}^1$ is the
elliptic fibration.
Then the map $\psi$ is an isomorphism, and thus a two to one cover of $[\mathbb{P}^1/C_2] \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta) \subset \StableModuli_{0,0} (S, \beta)$.
\end{lemma}

\begin{proof}
By genericity we may assume that the choice of map $\mathbb{P}^1 \rightarrow \Hilb$ avoids all point components and is transverse to the images of all curve components. We use the universal property of products to construct an inverse.

Let $S \xleftarrow{f} C \xrightarrow{\pi} B$ be a family of stable maps inside $\mathbb{P}^1 \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)$. Since $D$ is the restriction of the universal curve on $\MTwoOne$ to $\PP^1$ there is an induced morphism $C \rightarrow D$, and hence by properties of fibre products a unique morphism $C \rightarrow X$. This constructs an inverse to $\psi$, which we call it $\phi$.

We now need to prove that the two compositions $\phi \psi$ and $\psi \phi$ are in fact the identity maps. To show that $\psi \cdot \phi$ is the identity we show that $\iota \cdot \psi	 \cdot \phi = \iota$ and $pr_S \cdot \psi \cdot \phi = pr_S$. These are both automatic from the construction of $\phi$. Indeed, $\iota \cdot \psi \cdot \phi = d_* \cdot \phi = \iota$ and $pr_S \cdot \psi \cdot \phi = pr_X \cdot \phi = pr_S$.

Finally we show the composition $\phi \cdot \psi$ is the identity. Let $f: C \rightarrow X$ be a stable map in $\bigcup _{\alpha \in d^{-1} \beta} \bar{\mathcal{M}}_{0,0} (X, \alpha)$. The image $\psi (f:C \rightarrow X)$ is given by pushforward to $S$ and has a unique lift. But $f: C \rightarrow X$ is such a lift and so this composition too is the identity map.

\end{proof}

Since any degree two map from $[\mathbb{P}^1/C_2]$ to $[\mathbb{P}^2/C_2]$ is a regular embedding
there exists a Gysin map for the inclusion $\iota: \StableModuli (X, d^{-1}\beta) \rightarrow \StableModuli (S, \beta)$ covering $\mathbb{P}^1 \rightarrow \Hilb \cong [\mathbb{P}^2/C_2]$. We follow~\cite{VirtualPushforward} in constructing a compatibility datum between the two spaces.

\begin{theorem}[Compatibility of Gromov-Witten invariants betwen $S$ and $X$]

  For a generic choice of unravelling $d: X \rightarrow S$ there is an equality between the virtual classes associated to the obstruction theories on $\StableModuli_{0,0}(X, d^{-1} \beta)$ and $\StableModuli_{0,0} (S, \beta)$ under the Gysin map $\iota ^!$:
\[[\StableModuli_{0,0}(X, d^{-1} \beta)]^{vir} = 2 \iota^! [\StableModuli_{0,0} (S, \beta)] ^{vir}\]

\label{thm:SandXcomparisons}

\end{theorem}

\begin{proof}

We choose a generic unravelling which does not meet any point components and is transverse to any curve components. We have the following commutative diagram
\[
\xymatrix{X \ar[r]^{d} & S \\
\mathcal{C}_{0,0} (X,d^{-1} \beta) \ar [r]^{\iota} \ar[d]^{\pi_X} \ar [u]^{f_X} & \mathcal{C}_{0,0} (S, \beta) \ar[d]^{\pi_S} \ar [u]^{f_S} \\
\StableModuli_{0,0} (X,d^{-1} \beta) \ar [r]^{\iota} \ar[d]^{ram}  & \StableModuli_{0,0} (S, \beta) \ar[d]^{ram}\\
\mathbb{P}^1 \ar [r]^{\iota}& \MTwoOne
}
\]

The desired compatibility datum is a morphism of distinguished triangles
\[
\xymatrix{\cdots \ar[r] & (\iota^* R \pi_{S, *} f_S^* \Theta_{S/k})^{\vee} \ar[r]^{\phi} \ar[d]^{\gamma_1} & (R \pi_{X, *} f_X^* \Theta_{X/k})^{\vee} \ar[r] \ar[d]^{\gamma_2} & ram^* L_{\mathbb{P}^1 / \mathbb{P}^2} \ar[r] \ar[d]^{\gamma_3} & \cdots\\
\cdots\ar[r] & \iota^* L_{\StableModuli_{0,0}(S, \beta)/\mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)} \ar[r] & \cdots}
\]

To prove this we must show that the cone over $(\iota^* R \pi_{S, *} f_S^* \Theta_{S/k})^{\vee} \rightarrow (R \pi_{X, *} f_X^* \Theta_{X/k})^{\vee}$ is quasi-isomorphic to $ram^* L_{\mathbb{P}^1 / \mathbb{P}^2}$. First let us show that the map $\gamma_3 : ram ^* L_{\mathbb{P}^1/\Hilb} \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)}$ is an isomorphism. The morphism $\mathbb{P}^1 \rightarrow \Hilb$ is a local complete intersection morphism, defined by a single equation $F$ of degree two in $k[X,Y,Z]$. So in particular, $F$ forms a length one regular sequence
in $k[X,Y,Z]$. By genericity we may take the images of $\mathbb{P}^1$ and the curve components of $\StableModuli_{0,0}(S, \beta)$ to be transverse inside $\Hilb$. Therefore the sequence $F$ remains regular when pulled back to $\StableModuli_{0,0}(S, \beta)$ and $\StableModuli_{0,0}(X, d^{-1} \beta)$ is just the vanishing locus of the pullback of $F$. In the case of a regular embedding
there is an explicit formula for the cotangent complex, namely it is $\mathcal{I}/ \mathcal{I}^2[1]$ where $\mathcal{I}$ is the defining sheaf of ideals. Both source and target of $\gamma_3$ is $ram^*(\langle F \rangle / \langle F \rangle^2[1])$ and $\gamma_3$ is the identity.

Now taking the cone \Lawrence{over $\phi$} there is a morphism of triangles of the following form
\[
\centering
\xymatrix{... \ar[r] &(\iota^* R \pi_{S, *} f_S^* \Theta_{S/k})^{\vee} \ar[r] \ar[d] &(R \pi_{X, *} f_X^* \Theta_{X/k})^{\vee} \ar[r] \ar[d] & \Cone \ar[r] \ar[d]^{\delta} & ...\\
  ... \ar[r] & \iota^* L_{\StableModuli_{0,0}(S, \beta)/\mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)} \ar[r] & ...}
\]
\begin{equation}
\label{diag:bigone}
\end{equation}
To begin with we need a result allowing us to commute pull-back and push-forward. We claim that there is an isomorphism
\[\iota^* R \pi _{S \; *} f_S^* \Theta_{S/k} \cong R \pi_{X \; *} \iota ^* f_S^* \Theta_{S/k} \cong R \pi_{X \; *} f_X^* d^* \Theta_{S/k}\]
The second isomorphism is automatic, so we focus on the first. By~\cite{EGA4}, III.2, Corollary 6.9.9 and the fact that families of curves are flat, cohomology and base change commute. Applying~\cite{GromovWittenInvariantsInAlgebraicGeometry}, Proposition 5, $f^* _S \Theta_{S/k}$ admits a resolution by a two term complex of vector bundles $b: F^{-1} \rightarrow F^0$ such that $\pi_* F^i = 0$ and $R^1 \pi_* F^i$ is a vector bundle. 
Now $\pi_* f^* \Theta_{S/k}$ is just the cokernel of $R^1 \pi _* (F^0) \rightarrow R^1 \pi _* (F^1)$. But we have seen that cohomology and base change commute, thus we have an isomorphism $\iota^* R^1 \pi _* (F^i) \cong R^1 \pi_* (\iota ^* F^i)$ and $\iota^* R^0 \pi_* (F^i) = 0$. Now this is enough to show that there is the desired equality.

This allows us to calculate $\mathcal{C}one$ from the distinguished triangle on $X$
\begin{equation}
\label{exact-sequence-upstairs} \rightarrow \Theta _{X/k} \rightarrow d^* \Theta_{S/k} \rightarrow \MacBeth \rightarrow 
\end{equation}
The complex $\MacBeth$ is a line bundle in degree minus one and supported only on the critical locus of $X$ over $S$ in degree zero. The fibres of the map $d: X \rightarrow S$ are isomorphic to the fibres of $d:D \rightarrow \PP^1$. By an elementary calculation this is only singular over two points of $\PP^1$ where the corresponding double cover degenerates. The choice of $D$ was arbitrary so we can arrange that these two singular points do not occur over the critical values of the map $S \rightarrow \PP^1$, so over the images of the singular fibres. This ensures that $H^0 (\MacBeth)$ is supported only on two elliptic fibres of the fibration $\rho: S \rightarrow \PP^1$.

Let us describe the pullback of \Lawrence{$\mathcal{D}$} to fibres of the universal curve of stable maps. Let $f: C \rightarrow X$ be a stable map. By necessity such a stable map factorises through a (possibly singular) $K3$ double cover of $S$. This shows that $L^0 f^* \MacBeth$ is supported only on a finite set since the universal curve can have no components mapping into an elliptic fibre. By applying the Grothendieck spectral sequence we therefore see that both $H^1 (R\pi_*Lf^*\MacBeth)$ and $H^{-1}((R\pi_*Lf^*\MacBeth)^\vee)$ vanish.

By pushing forward \eqref{exact-sequence-upstairs} and dualizing in the
derived category, we obtain a long exact sequence

\begin{align*}
0\rightarrow & H^{-1} ((R\pi_*f^* d^* \Theta _{S/k})^\vee)
 \rightarrow  H^{-1}((R\pi_* f^* \Theta_{X/k})^\vee)
 \rightarrow   H^{0}((R\pi_*Lf^*\MacBeth)^\vee)\\
 \rightarrow & H^{0}((R\pi_*f^* d^* \Theta _{S/k})^\vee) \rightarrow H^{0}((R\pi_* f^* \Theta_{X/k})^\vee)
 \rightarrow  H^{1}((R\pi_*Lf^*\MacBeth)^\vee) \rightarrow 0
\end{align*}
we see that $H^{1}((R\pi_*Lf^*\MacBeth)^\vee)$ vanishes by mapping via the canonical isomorphisms to the non-negative part of the cotangent exact sequence:
\[  \iota^* L_{\StableModuli_{0,0}(S, \beta)/\mathfrak{M}} \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \mathfrak{M}} \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta) } \rightarrow \]
and applying the 5-lemma. We now have an exact sequence
\[0\rightarrow H^{-1} ((R\pi_*f^*d^* \Theta _{S/k})^\vee) \rightarrow H^{-1}((R\pi_* f^* \Theta_{X/k})^\vee) \rightarrow H^{0}((R\pi_*Lf^*\MacBeth)^\vee) \rightarrow \]
\[ \rightarrow H^{0}((R\pi_*f^* d^* \Theta _{S/k})^\vee) \rightarrow H^{0}((R\pi_* f^* \Theta_{X/k})^\vee) \rightarrow 0\]
and by the 5-lemma $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)$ maps surjectively to $ram^* L_{\PP^1/\PP^2}$. 
Taking stalks at a generic point $\eta=\Spec K$ of the moduli space,
we may apply the Riemann-Roch theorem to the two tangent sheaf terms and using additivity of Euler characteristics we find that $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)_{\eta}$ has dimension one as a vector space over $K$. Now $(R\pi_*Lf^*\MacBeth)^\vee$ has cohomology supported in degrees minus one and zero. Let $E^\bullet$ be a resolution of $R\pi_*Lf^*\MacBeth$, which is quasi-isomorphic to the truncation $\tau_{\leq 0} E^\bullet$ defined in~\cite{StacksProject} Tag 0118. Taking a projective Cartan Eilenberg double resolution of this we obtain a resolution of $R\pi_*Lf^*\MacBeth$ by projectives concentrated in non-positive degree. Therefore after dualising $R\pi_*Lf^*\MacBeth^\vee$ has a resolution concentrated in non-negative degree. This shows that $H^0(R\pi_*Lf^*\MacBeth^\vee)$ embeds into a vector bundle, and hence is torsion free. \Lawrence{We prove an analogous result to the statement that a surjective map of line bundles is an isomorphism}. Take the morphism
\[\delta: H^{0}((R\pi_*Lf^*\MacBeth)^\vee) \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)} \]
of the diagram \eqref{diag:bigone}.
After localising at a point $x$ we have an isomorphism 
\[ 
L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta), x}
\cong \mathcal{O}_{\StableModuli_{0,0}(X, d^{-1} \beta), x}
\]
 and we know that $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)$ is a rank one torsion free sheaf with $\delta$ is surjective. Therefore we have the following situation: a local ring $R$, a rank one torsion free $R$-module $M$ and a surjection $\delta:M \rightarrow R$. We claim that this implies that $M$ is isomorphic to $R$. Since $\gamma$ is surjective we can choose $m \in M$ mapping to $1 \in R$. Suppose that there were $n \in M$ not in the submodule of $M$ generated by $m$. The element $n$ maps to $f \in R$ and so at every generic point of $R$ one has $n = fm$. But $M$ is torsion free and so $n = fm$. Therefore $m$ generates $M$ freely, i.e., $R \cong M$ as $R$-modules. Therefore $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)$ is isomorphic to $ram^* L_{\PP^1/\PP^2}$.

\Lawrence{The Gysin map $\iota^!$ produces a class on $[\mathbb{P}^1/C_2] \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)$. However the moduli space $\StableModuli_{0,0} (X, d^{-1} \beta)$ is a two to one \'etale cover of $[\mathbb{P}^1/C_2] \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)$. Therefore by ~\cite{VirtualPushforward} we obtain the formula
\[ 2 \iota^! [\StableModuli_{0,0} (S, \beta)]^{virt} = \iota_* [\StableModuli_{0,0} (X, d^{-1} \beta)]^{virt}.\]
We call this \emph{the comparison formula for bisections on $S$ and sections on $X$}.}

\end{proof}

This leaves the question of relating the relative invariants on $S$ to the classical invariants. Bryan and Leung studied this problem for the tangent order one case. Their motivation was to apply point conditions to stable maps directly by restricting to those stable maps whose marked points map to $P$. The deformation theory then needs to be modified, restricting to those deformations which vanish at the marked point. In their Appendix A they prove that this produces the correct invariants and extend this to the case of certain divisors. We apply their ideas here, starting with the easiest case. Recall that we write $\LogModuli (S^\dagger, \beta)$ for the moduli of log stable maps to $S^\dagger$ in a class $\beta$ tangent to the boundary at a single point with maximal tangency. 

\begin{theorem}[Log invariants of sections are restrictions of Gromov-Witten invariants]

Let \Lawrence{$E$} be a section of a rational elliptic surface $S$, and let $\beta = E + nF$. Then there is an inclusion of components $\iota:  \LogModuli (S^\dagger, \beta) \rightarrow \StableModuli_{0,0} (S, \beta) $ and this induces an equality of virtual classes $\iota^* [\StableModuli_{0,0} (S, \beta)] =  [ \LogModuli (S^\dagger, \beta) ] $.
\label{thm:LogEqualityForSections}

\begin{proof}

In~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} the authors constructed the moduli space $\StableModuli_{0,0} (S, \beta)$: any curve is the union of the section $E$ together with different covers of the rational fibres. Since any curve in $\LogModuli(S, \beta)$ has a unique choice of marked point $\sigma$ the forgetful map is injective and since different components correspond only to different covers of the rational fibres this map is an inclusion of components.
Indeed by Proposition 4.3 of~\cite{TheTropicalVertex} the components of
$\LogModuli(S,\beta)$ are precisely those components of $\StableModuli_{0,0}
(S,\beta)$ where no component of the universal curve maps into the boundary.

Let us denote by $\mathfrak{M}_{0,0}$ the moduli Artin stack of prestable
genus zero curves and by $\mathfrak{M}^{\dagger}_{0,1}$ the moduli Artin
stack of all pre-stable genus one-pointed log curves. Recall that
the virtual fundamental class of $\StableModuli_{0,0}(S,\beta)$ is
defined via an obstruction theory relative to $\mathfrak{M}_{0,0}$
and recall
from \cite{LogarithmicGromovWittenInvariants} that the virtual
fundamental class of the moduli space $\LogModuli(S,\beta)$ is defined
by an obstruction theory relative to $\mathfrak{M}^{\dagger}_{0,1}$.

We consider the product $\StableModuli (S, \beta)_{0,1} \times_S F_0$ and restrict to those components of
$\StableModuli (S, \beta)_{0,1}$ where no component of the domain curve maps into $F_0$. We have inclusions $\LogModuli(S, \beta) \subset \StableModuli (S, \beta)_{0,1} \times_S F_0 \cong \StableModuli (S, \beta)_{0,0}$. Furthermore there is a virtual class on $\StableModuli (S, \beta)_{0,1} \times_S F_0$ relative to $\mathfrak{M}_{0,1}$ defined by the obstruction bundle induced by the pull-back
of the kernel of the natural morphism $\Theta_{S/k} \rightarrow{\mathcal N}_{F_0/S}$. Note that this kernel is rank two bundle away from the singular
points of $F_0$, and no curve in the moduli space passes through these
singular points.
Sections of the pull-back of this bundle then correspond to infinitesimal
deformations that deform the marked point in a direction along $F_0$.
It is a folklore result exposited in remark A.5 of the appendix of~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} that the virtual class this defines is equal to the pullback of the virtual class on $\StableModuli (S, \beta)_{0,0}$. \Lawrence{Their result holds so long as $F_0$ and $S$ are both smooth. This is not the case and we instead consider $\Spec k \in \PP^1$ which indeed does have smooth source and target. Then the Gysin map of $F_0$ in $S$ is compatible with the Gysin map of a point in $\PP^1$ for which we do have the desired equality.}

It remains to compare the virtual classes on $\StableModuli (S, \beta)_{0,1} \times_S F_0$ and on $\LogModuli(S, \beta)$. Write $D_i$ for the components of $F_0$. There is a standard exact sequence of sheaves on $S$ given by
\[0 \rightarrow \Theta_{S^\dagger/k}^\dagger \rightarrow \Theta_{S/k} \rightarrow \bigoplus \Lawrence{i_*} \mathcal{N}_{D_i/S} \rightarrow 0.\]
By definition the curves we wish to count do not map into the intersections $D_i \cap D_j$ and so the pull-backs of $\Theta^{\dagger}_{S^{\dagger}/k}$ or
$\ker(\Theta_{S/k}\rightarrow {\mathcal N}_{F_0/S})$ to any stable curve in $\LogModuli(S, \beta) \cong \StableModuli (S, \beta)_{0,1}$ are isomorphic. We consider the two cotangent complexes $L_{\LogModuli (S^{\dagger}, \beta) / \mathfrak{M}^{\dagger}_{0,1}}$ and $L_{\StableModuli_{0,1} (S, \beta) \times_S F_0 / \mathfrak{M}_{0,1}}$. Note the log structure on $\LogModuli (S^{\dagger}, \beta)$
is the pull-back of the basic log structure on $\mathfrak{M}_{0,1}$ under
the forgetful map because there are no components or nodes of the domain
curve mapping into $F_0$. Thus we have a strict factorisation 
$\LogModuli(S^{\dagger},\beta)\rightarrow \mathfrak{M}_{0,1}
\rightarrow \mathfrak{M}_{0,1}^{\dagger}$, with the second morphism
\'etale. Thus the two cotangent complexes are isomorphic.

Since we have an isomorphism of obstruction theories and an isomorphism of underlying schemes the corresponding virtual classes are equal.

\end{proof}

\end{theorem}

The techniques of Bryan and Leung do not apply directly to the tangency order two case. This means we have to work harder.

\begin{theorem}[Log invariants of bisections are restrictions of Gromov-Witten invariants]

Let $ i : \mathbb{P}^1 \rightarrow \Hilb \cong \mathbb{P}^2/C_2$ be the hyperplane of double covers ramifying over $0 \in \mathbb{P}^1$ and $\beta$ be a curve class with $\beta . K_S = -2$. We denote the product $\StableModuli_{0,0} (S,\beta) \times_{\Lawrence{[\mathbb{P}^2/C_2]}} \mathbb{P}^1$ by $\StableModuli^{ram}(S, \beta)$ with projection maps $pr$ to $\StableModuli_{0,0} (S, \beta)$ and $ram$ to $\mathbb{P}^1$. Then the moduli space of log stable maps to $S^\dagger$, $\LogModuli (S^\dagger,\beta)$ admits a morphism $j$ to $\StableModuli^{ram} (S,\beta)$ which is an inclusion of components and there is an equality of virtual classes

\begin{equation}
j^* i^! [\StableModuli_{0,0}(S,\beta)]^{virt} = [\LogModuli(S^\dagger,\beta)]^{virt}
\end{equation}

\label{thm:logtocclassicalcomparison}

\begin{proof}

The map $j$ is the forgetful map, forgetting the choice of marking. A log curve in this space only meets the boundary in a single point, so maps to the desired space. By construction of the space $\overline{\mathcal M}^{ram} (S, \beta)$ there is a canonical choice of marked point given by intersecting with the boundary $F_0$ so long as no component of the curve maps into $F_0$. Viewing this
point as a logarithmic marked point with contact order $2$ with one of the
boundary divisors $D_i$ or contact orders $1$ with adjacent divisors
$D_i$, $D_j$, 
we obtain a family of stable log maps. Thus $\LogModuli (S^\dagger, \beta)$ is open inside some components of $\StableModuli^{ram}$ but is also proper and hence closed. Thus $j$ is an inclusion of components. To be precise it is those components which do not cover the marked rational fibre and have precisely one point mapping to the boundary, which excludes any point components. 

As in the single section case we relate potentially distinct obstruction theories on $\LogModuli (S^\dagger, \beta)$ and $\StableModuli_{0,0}(S,\beta)$. Recall that the obstruction theories on $\LogModuli (S^\dagger, \beta)$ and $\StableModuli_{0,0}(S,\beta)$ are given by $R \pi_{S, *} f_S^* \Theta_{S^\dagger/k} ^{\dagger \: \vee}$ and $R \pi_{S, *} f_S^* \Theta_{S/k} ^{\vee}$ respectively. There is an exact sequence
\[ 0 \rightarrow  \Theta_{S^\dagger/k}^\dagger \rightarrow \Theta_{S/k} \rightarrow \bigoplus \mathcal{O}_{D_i}(D_i) \rightarrow 0 \]
where $F_0 = \bigcup D_i$. Pulling back to the universal curve over $\LogModuli (S^\dagger, \beta)$, pushing forward along $\pi$ and dualising we therefore obtain a distinguished triangle
\[ (R \pi_* f^* \bigoplus \mathcal{O}_{D_i}(D_i))^\vee \rightarrow (R \pi_*f^*\Theta_{S/k})^\vee \rightarrow (R \pi_*f^*\Theta_{S^\dagger/k}^\dagger)^\vee \rightarrow\]
Now $f^* \bigoplus \mathcal{O}_{D_i}(D_i)$ is supported only at a single point of each fibre of $\pi$, hence it pushes forwards to a line bundle on $\LogModuli (S^\dagger, \beta)$. Consider the cotangent triangle associated to the maps $\LogModuli (S^\dagger, \beta) \hookrightarrow \StableModuli (S, \beta) \rightarrow \mathfrak{M}_{0,0}$, noting that as before the map $\LogModuli (S^\dagger, \beta) \rightarrow \mathfrak{M}_{0,0}^\dagger$ factors through the \'etale map $\mathfrak{M}_{0,0} \rightarrow \mathfrak{M}_{0,0}^\dagger$
\[ i^*L_{\StableModuli (S, \beta)/\mathfrak{M}_{0,0}} \rightarrow L_{\LogModuli (S^\dagger, \beta)/\mathfrak{M}_{0,0}^\dagger} \rightarrow L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)} \rightarrow\]
Since $(R \pi_*f^*\Theta_{S/k})^\vee$ and $(R \pi_*f^*\Theta_{S^\dagger/k}^\dagger)^\vee$ are obstruction theories we obtain by~\cite{VirtualPullbacks} that $(R \pi_* f^* \bigoplus \mathcal{O}_{D_i}(D_i))^\vee[1]$ is an obstruction theory for $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$. But $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$ is a regular immersion defined by one equation, namely that defining $\PP^1 \subset \PP^2$. We therefore deduce that $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$ is a line bundle supported in degree minus one. But $(R \pi_* f^* \bigoplus \mathcal{O}_{D_i}(D_i))^\vee[1]$ is also a line bundle supported in degree minus \Lawrence{one} and maps surjectively and hence isomorphically to $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$. This produces an equality of virtual classes on $\LogModuli(S^\dagger, \beta)$
\[\iota^! ([\StableModuli_{0,0}(S,\beta)]^{vir}) = [\LogModuli(S,\beta)]^{vir}\]
as desired.
\end{proof}

\end{theorem}

We can combine this with our relation between the invariants of $X$ and $S$, producing the following theorem.

\begin{corollary}[A formula for bisections on $S$]

There are equalities
\[  deg \: (ram^* \Lawrence{(2H)} \cdot[\StableModuli_{0,0} (S, \beta)]^{virt}) = deg \:([\StableModuli_{0,0} (X, d^{-1}\beta)]^{virt}) \]
and
\[deg \: (j^* (ram^* \Lawrence{(H)} \cdot [\StableModuli_{0,0} (S, \beta)]^{virt})) = deg \: ([\LogModuli (S^\dagger, \beta)]^{virt}) \]
\end{corollary}

\section{Bounding Gromov-Witten invariants}

As we said at the start of the previous section there is no reason why when we pass to the log moduli space we don't remove components with high degree, potentially ruining our bound of the $J$-function. This section is dedicated to proving that restriction of the virtual class to curve components is positive, whilst the restriction to bubble components is not too negative. We begin by proving that the point components make no contribution to either of our invariants.

\begin{lemma}[Point components are trivial]

The restriction of $ram^* H \cdot [\StableModuli_{0,0} (S, \beta)]^{virt}$ to point components is zero.

\end{lemma}

\begin{proof}

This follows from conservation of number and choosing a different hyperplane to intersect with.

\end{proof}

By relating the invariants on $S$ to those on a $K3$ fibre of $X$ we can apply techniques from~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms}. The authors there prove positivity of the virtual class restricted to various components of the moduli space.

\begin{theorem}[Curve components are positive]

The restriction of $ram^* H \cdot [\StableModuli_{0,0} (S, \beta)]^{virt}$ to the preimage of a curve component $\mathcal{M}$ is positive.

\label{thm:CurvesArePositive}

\begin{proof}
  \Lawrence{By our compatibility result relating Gromov-Witten invariants on $S$ and $X$ it is enough to prove this positivity on the threefold $X$}. This problem was studied by Maulik and Pandharipande in~\cite{GromovWittenTheoryAndNoetherLefschetzTheory} but we can prove the equality here via a direct calculation. Recall that the reduced invariants on a $K3$ surface $K$ are given by an obstruction theory $T^\circ$ fitting into an exact triangle
  \[  \tau_{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K) \rightarrow R \pi_* f^* \Theta_{K/k} \rightarrow T^\circ \rightarrow\]
    where $\tau_{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K)$ is a line bundle supported in degree minus one. Suppose now that $i:K \hookrightarrow X$ is a fibre of the $K3$ fibration of $X$ into which all the stable curves in $\mathcal{M}$ map. There is an exact sequence on $K$
    \[ 0 \rightarrow \Theta_{K/k} \rightarrow \Lawrence{i^*} \Theta_{X/k} \rightarrow \mathcal{N}_{K/X} \rightarrow 0 \]
    where the normal bundle $\mathcal{N}_{K/X}$ is trivial since $K$ is a fibre. Pulling this back to the universal curve, pushing forwards to the moduli space and dualising we obtain a distinguished triangle on $\StableModuli_{0,0} (X, \beta)$
    \[ \mathcal{L} \rightarrow (R \pi_* f^* i^* \Theta_{X/k})^\vee \rightarrow (R \pi_* f^* \Theta_{K/k})^\vee  \rightarrow \]
    where $\mathcal{L}$ is a line bundle supported in degree zero. By composition we find a morphism $(R \pi_* f^* i^* \Theta_{X/k})^\vee \rightarrow T^\circ$ and let $\Cone$ be the cone over this. Applying the octahedral axiom we obtain a diagram
 \[ \begin{tikzpicture}
      \node (T) at (0,0) {$T^\bullet$};
      \node (L) at (0,2) {$\mathcal{L}[1]$};
      \node (C) at (3,0.5) {$\Cone[1]$};
      \node (K) at (-3,0.5) {$R \pi_* f^* \Theta_{K/k}^\vee$};
      \node (X) at (-6,-1) {$R \pi_* f^* \Theta_{X/k}^\vee$};
      \node (R) at (6,-1) {$\tau _{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K)[1]$};
      \node (E0) at (1.5,2.75) {};
      \node (E1) at (4.8,0.8) {};
      \node (E2) at (9,-1.5) {};
      \node (E3) at (9,-2.5) {};
      
      \draw [->] (L) -- (E0);
      \draw [->] (C) -- (E1);
      \draw [->] (R) -- (E2);
      \draw [->] (R) -- (E3);      
      \draw [->] (L) -- (C);
      \draw [->] (K) -- (L);
      \draw [->] (K) -- (T);
      \draw [->] (T) -- (C);
      \draw [->] (X) -- (K);
      \draw [->] (X) -- (T);
      \draw [->] (T) -- (R);
      \draw [->] (C) -- (R);
      \end{tikzpicture}\]
    showing that the following triangle is distinguished
    \[  \mathcal{L} \rightarrow \Cone \rightarrow  \tau _{\geq -1}R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K) \rightarrow \]
    Now the special property of the preimage of a curve components is that each component is the moduli space of stable maps to $K$. In particular the degree zero cohomology of $(R \pi_* f^* i^* \Theta_{X/k})^\vee$ and $T^\circ$ are isomorphic since both are isomorphic to the tangent space of $\mathcal{M}$. This is enough to show that in fact $\Cone$ vanishes since $\Cone$ has a two term resolution by line bundles, \Lawrence{the differential is a surjection of line bundles since the degree zero cohomology vanishes and hence this surjection is an isomorphism.}

    Now $\mathcal{M}$ is a component of the moduli space of stable maps both to $K$ and $X$ and the two induced obstruction theories are isomorphic. The result is known from~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} that each component of the moduli space contributes either one or zero to the total count.

\end{proof}

\end{theorem}

Now on the bubble component we prove that the virtual class cannot become too negative.

\begin{theorem}[Bubble components are bounded]
  
Let $X$ be a generic unravelling of $S$, $\StableModuli^{bub} \subset \StableModuli_{0,0} (X, d^{-1}\beta)$ the preimage of a bubble component, a family over $\iota : \mathbb{P}^1 \rightarrow \Hilb \cong \mathbb{P}^2/C_2$. Choosing a point $P: Spec \: k/C_2 \rightarrow \mathbb{P}^2/C_2$ in the image of $\iota$ defines a $K3$ surface $cov:K \rightarrow S$. This induces a map $P: \StableModuli_{0,0} (K, cov ^* \beta) \rightarrow \StableModuli^{bub}$ and generically fibres $\StableModuli^{bub}$ by moduli spaces of stable maps to $K3$ surfaces, $Kl: \StableModuli^{bub} \rightarrow \mathbb{P}^1$. We claim that there is a constant $q \in \mathbb{Q}$ with $P_* [\StableModuli_{0,0} (K, cov ^* \beta)] = q [_{0,0} (X, d^{-1} \beta)]$ and that $q$ is dependent only on the choice of $X$.\label{thm:BubblesArentTooNegative}
\begin{proof}

  This question is related to the comparison theory of Maulik and Pandharipande in~\cite{GromovWittenTheoryAndNoetherLefschetzTheory}. In~\cite{GromovWittenTheoryAndNoetherLefschetzTheory}, the authors consider a family of nonsingular $K3$ over a non-singular curve $C$, $p: X \rightarrow C$. The Gromov-Witten invariants of $X$ are related to those of a fibre $K$ by multiplication by the degree of the Hodge bundle of $X \rightarrow C$. The details of this calculation are contained in~\cite{HodgeIntegralsAndGromovWittenTheory}.

  In our situation however there are some singular fibres of the fibration $p: X \rightarrow \PP^1$. Twenty four of these fibres are ordinary double points, occurring where the double cover $\PP^1 \rightarrow \PP^1$ ramifies over the image of a rational fibre. One can take a resolution of this family as described in Example 5.1 of~\cite{GromovWittenTheoryAndNoetherLefschetzTheory} to obtain a new family of non-singular $K3$ surfaces with the same Gromov-Witten invariants. In their paper Maulik and Pandharipande move to an analytic space to construct the small resolution. \Lawrence{This is a strictly analytic construction and to apply techniques of log geometry we must remain in the algebraic world. We appeal to Theorem 7.3 of~\cite{AlgebraizationOfFormalModuliII} which says that any Moishezon manifold is the analytification of an algebraic space. This resolution is indeed a Moishezon manifold since it is a small resolution of another Moishezon manifold. Therefore we consider a small resolution $\pi: \tilde{X} \rightarrow X$ with $\tilde{X}$ a smooth algebraic space. The moduli stack of stable maps to a Deligne Mumford stack was constructed in~\cite{GromovWittenTheoryOfDeligneMumfordStacks}. In that paper the authors also proved that the usual technology for constructing virtual classes can be applied to such objects and so the techniques of~\cite{HodgeIntegralsAndGromovWittenTheory} continue to apply.} However there are also two singular fibres where the $K3$ surface degenerates to the union of two rational elliptic surfaces meeting along a smooth elliptic fibre. We augment this singular family to a smooth log family. Let $K_1$ and $K_2$ denote these two singular fibres and we write $X^\dagger$ for $X$ with the divisorial log structure induced by the $K_i$. Since we have not imposed any tangency conditions \Lawrence{and a generic curve is transverse} the classical and log Gromov-Witten invariants coincide. Now by construction the map $p: X^\dagger \rightarrow \PP^1$ is log smooth and we can define a relative theory by taking the cone over the natural morphism
    \[\tau_{\geq-1} R \pi_* (\omega_\pi^\dagger) ^\vee \otimes H^0 (X, \omega_{X/\PP^1}) \rightarrow R \pi_* f^* (\Theta_{X/\PP^1})\]
    This gives a relative obstruction theory which may play the role of the relative theory defined in section 2.2 of~\cite{GromovWittenTheoryAndNoetherLefschetzTheory}. In particular their equation
    \[ [\StableModuli _{0} (\pi, \epsilon)]^{vir} = c_1 (K^*) \cap [\StableModuli_{0}(\pi, \epsilon)]^{red}\]
from page 22 continues to hold, but where $K$ is the log canonical line bundle with fibre $H^0 (X_{\xi}, K^\dagger_{X_\xi})$ over a point $\xi$. This proves the desired result since we may take the Chern class to be supported away from the singular fibres.

\end{proof}

\end{theorem}

\section{Assembling the family}

We now have a description of the product formula on the mirror, of the terms appearing in the scattering diagram and of the piecewise linear function $\phi$. Therefore we can assemble all of these together to prove convergence of the mirror. This pits the growth of the function $\phi$ against the growth of the number of curves, so we begin with bounding the growth of $\phi$.

\begin{lemma}[Bounding the function $\phi$]
  We choose the lift of $\phi$ which is zero on the cone $C$ of Figure~\ref{fig:DualIntersectionRES}. Over a one-cell $v_i$ recall that the piecewise linear function $\phi$ changes by $D_i \otimes n_i$, where $n_i$ is defined in Definition
\ref{def:sheaf-of-monoids}. Then $\phi$ is given by the following formulae on the following cones. On $\langle (4n, 1), (4n+1, 1)\rangle$ it is given by
\[nx F - (n(2n-1)D_1 + 2n^2D_2 + n(2n+1)D_3 + n(2n+2)D_4)y\]
whilst on $\langle (4n+1, 1), (4n+2, 1)\rangle$ it is given by
\[nx F + x D_1 - ((n+1)(2n+1)D_1 + 2n^2D_2 + n(2n+1)D_3 + n(2n+2)D_4)y\]
and on $\langle (4n+2, 1), (4n+3, 1)\rangle$ it is given by
\[nx F + x(D_1 + D_2) - ((n+1)(2n+1)D_1 + 2(n+1)^2D_2 + n(2n+1)D_3 + n(2n+2)D_4)y\]
and finally on $\langle (4n+3, 1), (4n+4, 1)\rangle$ it is given by
\[nx F + x(D_1 + D_2 + D_3) - ((n+1)(2n+1)D_1 + 2(n+1)^2D_2 + (n+1)(2n+3)D_3 + n(2n+2)D_4)y\]

We now find rough bounds on $\phi$. Suppose that $m > n > 0$ are integers and we wish to compute $\phi ((m,1)) - \phi ((n,1))$. Since $\phi$ is strictly convex the difference between these is bounded by $\phi((m,1)) - \phi((m-1,1))$. From the formulae above we see that $\phi ((m,1)) - \phi ((m-1,1)) \geq (2\left \lfloor{m/4}\right \rfloor -1) D_{m-1}$ where the index of $D_{m-1}$ is taken mod $4$. The inequality $\phi ((m,1)) - \phi ((m-1,1)) \geq (1- 2\left \lfloor{m/4}\right \rfloor) D_{m+1}$ similarly holds for $m < n < 0$. Now suppose that $n,k > 0$ and we consider $\phi ((m,1)) - \phi ((n,1)) - \phi ((k,1))$ with $m \ge n+k$. By convexity this value is smallest if $n = m-1$ and $k=1$. In that case we can use the above bound to see that this is bounded also by $(2\left \lfloor{m/4}\right \rfloor -1)D_{m-1}$.

\end{lemma}

Recall the structure of the equations from Table~\ref{eqntable}. We will work through each term appearing and prove that they all converge in a neighbourhood of the origin.

\begin{proposition}

The coefficients $f_{(k,2)}, g_{(k,2)}$ and $r_{(k,2)}^i$ of these equations converge in a neighbourhood of the large complex structure limit point.

\begin{proof}

  By symmetry it is enough to prove convergence only of the first equation and of one of the products $\vartheta_{D_i}^2$.

  The coefficient $f_{2,2}$ is the sum over pairs which do not bend anywhere and end close to $(2,2)$. The pairs of pants are pairs of broken lines from $(4n,1)$ and $(-4n+2,1)$. Near $(2,2)$ these lines carry the monomials $z^{\phi (4n,1)}$ and $z^{\phi (-4n+2,1)}$. Therefore we want to study the convergence of
  \[\sum |z^{\phi (4n,1) + \phi (-4n+2,1)}| < \sum |z^{n^2F}|\]
  which converges since it decays at least exponentially. Exactly the same argument applies to show the convergence of $f_{(4,2)}$, $r_i^i$ and $r_{i+2}^i$. In all these cases none of the broken lines can bend, there is a $\ZZ$ indexed family of pairs of pants and the monomials are controlled entirely by the function $\phi$. For the product $\vartheta_{D_i}^2$ there is precisely one pair of broken lines which lie in only one maximal cell of $B$. This explains the leading coefficient of $1$ appearing in the formulae.

\end{proof}

\end{proposition}
We now study the contribution to the product of points with $E(P) = 1$, i.e.,
the theta functions $\vartheta_{D_i}$, restricting our analysis by symmetry to $\vartheta_{D_1}\vartheta_{D_3}$ and $\vartheta_{D_4}^2$. Let us begin with $\vartheta_{D_1}^2$ where we describe all pairs of broken lines which contribute to $\vartheta_{D_4}$. All such pairs contain one bend which for the purpose of proving convergence we can assume occurs on the first broken line. We may also assume that the first broken line is from $(4m + 4n, 1)$ with $m, n$ positive and scatters off of $(4n, 1)$ with a monomial $z^{E+kF}$ for $E$ a section meeting $D_1$ and $k$ non-negative. As a consequence the second line is from $(-4m, 1)$. This pair of lines contributes $I_{0,0,E+kF}^{\dagger}z^{E + \phi ((4m+4n,1)) - \phi ((4n,1)) + \phi ((-4m,1)) + kF}$ to the coefficient. This sum is taken over all $m,n > 0$, $k \geq 0$ and sections $S$ meeting $D_1$ in one point. 

\begin{lemma}[Single bends converge]

The following sum converges
\[\sum_{m,n,k, S} I_{0,0,S+K}^{\dagger}z^{S + \phi ((4m + 4n, 1))  + \phi ((-4n,1)) - \phi ((4m,1)) + kF}\]
\begin{proof}
We rewrite this sum as
\begin{equation}
\left( \sum _{S,k} I_{0,0,S+kF}^{\dagger} z^{S + kF}  \right) \left( \sum_{m,n}  z^{\phi ((4m + 4n, 1)) + \phi ((-4n,1)) - \phi ((4m,1))}\right)
\end{equation}

We know that the term $I_{0,0,S+kF}^{\dagger}$ has modulus at most $2^k$, therefore for $| z^F | < 1/2$ and $| z^S | < 1$ and after applying Lemma~\ref{thm:BoundedlyManySections} one has convergence of the left hand sum.

To prove convergence of the right hand sum we expand out the sum using the explicit formula for $\phi$ found above and bound it
\[\sum_{m,n} | z^{\phi ((4m+4n,1)) - \phi ((4m,1)) + \phi ((-4n,1))} | \le \sum_{m,n} | z^{\phi ((-4n,1)) + ((2m+2n-1)D_{3})} |\]
And this converges since $\phi ((-4n,1))$ is positive and grows quadratically in $n$.

\end{proof}

\end{lemma}

Therefore we have shown that all the functions appearing in the defining equations of the mirror family are in fact holomorphic in a neighbourhood of the origin.

\section{Recognising the family}

The above convergence result does not tell us very much about the family that one obtains. Fortunately the geometry of the singular locus allows us to say much more. Consider the equations~\ref{eqn:MirrorToRES1}-\ref{eqn:MirrorToRES3}, only looking at the terms of order two these simplify to:

\begin{align}
  \vartheta_{D_1} \vartheta_{D_3} & = f_{(2,2)} \vartheta_{2D_2} + f_{(6,2)} \vartheta_{2D_4}    \label{eqn:ReducedMirrorToRES1} \\
  \vartheta_{D_2} \vartheta_{D_4} & = f_{(0,2)} \vartheta_{2D_1} + f_{(4,2)} \vartheta_{2D_3}    \label{eqn:ReducedMirrorToRES2}\\
  \vartheta_{2D_i} & = \frac{\vartheta_{D_i}^2 (1+r_{i+2,2}^{i+2})-r_{i+2,2}^i \vartheta_{D_{i+2}}^2} {(1+r_{i,2}^i)(1+r_{i+2,2}^{i+2}) - r_{i,2}^{i+2} r_{i+2,2}^{i}}\label{eqn:ReducedMirrorToRES3}
\end{align}
These equations correspond to taking the projective closure of the above family and looking at the fibre at infinity. Substituting in the final equation into the first two we obtain the following pair of quadrics:

\begin{align}
  \vartheta_{D_1} \vartheta_{D_3} & = \frac{f_{(2,2)}(1+r_{4,2}^4) - f_{(6,2)}r^4_{2,2}}{(1+r^2_{2,2})(1+r^4_{4,2})-r^4_{2,2}r^2_{4,2}} \vartheta_{D_2}^2 + \frac{f_{(6,2)}(1+r_{2,2}^2) - f_{(2,2)}r^2_{4,2}}{(1+r^2_{2,2})(1+r^4_{4,2})-r^4_{2,2}r^2_{4,2}} \vartheta_{D_4}^2    \label{eqn:Ugh1} \\
  \vartheta_{D_2} \vartheta_{D_4} & = \frac{f_{(0,2)}(1+r_{3,2}^3) - f_{(4,2)}r^3_{1,2}}{(1+r^1_{1,2})(1+r^3_{3,2})-r^3_{1,2}r^1_{3,2}} \vartheta_{D_1}^2 + \frac{f_{(4,2)}(1+r_{1,2}^1) - f_{(4,2)}r^1_{3,2}}{(1+r^1_{1,2})(1+r^3_{3,2})-r^3_{1,2}r^1_{3,2}} \vartheta_{D_3}^2 \label{eqn:Ugh2}
\end{align}

There are some relations between the functions involved in this description. To simplify this description let us restrict to the locus where the areas of each of the $D_i$ are all $v$. Let us show that this curve is smooth over this locus. We will also be interested in some questions of modularity of certain constants, and we begin with generalities of equations of the form:

\[ X_1  X_3 = tX_2 ^2 + t X_4 ^2, \quad X_2  X_4= tX_1 ^2 + t X_3 ^2\]

Putting these equations into Sage we can arrange them into Weierstrass form (using the WeierstrassForm function). This produces the equation

\[v^2 - u^3 - u(-t^8/3 - 7t^4/24 - 1/768) - (2t^{12}/27 - 11t^8/72 - 11t^4/1152 + 1/55296)\]
A generically smooth elliptic curve with $j$-invariant
\begin{align}
\frac{16777216t^{24} + 44040192t^{20} + 38731776t^{16} + 11583488t^{12} + 151296t^8 + 672t^4 + 1}{65536t^{20} - 16384t^{16} + 1536t^{12} - 64t^8 + t^4}
\label{jinvariant} \end{align}
Making the substitution $s = 4t^2$ we obtain the following expression.
\[16 \frac{(s^4 + 14 s^2 + 1)^3} {s^{2} (s -1)^4 (s + 1)^4}\]
We compare this to the formula for the $j$-invariant in terms of the Jacobi modulus $k$. By definition the $j$-invariant is given by
\[ 256 \frac{(k^4-k^2 + 1)^3}{k^4 (k^2-1)^2} \]
After a change of coordinates $k = \frac{u + 1/4}{\sqrt{u}}$ we find that this is equal to
\[256 \frac{(u^4 + 7u^2/8 + 1/256)^3}{(u^{8} + (-1/4)u^{6} + 3/128u^4 + (-1/1024)u^2 + 1/65536)u^2}\]
and so lining this up with equation~\ref{jinvariant} we find that $u = s/4$. Substituting backwards we find $k = \frac{t^2 + 1/4}{t}$.

Now let us evaluate the function $t$ for the locus where the $z^{[D_i]}$ are all equal and we express the functions in terms of the area of a general smooth fibre, say $e^{i\pi \rho}$.

\[f_{(k,2)} (v) = \sum_{n} v^{(4n+1)^2} = \Theta_2 (0,\rho)\]
\[1 + r_{i,2}^i (v) = \sum_{n \: even} v^{4n^2} = (\Theta_3 (0,\rho) + \Theta_4(0,\rho))/2\]
and
\[r_{i+2,2}^i (v) = \sum_{n \: odd} v^{4n^2} = (\Theta_3 (0,\rho) - \Theta_4(0,\rho))/2\]
where $\Theta_i (z,\rho)$ are the Jacobi theta functions. Substituting these into equations~\ref{eqn:Ugh1} and~\ref{eqn:Ugh2} we find that $t$ is equal to the ratio $\frac{\Theta_2(0,\rho)}{2 \Theta_3 (0,\rho)}$. In particular this is not constant in $\rho$ and so for generic choices is smooth. Furthermore the Jacobi modulus $k$ of this curve is equal to $\frac{\Theta_2 (0,\rho) ^2 + \Theta_3 (0,\rho)^2}{2 \Theta_2 (0,\rho) \Theta_3 (0,\rho)}$. But by definition we have that $k$ is also $\Theta_2 (0,\tau) / \Theta_3 (0,\tau)$ where $\tau$ is the fundamental period of the elliptic curve. Of course there should be a relation between these two:

\begin{align*}
  \frac{\Theta_2 (0,\rho) ^2 + \Theta_3 (0,\rho)^2}{2 \Theta_2 (0,\rho) \Theta_3 (0,\rho)} & = \frac{1}{2} \left( \frac{\Theta_3 (0,\rho)}{\Theta_2 (0,\rho)} + \frac{\Theta_2 (0,\rho)}{\Theta_3 (0,\rho)}  \right) \\
  & = \frac{\Theta _3 (\rho /2 )^2} {\Theta_2 (\rho /2)^2}
  \end{align*}

The modular group acts on the Jacobi theta functions, with $\tau \mapsto \tau + 1$ swapping the pairs $\Theta_3$ and $\Theta_4$ and $\Theta_1$ and $\Theta_2$. The map $\tau \mapsto -1/\tau$ swaps $\Theta _2$ and $\Theta _4$ but preserves $\Theta_1$ and $\Theta_3$. Conjugating the first of these by the second we see that

\[ \frac{\Theta _3 (\rho /2 )^2} {\Theta_2 (\rho /2)^2} = \frac{\Theta _2 (\frac{\rho}{2-\rho})^2} {\Theta_3 (\frac{\rho}{2-\rho})^2}\]

But this should also be equal to $\frac{\Theta _2 (\tau)^2} {\Theta_3 (\tau)^2}$ and so we can deduce that $\tau$ and $\frac{\rho}{2-\rho}$ are equal up to conjugacy under $SL (2, \ZZ)$. But we have just seen that $\rho$ and $\frac{\rho}{2-\rho}$ are also conjugate. Therefore the two associated elliptic curves are isomorphic. The limit as the area of a curve approaches zero corresponds to the limit as the imaginary part of $\rho$ goes to positive infinity. In this limit the value of $\tau$ also approaches the cusp point, and so the corresponding elliptic curve degenerates to the Tate curve. Of course we have already seen this, in the limit that we turn off all corrections this elliptic curve degenerates to a cycle of four rational curves.

This construction connects very pleasantly to the story of mirror symmetry for elliptic curves as described by Dijkgraaf in~\cite{MirrorSymmetryAndEllipticCurves} or Polischuk and Zaslow in~\cite{CategoricalMirrorSymmetryTheEllipticCurve}. These describe the mirror to an elliptic curve with complexified K\"ahler class $\alpha$ as the elliptic curve with periods $\langle 1, \alpha \rangle$. We do not need to specify a complex structure on the first curve since the Fukaya category does not depend on this choice. The choice of volume $e^{i\pi \tau}$ corresponds to a choice of complexified K{\"a}hler class on the general fibre, and thus we see a philosophical explanation of the above equality.

If we restrict now to the locus where $z^{[C]}$ vanishes for $[C].[F] > 0$ the above surface degenerates further, only the above terms appear in the defining equation. Since now the equation is homogenous in affine space we recover the family as being a deformation of a cone over the mirror elliptic curve.

\section*{Future questions}

There are several questions that immediately spring from this project. Firstly what happens as one allows the areas of the different components of the fibre to vary. The elliptic curve at infinity deforms in a controlled manner. I do not believe that there is any reason to think that the periods should be as well behaved, but rather there will be some codimension one locus under which the previous description holds. It would be interesting to describe this locus. Secondly the reason we needed the $I_4$ fibre was simply to control the combinatorics and the relevant terms in the scattering diagram. Conjecturally one should be able to reconstruct the entire Gromov-Witten theory from our description. From this one should be able to describe the theory as one smooths the boundary to less degenerate configurations, whilst the generating functions remain holomorphic. If one could do this then the construction would converge relative to any boundary fibre. Thirdly, following the previous point, one could attempt to understand how to apply this construction to the central fibre of a type II degeneration, similarly to the work of Atsushi Kanazawa in~\cite{DoranHarderThompsonConjectureViaSYZMirrorSymmetryEllipticCurves} in the elliptic curve case.

\quad \newline
\noindent
\texttt{National Center for Theoretical Sciences\\ No. 1 Sec. 4
Roosevelt Rd., National Taiwan University\\ Taipei, 106, Taiwan}

\end{document}

\vspace*{\fill}
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\begin{quotation}
\em 
“The mountains are calling and I must go.”

\raggedleft
- John Muir
\end{quotation}
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\quad

\quad

\chapter*{Abstract}

The construction introduced by Gross, Hacking and Keel in~\cite{MirrorSymmetryforLogCY1} allows one to construct a mirror family to $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. To do so one constructs a formal smoothing of a singular variety they call the $n$-vertex. By arguments of Gross, Hacking and Keel one knows that this construction can be lifted to an algebraic family if the intersection matrix for $D$ is not negative semi-definite. In the case where the intersection matrix is negative definite the smoothing exists in a formal neighbourhood of a union of analytic strata. A proof of both of these is found in~\cite{MirrorSymmetryforLogCY1}.

In our first project we use these ideas to find explicit formulae for the mirror families to low degree del Pezzo surfaces. In the second project we treat the remaining case of a negative semi-definite intersection matrix, corresponding to $S$ being a rational elliptic surface and $D$ a rational fibre. Using intuition from the first project we prove in the second project that in this case the formal family of their construction lifts to an analytic family.

\tableofcontents

\pagenumbering{arabic}

\chapter{The Saga of Mirror Symmetry}

Algebraic geometers have long been interested in enumerative questions on varieties. These problems have led to the development of techniques in intersection theory and deformation theory which have then been applied to solve many other problems. As an example let us consider the following problem

\begin{center}
How many degree one curves are there on a smooth cubic surface $S$ in $\PP^3$?
\end{center}

To understand when we expect these problems to have a finite number of solutions we must turn to deformation theory and the Riemann-Roch theorem. By~\cite{DeformationTheory} the deformations of a smooth subvariety $Z$ of a smooth variety $V$ are controlled by the normal bundle $\mathcal{N}_{Z/V}$ to $Z$ in $V$. The global sections of this sheaf correspond to first order deformations of the embedding whilst elements of the module $H^1 (Z, \mathcal{N}_{Z/V})$ correspond to obstructions to lifting these deformations to order two. By the Riemann-Roch theorem applied to the deformations of a smooth curve $C$ inside a smooth variety $V$ the difference of the ranks of these two modules, or \emph{expected dimension}, is given by
\[(1-g(C))(\dim V - 3) - K_V . [C]\]
	In the above problem this expected dimension is zero. One can show that the normal bundle $\mathcal{N}_{Z/V}$ has no higher cohomology, so the solution space, in this case the moduli space of maps from $\PP^1$ to $S$, genuinely is zero dimensional and has length 27, corresponding to the 27 lines on the cubic surface. Now looking at the above formula one notices that on a Calabi-Yau threefold, that is to say a threefold with trivial canonical class, any class $\beta$ \Lawrence{ought to support} a finite number of rational curves\Lawrence{, although the actual picture is more subtle}. \Lawrence{Throughout the history of algebraic geometry many mathematicians for many reasons have attempted to calculate these invariants for low degree rational curves} lying on a smooth quintic in $\PP^4$. This question rapidly becomes more intractable as the degree grows, degree one was known to mathematicians in the $19^{th}$ century, degree two was calculated in~\cite{OnTheFinitenessOfRationalCurvesOnQuinticThreefolds} by Sheldon Katz and degree three by Ellingsrud and Str{\o}mme in~\cite{TheNumberofTwistedCubicCurvesontheGeneralQuinticThreefold}. This question however was catapulted to fame by the work of physicists in~\cite{APairOfCalabiYauManifoldsAsAnExactlySolubleSuperconformalTheory}. Using this work a general formula was given by Givental in~\cite{TheMirrorFormulaForQuinticThreefolds}.

\section{The mathematical framework}

\Lawrence{During the 80's and 90's physicists had explored a great number of topological conformal field theories, string theories built from maps from a Riemann surface into a target space. Of these two, the $A$ and $B$-models are twists of the same theory, depending on symplectic and complex data respectively. Therefore all the physical data, boundary conditions, partition functions and so on, should be equal up to a change of basis. The driving motivation for Mirror Symmetry is that to any target space $X$ there should be another target space $\check{X}$, called the \emph{mirror}, interchanging these two models. We will not formally attempt to describe these theories as this is purely motivational. Instead we will talk about what data goes into building them.}

Let us begin this section by recalling some motivation from symplectic geometry. This is relevant to us as the Gross-Siebert program is an attempt to recreate algebro-geometrically this symplectic data. Since this relies on the reader being familiar with symplectic geometry we will not attempt to build examples, rather we jump in with the definitions. 

\begin{definition}

  A \emph{symplectic manifold} is a $2n$-dimensional smooth manifold $M$ together with a choice $\omega$ of closed non-degenerate two-form on $M$. An \emph{almost complex structure} on $M$ is a choice of automorphism $J$ of the tangent bundle $T_M$ such that $J^2 = -1$. Such a structure is called \emph{compatible} if $\omega (v, Jv)$ defines a metric on $M$.

  Given $(M, \omega)$ a symplectic manifold a submanifold $N \subset M$ is a \emph{symplectic} submanifold if $\omega|_N$ is closed and non-degenerate. It is \emph{isotropic} if $\omega|_N$ is zero and it is \emph{Lagrangian} if it is both isotropic and of dimension $n$. 
\end{definition}

Examples of \Lawrence{symplectic} manifolds include complex K\"ahler manifolds, so include smooth complex projective varieties with the restriction of the Fubini-Study symplectic form on $\PP^n$. Let $(M, \omega)$ be such a variety. The Lagrangian submanifolds of $(M, \omega)$ naturally arise in the construction of the $A$-model as the boundary conditions of strings, \Lawrence{called} $D$-branes. Mathematically these boundary conditions should form a category, in this case the \emph{Fukaya category}. There are many different constructions of this category dependent on what properties one wants of it, see~\cite{LagrangianIntersectionFloerTheoryAnomalyAndObstruction} for the initial definitions. In broad brush-strokes the objects of this category are Lagrangian submanifolds with a choice of additional data\Lawrence{, in our later example a flat $U(1)$ bundle over that Lagrangian}. Morphisms $L_1 \rightarrow L_2$ in this category correspond to elements of a homology theory built from intersection points of $L_1$ and $L_2$. In the philosophy of Kontsevich and Soibelman there should be a whole system of operations defining an $A_{\infty}$ structure hidden behind these definitions as described in~\cite{NotesOnAInfinityAlgebrasAInfinityCategoriesAndNonCommutativeGeometry}.

Associated to this $k$-linear category is its Hochschild cohomology. This is a canonical construction of a ring from a $k$-linear category given by the bar complex \Lawrence{and in many cases the Hochschild cohomology of the (wrapped) Fukaya category of a symplectic manifold $(V, \omega)$ is isomorphic to the \emph{symplectic cohomology}, written $SH^\circ (V, \omega)$. We won't precisely define the symplectic cohomology here, but will introduce part of the definition later as motivation}. The properties of this construction are not geometrically clear, but following ideas of Seidel in~\cite{FukayaCategoriesAndDeformations} there is a morphism from this to the quantum cohomology of the space. This morphism is conjecturally an isomorphism following promising results of Bourgeois, Ekholm and Eliashberg in ~\cite{EffectOfLegendrianSurgery}, although the technical difficulties of calculating the Hochschild cohomology makes proving this hard. We will follow this conjecture and define the quantum product.

The \Lawrence{(small)} quantum cohomology of a smooth variety $X$ is a deformation of the singular cohomology incorporating the three point functions arising in the physics. Letting $H_i$ be cohomology classes, the product $H_1 * H_2$ is defined by

\[ \langle H_1 * H_2, H_3 \rangle = \sum _{\alpha \in H_2 (X)} I_{0,3} (X, \alpha, H_1, H_2, H_3) z^{\alpha} \]
where $I_{0,3} (X, \alpha, H_1, H_2, H_3)$ counts rational curves in class $\alpha$ with three marked points lying on \Lawrence{classes Poincar\'e dual} to $H_1$, $H_2$ and $H_3$. This produces a product on the cohomology of $X$ with coefficients in $\CC$ deforming the classical product over a formal neighbourhood of the origin in $k[H_2(X)]$. \Lawrence{In fact this product remains associative, and the details of this construction are described in~\cite{MirrorSymmetryAndAlgebraicGeometry} Theorem 8.1.4.}

On the complex geometric side the $D$-branes correspond to elements in the derived category of the target and strings correspond to morphisms between them. Here the $A_{\infty}$ structure suggested by Kontsevich can be seen more explicitly as in~\cite{UniquenessOfDgEnhancementsForTheDerivedCategoryOfAGrothendieckCategory}. The equivalence between these two constructions is made explicit by the following famous conjecture of Kontsevich as described in~\cite{HomologicalAlgebraOfMirrorSymmetry}

\begin{conjecture}

  Let $X$ and $\check{X}$ be a mirror pair. After taking split completions there is an isomorphism

  \[ \mathcal{F}uk (X) \cong \mathcal{D}^b (Coh (\check{X}))\]
  
\end{conjecture}

Split completions preserve the Hochschild cohomology, so there is an induced isomorphism of Hochschild cohomology rings. In general the Hochschild cohomology of a derived category is hard to compute. \Lawrence{Fortunately when working with the category of derived category of coherent sheaves on an affine scheme $X$ we can say much more}.

\begin{lemma}

Let $\Spec A$ be an affine variety, then there is an natural isomorphism of rings $A \cong HH^0 (\mathcal{D}^b \; Coh (\Spec A ))$

\begin{proof}

  The Hochschild cohomology of a category is described in terms of the endomorphism ring of the identity functor, $E$ which is canonically isomorphic to $A$ via its action on $A$. To be precise it is the cohomology of the bar complex applied this functor with multiplication given by cup product. Taking the dual of the bar complex

  \[ 0 \rightarrow E \rightarrow \Mor (E,E) \rightarrow \Mor (E, E \otimes E)  \rightarrow \ldots\]

The first differential is given by $\partial: e_1 \mapsto (e_2 \mapsto e_1 e_2 - e_2 e_1)$. Since $E$ is a commutative ring this differential is zero and we obtain the claimed isomorphism.
  
\end{proof}

\end{lemma}

One prediction of mirror symmetry therefore would be that the two Hochschild cohomologies are isomorphic.

\section{Constructing mirror pairs}

There are many flavours of mirror symmetry constructions, work of Dolgachev on mirror symmetry for K3 surfaces~\cite{MirrorSymmetryForLatticePolarizedK3Surfaces}, of Borcea and Voisin~\cite{BorceaVoisin}, on LG/CY mirror symmetry of Berglund, H\"ubsch and Krawitz for hypersurfaces in weighted projective spaces~\cite{FJRWRingsAndLandauGinzburgMirrorSymmetry}, and this list is far from exhaustive. For our interests we will follow Batyrev and Borisov, who studied toric varieties associated to polytopes. The best reference to learn about toric geometry is~\cite{ToricVarieties}.

Recall that from a lattice $M$ with dual lattice $N$ and a rational polytope $\Delta \subset M \otimes \RR$ one can construct a \emph{dual polytope} inside $N \otimes \RR$. This is constructed as

\[ \Delta ^\vee = \{ n \in N \mid \langle m, n \rangle \geq -1 \quad\forall m \in \Delta \}\]
and the polytope $\Delta$ is called \emph{reflexive} if $\Delta ^\vee$ is a lattice polytope in $N$. From any rational reflexive polytope one can construct a toric variety $\PP_\Delta$ and inside this a canonical family of Calabi-Yau hypersurfaces $CY (\Delta)$. Entries in the Hodge diamond of a generic \Lawrence{hypersurface} can be read off from the combinatorics of the polytopes. Batyrev exploited this in~\cite{DualPolyhedraAndMirrorSymmetryForCalabiYauHypersurfacesInToricVarieties} to prove that one has an equality $h^{1,1} (CY (\Delta)) = h^{2,1} (CY (\Delta ^\vee))$ for \Lawrence{``nice'' families} of anti-canonical hypersurfaces inside \Lawrence{a large class of projective} toric varieties. Together with Borisov he extended this result to complete intersections in Gorenstein Fano toric varieties. This approach is relevant to us in that it is the natural testing ground for the following conjecture of~\cite{MirrorSymmetryIsTDuality}

\begin{conjecture}[The SYZ conjecture]

Let $X, \check{X}$ be a mirror pair of \emph{Calabi-Yau varieties}. Then there is an affine manifold with singularities $B$ and maps $\phi: X \rightarrow B$, $\check{\phi}: \check{X} \rightarrow B$ which are dual special Lagrangian fibrations. This is shown schematically below

\end{conjecture}

Let us use this to explore fully mirror symmetry for the most simple example, an elliptic curve.

\begin{example}

\Lawrence{Let $E \cong T^2$ be an elliptic curve over $\CC$ with projections $\pi_L$ and $\pi_R$. Appropriately identified the projection to the first factor $\pi_L: E \rightarrow S^1$ gives a special Lagrangian fibration. Therefore the mirror manifold should be fibred by the dual circles, i.e., it is the dual elliptic curve $\check{E}$ and has two basic types of Lagrangians, Lagrangian sections and Lagrangian fibres.

  Now the derived category of quasi-coherent shaves on $E$ is generated by the structure sheaf $\mathcal{O}_E$ and skyscraper sheaves, so it is enough to find the Lagrangians mirror to these objects. Conjecturally a line bundle should be mirror to a Lagrangian section of the fibration $\check{E} \rightarrow S^1$. Under this the structure sheaf is dual to a Lagrangian section with winding number zero under the projection $\pi_R$ to $S^1$. The associated $U(1)$ bundle is the trivial bundle. The skyscraper sheaves should be mirror to a special Lagrangian fibre of this fibration, together with a choice of flat $U(1)$ bundle. Again conjecturally the mirror to a skyscraper sheaf is a fibre of the special Lagrangian fibration. There is a natural candidate for the fibre mirror to $k_P$, the fibre over the image of $P$. A choice of flat $U(1)$ bundle on $S^1$ is equivalent to a choice of phase in $\CC^*$. This choice of phase is the value of $P$ under $\pi_R$.

  This correspondence extends to the entirety of both categories by taking exact sequences. To test that this correspondence is correct we consider the homomorphisms between objects, which also should be preserved. Given two distinct points $P$ and $Q$ of $E$ lying on distinct Lagrangian fibres it is immediate that $\Mor (k_P, k_Q)$ is trivial. On the symplectic side the homomorphisms between transverse Lagrangians $L_1$ and $L_2$ are given by elements of the vector space $\CC^{\{L_1 \cap L_2\}}$. In this case the mirrors to $k_P$ and $k_Q$ are disjoint and so there is only the zero homomorphism. Similarly the space $\Mor (\mathcal{O}_E, k_P)$ is one dimensional and the corresponding Lagrangians meet transversely at a single point. Finally if the two Lagrangians are identical one needs
to apply a Hamiltonian isotopy to one of them so that the Lagrangians
become transverse. In this case one can check that the Hom space between
the two Lagrangians is dimension one in degrees $0$ and $1$, which agrees
with the derived Hom spaces between $\mathcal{O}_E$ and $\mathcal{O}_E$ or
between $k_P$ and $k_P$.}
\end{example}

In the case of toric varieties there is a natural choice of fibration given by the action coordinates. One can show that this conjecture is in fact realised for toric varieties under the above construction, \Lawrence{with the fibres degenerating over the boundary of the polytope.} It was this understanding of mirror symmetry for toric varieties that is at the core of the Gross-Siebert program. In its full generality the Gross-Siebert program is believed to be a mirror construction for a pair $(V,D)$ where $V$ is any smooth variety and $D$ is a divisor supporting an ample or anti-ample class. Let us review the philosophy of the program for Fano varieties since this will occupy us later.

\section{The Gross-Siebert program}

According to \Lawrence{Givental's 1994 ICM lecture} mirror symmetry predicts that associated to a toric Fano variety $V$ there should exist a mirror Landau-Ginzburg model. This is a pair $(\check{V}, \check{W})$ where $\check{V}$ is a smooth variety and $\check{W}: \check{V} \rightarrow \CC$ the \emph{super-potential}, a regular function. \Lawrence{This was extended in work of Cho and Oh, who in ~\cite{FloerCohomologyAndDiscInstantonsOfLagrangianTorusFibersInFanoToricManifolds} realised that a choice of effective anti-canonical divisor played a role in determining the Landau-Ginzburg model.}

To construct this we work in two stages, Gross, Hacking and Keel suggest the mirror to the complement $V \setminus D$ should be the underlying scheme $\check{V}$, which under the conditions on $D$ is affine. \Lawrence{The intuition for this is not clear and indeed is false in higher dimensions .} Following our discussion about Hochschild cohomology if we could describe algebraically the symplectic cohomology of $V \setminus D$ we could recover $\check{V}$ as being the spectrum of the ring \Lawrence{$SH^0 (V, \omega)$}. A technical difficulty that we will gloss over is that $V \setminus D$ is not proper, so that one should work with the wrapped Fukaya category. The super-potential is then defined in terms of a canonical basis of the ring \Lawrence{$SH^0(V,\omega)$}, the $\vartheta$-functions. The mirror category to the Fukaya category of $(V,D)$ should be the category of matrix factorisations of $W$. The corresponding Hochschild cohomology ring (in the case that $\check{W}$
has isolated singularities) is given by the Jacobian ring
$Jac(\check{V},\check{W})$ of the singularity.
One interpretation of the statement that these two are \emph{mirror pairs} is the statement that there is an isomorphism of rings

\[ QH (V) \cong Jac (\check{V}, \check{W})\]
but any proof of this requires a stronger understanding of the glueing properties of Gromov Witten invariants. We expect a proof of this to follow from future work of Gross and Siebert but for now we leave this as a conjecture.

So how does one calculate algebraically the symplectic cohomology? Here the SYZ \Lawrence{heuristic enters the picture. If we suppose the existence of an SYZ fibration,}
the image of a stable map under this fibration is called a tropical amoeba. This is a space retracting to a piecewise linear graph, as described in~\cite{AmoebasOfAlgebraicVarietiesAndTropicalGeometry}. As one moves closer and closer to a large complex limit point of a variety $V$ this space collapses onto its skeleton, a tropical curve in the base. Taking this as a stand point we should count all possible tropical curves on the base \Lawrence{of the SYZ fibration} and then following~\cite{TropicalGeometryAndMirrorSymmetry} reconstruct the original invariants using log geometry. Due to the presence of singularities of the SYZ fibration these piecewise linear objects can bend in a highly controlled manner, corresponding to glueing Maslov index zero disks. \Lawrence{Gross, Hacking and Keel encode this symplectic data in the definition of the canonical scattering diagram, and from this algebraic variant of the symplectic cohomology of $X$ they attempt to construct the underlying scheme of the mirror family}. The techniques themselves were introduced by Kontsevich and Soibelman in their paper~\cite{AffineStructuresAndNonArchimedeanAnalyticSpaces} to study $K3$ surfaces. These were applied by Gross in~\cite{TropicalGeometryAndMirrorSymmetry} to study the prototype example of the construction, mirror symmetry for $\PP^2$, and by Gross and Siebert in~\cite{AnInvitationToToricDegenerations} to construct mirror partners to Calabi-Yau threefolds. It was expanded upon in work of both~\cite{TheTropicalVertex} and ~\cite{ATropicalViewOfLandauGinzburgModels} to construct mirrors to Fano varieties.

This just leaves how to find the base $B$ of the SYZ fibration. It should be invariant topologically under small deformations of $X$ and $\check{X}$. This inspired Mark Gross and Bernd Siebert to study toric degenerations of varieties as described in~\cite{AnInvitationToToricDegenerations}. The idea is to deform to a degenerate central fibre, construct the base there and attempt to deform the algebra structure to incorporate the missing singular fibres.

This construction depends on a choice of degeneration to a toric variety. Let us define the objects of study.

\begin{definition}

A \emph{toric degeneration} is a flat family $\mathfrak{X} \rightarrow C$ with
$C$ a non-singular curve with a point $0\in C$ such that the fibre $\mathfrak{X}_0$ is a union of toric varieties glued along toric strata and \'etale locally at zero-dimensional strata of the central fibre, the morphism $\mathfrak{X}
\rightarrow C$ is given by a monomial.
\end{definition}

\noindent These were initially studied by Mumford in~\cite{AnAnalyticConstructionOfDegeneratingAbelianVarietiesOverCompleteRings}. Of the many different degenerations possible the philosophy of Gross and Siebert is that we should study those with maximally unipotent monodromy as described in~\cite{MaximalUnipotentMonodromyForCompleteIntersectionCYManifolds}. 

In fact, instead of working with toric degenerations,
Gross, Hacking and Keel work with pairs $(S,D)$
where $S$ is a smooth Fano surface and $D$ a particular type of anti-canonical divisor on $S$. From this data we construct a base $B$ from the dual intersection complex of $D$. Given this we describe the mirror family via the data of a \emph{scattering diagram} on the base $B$. This encodes data necessary to construct
a smoothing of a union of affine planes. The scattering diagram involves relative Gromov-Witten invariants by assigning to each rational line in $B$ a generating
power series for these invariants. \Lawrence{In the symplectic cohomology the product comes from counts of Maslov index two disks. The tropical analogue of these are \emph{broken lines}, decorated balanced piecewise linear graphs and Gross, Hacking and Keel define the product analogously.}

The super-potential has a simple form in terms of these theta functions: it is a sum of the theta functions attached to each one-cell of $B$. Furthermore the theta functions should induce an embedding of the mirror into affine space. Therefore if we have a generating basis then we obtain a formal family smoothing a central fibre, the union of affine planes. By controlling the combinatorics of the products of these theta functions we will show that this family is in fact algebraic or analytic dependent on the surface.

\section{Log Geometry}

A key technical tool in mirror symmetry is log geometry, which formalises the open sector of the Gromov-Witten theory and systematises the study of toric degenerations. These were first introduced by Kato in~\cite{Kato}, and were studied by Deligne and Illusie in~\cite{LogarithmicSpacesAccordingToKKato}. The foundational properties were explored in~\cite{LecturesOnLogarithmicGeometry} which will form a basis for the entire theory. In~\cite{TropicalGeometryAndMirrorSymmetry} Mark Gross used them to more strongly relate the tropical and algebraic pictures of~\cite{AmoebasOfAlgebraicVarietiesAndTropicalGeometry}. The philosophy
is that log geometry provides a connecting link between algebraic 
and tropical geometry. We will introduce these log schemes.

\begin{definition}

  A \emph{pre-log scheme} $X$ is a pair $(\underline{X}, \alpha_X: \mathcal{M}_X \rightarrow \mathcal{O}_{\underline{X}})$ of a scheme $\underline{X}$ (the \emph{underlying scheme}), a sheaf of monoids on $\underline{X}$ in the \'etale topology $\mathcal{M}_X$ and a homomorphism $\alpha_X$ of sheaves of monoids from $\mathcal{M}_X$ to the multiplicative monoid $\mathcal{O}_{\underline{X}}$. Saying that this is a \emph{log scheme} means that the restriction of $\alpha_X$ to the inverse image of $\mathcal{O}_{\underline{X}}^\times$ is an isomorphism.

  To any pre-log scheme there is an associated log structure given by taking the amalgamated sum

  \[ \mathcal{M}_X \oplus_{\alpha^{-1} \mathcal{O}_{\underline{X}}^{\times}} \mathcal{O}_{\underline{X}}^\times\]

  A morphism of log schemes $\phi : X \rightarrow Y$ is a pair $(\underline{\phi}, \phi^\#)$ with $\underline{\phi}: \underline{X} \rightarrow \underline{Y}$ a morphism of schemes and $\phi^\#$ fitting into a commutative diagram
\[    \begin{tikzcd}
      \underline{\phi} ^{-1} \mathcal{M}_Y \arrow{r}{\phi^\#} \arrow{d} & \mathcal{M}_X \arrow{d}\\
      \underline{\phi}^{-1} \mathcal{O}_{\underline{Y}} \arrow{r}{\phi^*} & \mathcal{O}_{\underline{X}}
    \end{tikzcd} \]

  Given a morphism of schemes $\underline{\phi}: \underline {X} \rightarrow \underline{Y}$ and the structure of a log scheme on $\underline{Y}$ the inverse image sheaf $\underline{\phi}^{-1} (\mathcal{M}_Y)$ on $X$ is naturally a pre-log structure. We call the associated log structure the \emph{pull-back log structure},
written $\phi^*\mathcal{M}_Y$. A morphism of log schemes $\phi: X \rightarrow Y$ is \emph{strict} if the natural
map $\phi^\#:\phi^*{\mathcal{M}_Y} \rightarrow \mathcal{M}_X$ is an isomorphism.

There is an interesting invariant of a log scheme which contains the combinatorial data not seen by classical geometry. The \emph{ghost} or \emph{characteristic sheaf} is the quotient $\mathcal{M}_X/\mathcal{O}_X^\times$.
  
\end{definition}

We introduced log structures to study degenerations of pairs $(X,D)$, so we now explain how to construct a log structure from such a pair.

\begin{example}

  Let $\underline{X}$ be a smooth scheme and $\underline{D}$ be a simple normal crossings divisor. Write $\underline{U}$ for the complement of $\underline{D}$ in $\underline{X}$ and let $j$ be the inclusion of $\underline{U}$ into $\underline{X}$. Let $\mathcal{M}$ be the sheaf $j_* \mathcal{O}_{\underline{U}}^* \cap \mathcal{O}_{\underline{X}}$, and $\alpha_X$ the canonical inclusion. This defines a log scheme $X$, the \emph{divisorial log structure} on $\underline{X}$.

\end{example}

General log schemes exhibit pretty wild behaviour so we restrict those we consider. The objects we want to study are localled modelled on toric varieties together with their toric boundary, which has another intrinsic definition which we give below.
\begin{example}

  Let $P$ be a finitely generated integral monoid. Take $\underline{P}$ to be the constant sheaf on $\Spec k[P]$, which maps to $\mathcal{O}_{\Spec k[P]}$ via $p \mapsto z^{p}$. Let $\LogSpec k[P]$ denote the corresponding log scheme, the \emph{toric log scheme}, whose log structure is referred to as the \emph{toric log structure}. \Lawrence{If $\mathbb{T}$ is any toric variety then each 
toric Zariski open affine subset carries a log structure as just described.
These turn out to be canonically isomorphic on intersections of these
Zariski open subsets, yielding a log structure on $\mathbb{T}$. The corresponding characteristic sheaf is only trivial over the big torus $\mathbb{G}_m^n \subset \mathbb{T}$.}

  Let us describe the ghost sheaf on $\AA^2 \cong \Spec k[\NN^2]$ with its toric log structure. On the big torus the toric monomials are certainly invertible and so the ghost sheaf has trivial stalks at these points. Now along the two toric strata $z^{(1,0)} = 0$ and $z^{(0,1)} = 0$ those same monomials are not invertible. Since an an element of $\mathcal{M}_{\AA^2}$ in a neighbourhood of one 
of these strata takes the form $z^{(a,b)} f$ for $f$ invertible and either
$b=0$ or $a=0$ in these two cases. Hence  each contribute an $\NN$ to the ghost sheaf. Drawing the stalks of the ghost sheaf we obtain the following picture:

  
  
\end{example}

Now the actual objects we want are locally modelled on this construction

\begin{definition}

  A log scheme $X$ is called \emph{fine saturated} (abbreviated as $fs$)
if \'etale locally on $X$ there are strict morphisms to toric log schemes.

  \end{definition}

A surprisingly important example is the \emph{standard log point}, $\Spec k ^\dagger = (\Spec k, \alpha: k^\times \oplus \NN \rightarrow k)$ where $\alpha$ sends $(a, 0)$ to $a$ and $(a, 1)$ to $0$. This is isomorphic to the origin inside $\AA^1$ with its toric log structure.

Now we will only work with fs log schemes and so it is worth a little revision of history: from now on whenever we refer to log schemes we mean fs log schemes. The category of fs log schemes has many fantastic properties

\begin{theorem}[Log schemes have fibre products]

  The category of fs log schemes possesses fibre products. Given a diagram $X \rightarrow Z \leftarrow Y$ there is a canonical morphism $\underline {X \times_Z Y} \rightarrow \underline{X} \times_{\underline{Z}} \underline{Y}$ which is the composition of a finite surjective map followed by a closed immersion. If the map $X \rightarrow Z$ is strict then the above canonical morphism is an isomorphism and the induced map $X \times_Z Y \rightarrow Y$ is strict.
  
  \end{theorem}

\begin{proof}

  This is Corollary 2.1.6 of~\cite{LecturesOnLogarithmicGeometry}.
  
\end{proof}

Let us give an example where the above map is not an isomorphism

\begin{example}

  Let $\AA^2_\dagger$ denote the plane with its toric log structure. There is a toric blowup of this, blowing up the origin which we denote $\pi: Bl \rightarrow \AA^2_\dagger$. Choose two different points of the exceptional curve $P \cong Q \cong (\Spec k, k^\times \oplus \NN)$. We claim that the product $P \times_{\AA^2_\dagger} Q$ is empty, but the product of the underlying schemes is a point.

  To show that the product is empty note that it would be enough to prove that it doesn't have any $k$-points. Now any fs $k$-valued point would have log structure $k^\times \oplus Q$, which by choosing an element we can reduce to having log structure $k^\times \oplus \NN$. This reduces the problem to one of algebra, what is the push-out of the diagram

  

This push-out is zero, so the sections coming from $\NN^2$ map to zero. But then the maps to $k$ do not commute and there is no such point.
  
\end{example}

The geometry of log schemes is highly related to the combinatorial data of the log structure and hence to the combinatorics of toric varieties. \Lawrence{One way to organise the data of toric varieties and their morphisms} is through fans and refinements of fans. Suppose we have lattices $M = \ZZ^n$ and $N = M^\vee$ contained inside vector spaces $M_\RR = M \otimes_{\ZZ} \RR$ and $N_\RR = M_\RR ^\vee$. A cone $\sigma$ inside $N$ is a subset consisting of positive linear combinations of a finite collection of rational vectors $\phi_1, \ldots, \phi_n$. Such a cone is convex and rational polyhedral and has an associated dual cone given by
\[\sigma^\vee = \{ v \in M_\RR \mid \phi (v) \geq 0 \forall \phi \in N_\RR \}\]
Associated to $\sigma^\vee$ is the monoid $\sigma^\vee \cap M$ and hence a monoid ring $k[\sigma^\vee \cap M]$. After passing to $\Spec$ we obtain a scheme and thus a covariant functor from cones and inclusions of cones to toric varieties and morphisms of toric varieties. This is not an equivalence as the image consists only of affine toric varieties. To include the glueing conditions we introduce the notion of a fan:

\begin{definition}

  A \emph{fan} $\Sigma$ inside $N$ is a simplicial complex of cones. It is \emph{complete} if the collection covers all of $N$. A morphism of fans $f: \Sigma_1 \rightarrow \Sigma_2$ is a linear map on underlying vector spaces mapping each cone of $\Sigma_1$ into a cone of $\Sigma_2$.
  
\end{definition}

According to~\cite{ToricVarieties} this produces an equivalence of categories between the category of toric varieties and toric morphisms and the category of fans and their morphisms.

Log geometry was introduced to make sense of degeneration and moduli problems, which are two sides of the same coin. Classically the local deformations of a scheme are given by the tangent bundle, or perhaps normal bundles constructed from this. To a morphism of log schemes $f: X \rightarrow Y$ there is an induced sheaf of log differentials whose sections correspond to infinitesimal deformations. I learnt the definition from~\cite{LogBettiCohomologyLogEtaleCohomologyAndLogDeRhamCohomologyOfLogSchemesOverC} and we repeat this here. This plays the role of the usual tangent sheaf, for instance a morphism is log smooth just when this sheaf is locally free.

\begin{definition}

  Let $f: X \rightarrow Y$ be a morphism of log schemes. A log derivation is a pair of maps $D, Dlog$ from $\mathcal{O}_{\underline{X}}$ or $\mathcal{M}_X$ respectively to a sheaf $\mathcal{E}$ satisfying:

  \begin{enumerate}
  \item $D$ is a derivation in the classical sense.
  \item $Dlog$ vanishes on sections of $f^{-1} (\mathcal{\mathcal{M}}_Y)$.
  \item For every element $m \in \mathcal{M}_X (U)$ one has an equality $D (\alpha_X(m)) = \alpha_X (m) Dlog(m)$.
  \end{enumerate}

There is a universal sheaf classifying these, the logarithmic cotangent bundle $\Omega^{1, \dagger}_{X/Y}$. There is a natural pullback map, given $f:X \rightarrow Y$ and $g: Y \rightarrow Z$ there is a canonical morphism $f^* \Omega^{1, \dagger}_{Y/Z} \rightarrow \Omega^{1,\dagger}_{X/Z}$. We write $\omega^\dagger_{X/Y}$ for the top tensor power in the case where $\Omega^{1, \dagger}_{X/Y}$ is a vector bundle.
  
\end{definition}

We will need more results on log geometry in our second project and we will introduce more results as we need them.

\section{Outline of this thesis}

In the first project we study the mirror construction of~\cite{MirrorSymmetryforLogCY1} for low degree del Pezzo surfaces. These were studied symplectically in~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves} and predictions made for the structure of the mirror family. We construct an algebraic family smoothing the central fibre using the combinatorics of the scattering diagram. We will see that these agree with the predictions of~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves}, and obtain descriptions of the mirror family. This is novel for the case of del Pezzo surfaces of degree two.

In the second project we again study this construction, this time using the intuition from the first project to prove that the formal smoothing to the mirror family of a rational elliptic surface lifts to an analytic smoothing. To do this we identify an infinite set of coefficients which appear in the relations between the theta functions. Whereas in the previous cases we could explicitly calculate the finite set of relevant coefficients here we attempt to bound those coefficients. To do this we construct a larger target space, an \emph{unravelled threefold}, which lifts the desired classes. This space occurs as a nef complete intersection inside a toric variety so we can apply the techniques of Givental~\cite{AMirrorTheoremforToricCompleteIntersection} and calculate the invariants on the threefold. Careful analysis relates these invariants to the original desired invariants, producing the convergence result.

\section{Conventions}

Throughout this paper we will work over $\CC$, although it is clear that the first project makes sense over arbitrary base fields. Now let us collect some terminology here, so that one can check here rather than searching through the bulk of the text.

Recall the notion of a virtual fundamental class from~\cite{TheIntrinsicNormalCone}. In full generality given a morphism $f: X \rightarrow Y$ this associates to an embedding $C \hookrightarrow E$ of a cone stack into an abelian cone stack $E$ over $X$ (an \emph{obstruction theory}) a class in the bivariant Chow theory of $f: X \rightarrow Y$ of dimension $rk(E)$. We supress the choice of embedding and write $[E, C]$ for the associated class. Given a variety $V$ the moduli space of stable maps of genus $g$ and class $\beta$ and with $n$ marked points sits over the moduli stack $\mathfrak{M}_{g,n}$. If $V$ is smooth then there is a canonical obstruction theory with target $h^0 / h^1 (R\pi_* f^* \Theta_{X/Spec \: k})$. We write $I_{g,n,\beta} (\ldots)$ for the associated Gromov-Witten invariant. If there is a divisorial log structure on $V$, $D = D_1 \cup D_2 \ldots D_n$ we will write $I_{0, 1, \beta}^\dagger (D_i)$ for log Gromov Witten invariant tangent to full order to the component $D_i$ not meeting any other component of $D$.

Often we will be interested in connected components of a given stack of stable maps. To describe this situation we introduce the following terminology

\begin{definition}

Let $f: X \rightarrow Y$ be a morphism of stacks. We say that it is an \emph{inclusion of components} if it is an isomorphism between $X$ and the union of a set of connected components of $Y$.

\end{definition}

In the second project we will study rational elliptic surfaces. Here $\rho: S \rightarrow \PP^1$ is a rational elliptic surface and $[F]$ is the class of a fibre. We will write $F_0$ for a fixed rational fibre, of type $I_4$, over a point $0 \in \PP^1$. This says that $F_0$ is a cycle of four rational curves. We write $S^\dagger$ for the log structure on $S$ given by $F_0$.

We will study the moduli stack of genus zero log stable maps to $S^\dagger$ in a class $\beta$ in our second project. We write $\LogModuli(S^\dagger, \beta)$ for this moduli space, requiring that the curve meets $F_0$ at a single point to maximal tangency order. This space carries a universal family which we denote by $\mathcal{C}^\dagger$, it comes with two maps, $\pi: \mathcal{C}^\dagger \rightarrow \mathcal{M}^\dagger(S^\dagger, \beta)$ and $f: \mathcal{C}^\dagger \rightarrow S^\dagger$. This stack admits a map down to the moduli space of stable maps with basic log structures constructed in~\cite{LogarithmicGromovWittenInvariants}, we denote this space $\mathfrak{M}_{0,0,S^\dagger}$.

\chapter{Mirror families to low degree del Pezzo surfaces}

Mathematicians have over many centuries come to understand the geometry of algebraic surfaces in great detail. Of the many classification results known we recall the following

\begin{theorem}[Classification of del Pezzo surfaces]

Let $S$ be a smooth rational Fano surface, that is to say a smooth projective surface $S$ with $\omega_S ^\vee$ ample. Then $S$ is isomorphic to one of the following surfaces

\begin{itemize}
\item $\PP ^1 \times \PP^1$.
\item The blow-up of $\PP^2$ in zero through eight points in general position.
\end{itemize}

In the second case the surface is described by the degree of the anti-canonical class, also called the \emph{degree of $S$} and we write $dP_d$ for a generic degree $d$ surface. The respective Chow groups $A_1 (S)$ are generated by
\begin{itemize}
\item The pullback of a point under the two projection maps.
\item A hyperplane class $H$ and the exceptional curves $E_1, \ldots , E_{9-d}$.
\end{itemize}
whilst the groups $A_0(S)$ and $A_2 (S)$ are isomorphic to $\ZZ$ generated by a point and a fundamental class respectively.

\begin{proof}

  See ~\cite{ElementaryTheoryOfDelPezzoSurfaces}.
  
\end{proof}

\end{theorem}

Of these the surfaces $\PP^1 \times \PP^1$, $\PP^2, dP_8, dP_7$ and $dP_6$ are all toric and mirror families can be constructed using techniques we will introduce to motivate the Gross-Siebert construction. The next three surfaces were studied in~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves}. To build up our intuition with the Gross-Siebert program we will show that these agree with the predictions there. We will then handle one of the remaining two cases, the blow up in seven points. Explicit formulae for these families are new. Precisely the same techniques can be applied in the case of $dP_1$ but given the frankly atrocious nature of the equations for $dP_2$ we do not pursue this. First we introduce a method to systematically study toric degenerations.

\section{Looijenga Pairs}

To study moduli problems in geometry one often needs a method to rigidify the problem and remove automorphisms. In studying the moduli space of Fano varieties this is most naturally done by considering a pair $(X,D)$ of a variety $X$ and an anti-canonical divisor $D$. This produces a moduli space with a boundary component consisting of degenerate divisors $D$. One locus within this boundary is particularly well suited to our needs. A \emph{Looijenga pair} is a pair $(X,D)$ where $X$ is a smooth rational surface and $D$ is an anti-canonical divisor consisting of a cycle of $-1$-rational curves.

Such a pair can be viewed in many different ways. The most enlightening for us is that these are log Calab\Lawrence{i}-Yau varieties, log varieties with trivial log canonical sheaf. This is calculation is a specific case of the following general calculation

\begin{lemma}

Let $(X,D)$ be a pair with $X$ smooth and $D$ having simple normal crossings. Write $X^\dagger$ for the divisorial log structure on $X$. Then there is an isomorphism $\omega_{X^\dagger}^\dagger \cong \omega_{X} \otimes \mathcal{O}(D)$.
  
\end{lemma}

When $X$ is a toric surface and $D$ the union of toric strata this structure allows us to recover the fan for $X$, at least up to a choice of embedding into $\RR^2$. This is done using the \emph{dual intersection complex} $\Delta_{(S,D)}$ 
of $D$, an abstract fan, which we now construct.

\begin{construction}

Let $D = D_1 \cup D_2 \cup \ldots \cup D_n$ be a cycle of rational curves on a smooth surface $S$ such that $D_i \cap D_j$ is a single point just when $i$ and $j$ differ by 1 mod $n$ and otherwise is empty (in the case $n=2$ we relax this to saying that there are two points in the unique intersection $D_1 \cap D_2$). Then $\Delta_{(S,D)}$ contains precisely one zero-dimensional cell 
$\{0\}$ corresponding to the interior $S \setminus D$. For each component $D_i$, $\Delta_{(S,D)}$
contains a one-dimensional cone with $v_i$ its primitive generator.
Attach the zero-dimensional cone as 0 inside each of these rays. Now introduce a two-dimensional cone in $\Delta_{(S,D)}$ for each intersection point of $D_i \cap D_j$, spanned by $v_i$ and $v_j$. This produces the \emph{dual intersection complex} $\Delta_{(S,D)}$ as a cone complex but it carries more structure. 
We write $B$ to be the underlying topological space of the cone complex
$\Delta_{(S,D)}$. It is homeomorphic to $\RR^2$. We give $B\setminus\{0\}$
an affine manifold structure by defining an affine coordinate chart by embedding
the union of the cones $\RR^{\geq 0} v_{i+1} \oplus \RR^{\geq 0} v_i $ and $\RR^{\geq 0} v_{i} \oplus \RR^{\geq 0} v_{i-1}$ into $\RR^2$ via the relations $v_{i-1} \mapsto (1,0)$, $v_i \mapsto (0,1)$ and $v_{i+1} \mapsto (-1,-D_i ^2)$. This expresses $\Delta_{(S,D)}$ not just as a complex of sets but an affine manifold with singularities (indeed a single singularity at the origin).

\end{construction}

An integral affine manifold $M$ carries a sheaf of integral vector
fields, which we write $\Lambda_M$. On a chart this is isomorphic to the 
constant sheaf with coefficients $\ZZ^n$. 
The transition functions naturally give identifications of these constant
sheaves. In addition, we define $M(\ZZ)$ to be the set of points of $M$
with integer coordinates in some, hence all, integral affine coordinate
chart.

This construction is part of a general philosophy: the base of the SYZ fibration should be an affine manifold with singularities, a topological manifold with charts such that the transition functions lie in the affine transformation group $\RR^n \ltimes Sl (\ZZ^n)$. Of course the fan for a toric surface is a subdivision of $\RR^2$ whereas the dual intersection complex is just an abstract fan. The following lemma makes sense of the situation.

\begin{lemma}[Toric reconstruction]

Let $S$ be a \Lawrence{compact} toric surface and $D$ the toric boundary. Then $(S,D)$ is a Looijenga pair, $\Delta_{(S,D)}$ embeds into $\RR^2$ and any choice of embedding gives the fan for a toric variety isomorphic to $S$.

\begin{proof}

  See ~\cite{ToricVarieties}.
  
\end{proof}

\end{lemma}

Given a choice of ample divisor $H$ on $S$ toric, the fan $\Delta_{(S,D)}$
carries a natural choice of piecewise linear function up to a choice of global linear function and we denote this $\phi$. Such a function is defined 
by how it changes where it is non-linear. If $\phi_{v_i,v_{i+1}}$ denotes
the linear extension of $\phi$ restricted to $\RR^{\ge 0}v_i+\RR^{\ge 0}v_{i+1}$, then we determine $\phi$ by insisting that
$\phi_{v_i,v_{i+1}}-\phi_{v_{i-1},v_i}=(H.D_i)n_i$, where $n_i$ is a primitive element of the dual space vanishing on $v_i$ and positive on $v_i \RR^{\ge 0} + v_{i+1} \RR^{\ge 0}$.

Now we ask how we can reconstruct the mirror from this data? This is surprisingly easy, for each point $P$ of the integral structure introduce a variable $\vartheta _P$ together with the multiplication rule that for points $P$ and $Q$ the product $\vartheta_P \vartheta_Q$ is $z^{\phi (P+Q) - \phi (P) - \phi (Q)}\vartheta_{P+Q}$\label{ToricThetaProduct}. This recreates the counts of holomorphic triangles as suggested by Auroux; in this toric case there are no additional singularities to account for. For non-zero choices of coefficients $z^{\phi(P)}$ the \Lawrence{spectrum} $\check{W}$ of the ring spanned by the $\vartheta$ is then an algebraic torus and the mirror should be the pair $(\check{W}, \sum \vartheta_{v_i})$. Let us perform this in the case of $\PP^2$.

\begin{example}

  Our first goal is to find the fan. Fortunately we learnt a fan for $\PP^2$ back in infancy: it has one cells generated by $(1,0), (0,1)$ and $(-1,-1)$.

  
  
  Calculating the product defined above, using the ample divisor
$H$ being the class of a line in $\PP^2$, one finds $\vartheta_{(1,0)} \vartheta_{(0,1)} = z \vartheta_{(-1,-1)}$. As in the basis constructions of~\cite{TheGreedyBasisEqualsTheThetaBasis} these functions morally ought to produce an embedding of the mirror into affine space and their sum should be the super-potential. Doing this we obtain the mirror as being $\mathbb{G}_m^2$ with superpotential $x + y + z / xy$. As expected this is the mirror predicted by other constructions. Calculating the Jacobian ring of the singularity we find that it is $\ZZ [X, z] / \langle X^3 - z \rangle$, isomorphic to the quantum cohomology of $\PP^2$.
  
\end{example}

\Lawrence{There were several steps involved in the above construction which we need to generalise. In order we
\begin{enumerate}
\item Constructed a base $B$.
\item Found a canonical piecewise linear function $\phi$ on $B$.
\item Introduced functions coming from integral points of $B$ and a multiplication rule on these functions.
\item Used these functions to embed the mirror into affine space. \label{generalisationtable}
\end{enumerate}

We will now set about generalising these in order to the case of a Looijenga pair $(S,D)$.} We will want to keep track of the data \Lawrence{of the pair (S,D)} we have chosen later, so let us record here a choice of boundary divisor and affine manifold for all the non-toric del Pezzo surfaces. Here we record the boundary divisor in $\PP^2$ before blowing up.
The circles represent points to be blown up, with the exceptional divisor included in the boundary if the circle is red. \Lawrence{Since we can't embed
the dual complex as an affine manifold in $\RR^2$, we instead provide a list
of cones in $\RR^2$ and an integral affine isomorphism between two of these
cones. The dual intersection complex is then obtained by identifying these
two cones using this isomorphism.
In the pictures below the cone shaded grey is identified integral linearly with the standard first quadrant}, whilst the dotted region is removed entirely.

\begin{center}
\begin{tabular}{c|ccc}
  Surface & Boundary & Class of boundary & Dual complex \\
  $dP_5$ & \begin{tikzpicture} \draw [red] (0,0) circle (1pt); \draw [red] (1,0) circle (1pt); \draw (0,1) circle (1pt); \draw (0.2,0.2) circle (1pt); \draw [red] (-0.5,0) -- (1.5,0); \draw [red] (0,-0.5)--(0,1.5); \draw [red] (1.5, -0.125) -- (-0.5, 0.375); \end{tikzpicture} & \makecell{$\cup D_i = E_1 \cup (H - E_1 - E_2)$ \\ $ \cup E_2 \cup (H-E_2 - E_3) \cup (H-E_1 - E_4)$}  & \begin{tikzpicture} \fill[lightgray] (0,0) -- (1,0) -- (1,-1); \draw (0,0) -- (1,0); \draw (0,0)--(0,1); \draw (0,0) -- (-1,1); \draw (0,0) -- (-1,0); \draw (0,0) -- (0,-1); \draw[dashed] (0,0) -- (1,-1); \end{tikzpicture} \\
  $dP_4$ & \begin{tikzpicture} \draw [red] (0,0) circle (1pt); \draw (1,0) circle (1pt); \draw (0,1) circle (1pt); \draw (0.5,0.2) circle (1pt); \draw (0.2,0.5) circle (1pt); \draw [red] (-0.5,0) -- (1.5,0); \draw [red] (0,-0.5)--(0,1.5); \draw [red] (-0.5,1.2) -- (1.2,-0.5); \end{tikzpicture} & \makecell{$\cup D_i = E_1 \cup (H - E_1 - E_2) \cup$ \\ $(H- E_3 - E_4) \cup (H - E_1 - E_5)$} & \begin{tikzpicture} \fill[lightgray] (0,0) -- (0,-1) -- (1,-1); \draw (0,0) -- (1,0); \draw (0,0)--(0,1); \draw (0,0) -- (-1,1); \draw (0,0) -- (-1,0); \draw[dashed] (0,0) -- (0,-1); \fill[pattern = dots] (0,0) -- (1,0) -- (1,-1); \end{tikzpicture} \\
  $dP_3$ & \begin{tikzpicture} \draw (0.8,0) circle (1pt); \draw (-0.8,0) circle (1pt); \draw (0.3,0.866) circle (1pt); \draw (0.5,-0.416) circle (1pt); \draw (-0.5,-0.416) circle (1pt); \draw (-0.3,0.866) circle (1pt); \draw [red] (-1.2, -0.416) -- (1.2,-0.416); \draw [red] (-1.1,-0.520) -- (0.066, 1.5); \draw [red] (1.1,-0.52) -- (-0.066, 1.5); \end{tikzpicture} & \makecell{$\cup D_i = (H - E_1 - E_2) \cup$ \\ $(H- E_3 - E_4) \cup (H - E_5 - E_6)$} & \begin{tikzpicture} \fill[lightgray] (0,0) -- (0,-1) -- (-1,-1) -- (-1,0); \draw (0,0) -- (1,0); \draw (0,0)--(0,1); \draw (0,0) -- (-1,1); \draw[dashed] (0,0) -- (-1,0); \fill[pattern = dots] (0,0) -- (1,0) -- (1,-1) -- (0,-1); \end{tikzpicture} \\
  $dP_2$ & \begin{tikzpicture} \draw [red] (0.6,1) .. controls (-1.5, 0) .. (0.6, -1); \draw [red] (0,1)--(0,-1); \draw (-0.975, 0) circle (1pt); \draw (-0.6, 0.4) circle (1pt); \draw (-0.6, -0.4) circle (1pt); \draw (0, 0.3) circle (1pt); \draw (0, -0.3) circle (1pt); \draw (0.38, 0.9) circle (1pt); \draw (0.38, -0.9) circle (1pt); \end{tikzpicture} & \makecell {$\cup D_i = (H - E _1 - E_2) \cup$ \\ $(2H - E_3 - E_4 - E_5 - E_6 - E_7)$} & \begin{tikzpicture} \fill[lightgray] (0,0) -- (-1,0) -- (-1,1); \draw (0,0) -- (1,0); \draw (0,0)--(0,1); \draw [dashed] (0,0) -- (-1,1); \fill[pattern = dots] (-1,0) -- (1,0) -- (1,-1) -- (-1,-1); \end{tikzpicture} \\

\end{tabular}
\end{center}

\subsection{The Mumford degeneration}

Here we attempt to generalise bullet points 1 and 2 of Table~\ref{generalisationtable}. The construction of the mirror family was inspired by the Mumford degeneration of an Abelian variety to the union of toric varieties. Let us recall the construction of~\cite{AnAnalyticConstructionOfDegeneratingAbelianVarietiesOverCompleteRings}

\begin{example}

  Let $B \subset N_\RR$ be a lattice polyhedron, $\mathcal{P}$ a lattice polyhedral decomposition of $B$ and $\phi: B \rightarrow \RR$ a strictly convex piecewise linear integral function. One takes the graph over $\phi$ to produce a new polyhedron

\[\Gamma (B, \phi) := \{ (n, r) \in N_\RR \oplus \RR \mid n \in B, r \geq \phi (n)\}\]

This produces a lattice polyhedron unbounded in the positive direction on $\RR$. To construct a family from this we perform a cone construction, let $C (\Gamma)$ be the closure of the cone over $\Gamma (B, \phi)$:
\[
C(\gamma) = \overline{\{(n,r_1,r_2)\in N_{\RR}\oplus\RR\oplus\RR\,|\,
(n,r_1)\in r_2 \Gamma(B,\phi)\}}.
\]
 This carries an action of $\RR^+$ given by translating the second component of $\Gamma(B, \phi)$. The integral points of $C (\Gamma)$ form a graded monoid and we can take $Proj$ of this to produce a variety projective over $\AA^1$. We write this $\PP_{\Gamma (B,\phi)}$. The general fibre is a toric variety, 
whilst the fibre over the origin is a union of toric varieties.

\end{example}

The construction of~\cite{MirrorSymmetryforLogCY1} mimics this in reverse: it claims that the central fibre should be the $n$-vertex, which for $\Lawrence{n>2}$ is a cycle of $n$ copies of $\AA^2$ glued adjacently along their axes, and then attempts to smooth. We cannot hope to have global coordinates as in the toric Mumford degeneration, only local coordinates. To incorporate this data the authors of~\cite{MirrorSymmetryforLogCY1} define a twist of the tangent sheaf which has enough global sections, \Lawrence{in particular the function $\phi$ lifts to a section of this bundle}.

\begin{definition}
\label{def:sheaf-of-monoids}
  Let $(S,D)$ be a Looijenga pair with associated dual intersection complex $\Delta_{(S,D)}$ and let $\eta: NE (S) \rightarrow M$ be a homomorphism of monoids. We want to construct a multi-valued piecewise linear function on $\Delta_{(S,D)}$ which will bend only along the one-cells. This will be a collection of
piecewise linear functions defined on open subsets of $B$ which differ by linear
functions on overlaps. Such functions are determined by their bends at
one-cells, which are encoded as follows. For a
one-cell $\tau=\RR^{\geq 0} v_i$, choose an orientation $\sigma_+$ and 
$\sigma_-$ of the two two-cells separated by $\tau$ and let $n_\tau$ be the unique primitive linear function positive on $\sigma_+$ and annihilating $\tau$. We want
to construct a representative $\phi_i$ for $\phi$ on $\sigma_+\cup \sigma_-$.  Writing $\phi_+$ and $\phi_-$ for the linear function defined by $\phi_i$ on $\sigma_+$ and $\sigma_-$, the function $\phi_i$ is then defined up to a linear
function by the requirement that
\[ \phi_+ - \phi _- = n_\tau \otimes \eta ([D_i])\]
Such a function is convex in the sense of~\cite{MirrorSymmetryforLogCY1} Definition 1.11, and one says that it is \emph{strictly convex} if $\eta([D_i])$ is not invertible for any $i$. 

This function determines a $M^{gp}\otimes\RR$-torsor $\PP$ as defined 
in~\cite{GrossSiebert}, Construction 1.14, on $B\setminus \{0\}$
which is trivial on each $(\sigma_+\cup\sigma_-)\setminus\{0\}$, i.e.,
is given by $((\sigma_+\cup\sigma_-)\setminus \{0\})\times (M^{gp}\otimes\RR)$.
These trivial torsors are glued on the overlap of two adjacent such sets,
namely on $\RR^{\ge 0} v_{i} \oplus\RR^{\ge 0}v_{i+1}$, via the map
\[ (x,p) \mapsto (x, p + \phi_{i+1} (x) - \phi_i (x))\]
which induce isomorphisms on the monoid of points lying above $\phi$.
This construction is designed to allow us to run the Mumford construction locally, even if we cannot run it globally. 
Write
$\pi: \PP \rightarrow B\setminus\{0\}$, and we write $\mathcal{M}$ for the 
sheaf $\pi_* (\Lambda _{\PP})$, bearing in mind that $\PP$ also has the
structure of an affine manifold. Then there is a canonical exact sequence
  \[ 0 \rightarrow \underline{M^{gp}} \rightarrow \mathcal{M} \xrightarrow{r} \Lambda_{B} \rightarrow 0 \]
  We write $r$ for the second map in this sequence. This will not be mentioned again until we define the canonical scattering diagram so keep this in mind until then.
Furthermore, \cite{GrossHackingSiebert}, Definition 1.16 gives a subsheaf
of monoids $\mathcal{M}^+\subset \mathcal{M}$.
 
\end{definition}

If one performs this construction in the case of a toric variety with the \Lawrence{function} $\phi$ and pairs it with an ample class one obtains the height function we used in the Mumford degeneration. After introducing this Gross, Hacking and Keel continue to explain how such a $\phi$ gives rise to a canonical deformation of the $n$-vertex. This concerns their proof that the construction smooths the $n$-vertex and we wait until later on to introduce this.

\section{Scattering diagrams}

\Lawrence{The next step to move from the toric world to the non-toric world is bullet point 3 of Table~\ref{generalisationtable}} is to modify the construction of the $\vartheta$-functions. To do so in a systematic way we must introduce scattering diagrams.

In Mumford's degeneration of an Abelian variety above all the fibres of the SYZ torus fibration on a generic member of the family are smooth. This can not be the case for a special Lagrangian fibration of a general variety as there must be some singular fibres and around these singular fibres there will be some monodromy action. \Lawrence{This monodromy is an obstruction to a naive definition
of theta functions being well-defined on the mirror. Thus one needs a
corrected notion of theta functions, which from symplectic geometry can be done by counting so-called Maslov index two disks as described in~\cite{MirrorSymmetryAndTDualityInTheComplementOfAnAnticanonicalDivisor}. These can
be generated by gluing on Maslov index zero disks onto  standard 
Maslov index two disks. Rather than trying to make this symplectic
heuristic precise, we instead are motivated by this picture as follows.
If we stand well away from the singularities and tropicalise these Maslov
index zero disks they appear to be a collection of lines passing out from the origin together with the information of their class. This data is the inspiration for the definition below of a scattering diagram on the base $B$.}

\begin{definition}

  Let $B$ be an affine manifold with a single singularity, with 
$B$ homeomorphic to $\RR^2$ with singularity at the origin, so that $B^*:=
B\setminus \{0\}$ is an affine manifold.
Let $\mathcal{M}$ be a locally constant sheaf of abelian groups
on $B^*$ with a subsheaf of monoids $\mathcal{M}^+\subset\mathcal{M}$
and equipped with a map $r:\mathcal{M} \rightarrow \Lambda_B$.
Let $\mathcal{J}$ be a sheaf of ideals in $\mathcal{M}^+$ with stalk $\mathcal{J}_x$ maximal in $\mathcal{M}^+_x$ \Lawrence{for all $x \in B^*$}.
Let
$\mathcal{R}$ denote the sheaf of rings locally given by the completion of 
$k[\mathcal{M}^+]$ at $\mathcal{J}$. 
 A \emph{scattering diagram with values in the pair $(\mathcal{M}, \mathcal{J})$ on $B$} is a function $f$ which assigns to each rational ray from the origin an section of the restriction of $\mathcal{R}$ to the ray.
We require the following properties of this function:

  \begin{itemize}
    \item For each $\mathfrak{d}$ one has $f(\mathfrak{d}) = 1$ mod 
$\mathcal{J}|_{\mathfrak{d}}$.
    \item For each $n$ there are only finitely many $\mathfrak{d}$ for which $f(\mathfrak{d})$ is not congruent to $1$ mod $\mathcal{J}|_{\mathfrak{d}}^n$. These $\mathfrak{d}$ are called \emph{walls}.
    \item \Lawrence{For each ray $\mathfrak{d}$ and for each monomial $z^{p}$ appearing in $f(\mathfrak{d})$ one has $r(p)$ tangent to $\mathfrak{d}$. A line for which $r(p)$ is a positive generator of $\mathfrak{d}$ for all $p$ with $c_p\not=0$
 is called an \emph{incoming ray}. If instead
$r(p)$ is a negative generator of $\mathfrak{d}$ for all $p$ with $c_p\not=0$,
it is called an \emph{outgoing ray}.}
  \end{itemize}
  
  We denote such an object by the tuple $(B, f, \mathcal{M}, \mathcal{J})$. We say that $(B,f,\mathcal{M},\mathcal{J})$ is obtained from $(B,f',\mathcal{M},\mathcal{J})$ by \emph{adding outgoing rays} if for each ray $\mathfrak{d}$ one can write $f(\mathfrak{d}) = f' (\mathfrak{d}) (1 + \sum (c_p z^p))$ where for each monoid element $p$ with $c_p\not=0$ the vector $-r(p)$ is a generator of $\mathfrak{d}$.

\end{definition}

In our case the sheaf of monoids $\mathcal{M}^+$ will be as given in
Definition \ref{def:sheaf-of-monoids}, with the monoid $M$ being
 a finitely generated sharp submonoid of $H_2(S,\ZZ)$
containing $NE(S)$, the monoid generated by effective curves on $S$. Being sharp means the only invertible element of $M$ is the identity element, and so the maximal ideal is just the complement of the identity. We introduce the choice of scattering diagram, the \Lawrence{\emph{canonical scattering diagram},} and then motivate and define each term appearing. A ray $\mathfrak{d} = (a v_i + b v_{i+1}) \RR^{\geq 0}$ specifies a blow up of $S$ given by refining the fan until $\mathfrak{d}$ is a one-cell and all the two-cells are integrally isomorphic to $(\RR^{\geq 0})^2$. The ray $\mathfrak{d}$ then corresponds to a component $C$ in the inverse image of $D$. To this ray then we assign the power series
\[f(\mathfrak{d}) := \exp \left[ \sum_\beta k_{\beta} N_\beta z^{\eta (\pi_* (\beta)) - k_\beta m_{\mathfrak{d}}'}\right],\]
where $m_{\mathfrak{d}}'$ is the unique lift of a primitive outward
pointing tangent vector $m_{\mathfrak{d}}$ to $\mathfrak{d}$ not contained
in the relative interior of $\mathcal{M}|_{\mathfrak{d}}$.
The number $N_{\beta}$ counts the number of relative curves mapping to $(S,D)$ tangent to $C$ to maximal order $k_\beta$ at a single point as outlined in~\cite{MirrorSymmetryforLogCY1} at the beginning of section 3.
 By the Riemann-Roch formula this is of dimension
\[ dim \: S - 3 - K_S. \beta - k_\beta + 1 = 0.\]
By~\cite{GromovWittenInvariantsInAlgebraicGeometry} this produces a Gromov-Witten invariant. This construction may also be run using logarithmic Gromov-Witten invariants rather than these blow ups, see~\cite{LogarithmicGromovWittenInvariants} for the definition, and this will be explored in future work of Gross and Siebert.

Heuristically this encodes the glueing data for a generating set of the open Gromov-Witten invariants. On the symplectic side the instanton corrections appear as pseudo-holomorphic disks of Maslov index zero. Given a Maslov index two disk, one can attach Maslov index zero disks to produce new index two disks and it is the combinatorics of this glueing that we encode using these diagrams.

\subsection{Broken lines}

To encode the Maslov index two disks themselves we collapse them onto a skeleton in the base of the SYZ fibration, having pushed all the singularities to the origin. Then following~\cite{AmoebasOfAlgebraicVarietiesAndTropicalGeometry} the disks appear as balanced tropical curves together with a mark of the class that they lie in. The skeleton of these disks are trivalent graphs with one leg ending at the origin and the other two legs passing to infinity. The definition below defines how one of these legs behave, the full picture only arising once we have discussed pairs of pants and even then we supress the third leg to the origin.

\begin{definition}

  A \emph{broken line from $v\in B(\ZZ)$ to $P \in B$} on a scattering diagram $(B, f, R, J)$ is a choice of piecewise linear function $l: \RR^{\leq 0} \rightarrow B$ and a map $m: \RR^{\leq 0} \rightarrow \prod_{t\in\RR^{\leq 0}}
\mathcal{R}_{l(t)}$ such that the following hold:

  \begin{itemize}
  \item $l$ has only finitely many points where it is non-linear and these only occur where $l$ maps into a rational ray of $B$.
  \item $l (0) = P$.
  \item $l(t)$ lies in the same cone as $v$ and is parallel to $v$ for all $t$ sufficiently negative.
  \item $l$ does not map an interval to a ray through the origin.
    \item $m(t)\in \mathcal{R}_{l(t)}$ for all $t$ and is a monomial in
this ring, written as $c_t z^{m_t}$. Further, on each domain of linearity
of $l$, $m(t)$ is given by a section of $\mathcal{R}$ pulled back to 
this domain of linearity.
  \item For $t$ very negative, $m_t\in \mathcal{M}^+_{l(t)}$ is the
unique element not lying in the interior of this monoid satisfying
$r(m_t) = v$.
  \item $r(m(t_0)) = -{ \partial l \over \partial t}\big|_{t=t_0}$ wherever $l$ is linear.
  \item $m$ only changes at those points where $l$ is non-linear.
  \item Let $t \in \RR_{\leq 0}$ be a point where $l$ is non-linear. We write $\partial(l_+)$, $\partial(l_-)$, $m_+$ and $m_-$ for the values of $\partial l \Lawrence{/} \partial t$ and $m$ on either side of $t$. Suppose that $l(t)$ lies on a ray $\mathfrak{d}$ with primitive normal vector $n_{\mathfrak{d}}$, negative on $\partial (l_-)$. Then $m_+$ is a monomial term of $m_- f_{\mathfrak{d}}^{\langle n_{\mathfrak{d}}, \partial (l_-)\rangle}$.
\end{itemize}

\end{definition}

There is more technical content in~\cite{MirrorSymmetryforLogCY1} exploring how to deform the Mumford construction to produce a formal smoothing of the $n$-vertex. We do not concern ourselves with those results at the moment since we now have enough to define the multiplication rule on the $\vartheta$-functions. In ~\cite{GrossHackingSiebert} Gross, Hacking and Siebert introduce an entirely abstract construction of the $\vartheta$-functions. See Section \ref{sec:explicit-GW} for an overview of these results.

The key data we will need is the count of pairs of pants, which are expressed in terms of broken lines as pairs of broken lines $(l_P, m_P)$  and $(l_Q, m_Q)$ from $P$ and $Q$ respectively to an irrational point near to $R$ such that 
$(\partial l_P/\partial t)|_{t=0} + (\partial l_Q/\partial t)_{t=0} = R$. Let $T_{P,Q \rightarrow R}$ denote the set of such pairs.

We have seen on page~\pageref{ToricThetaProduct} how to define a product on the $\vartheta$-functions in the toric case, so let us now study the general case. The outcome of our conversation about Hochschild cohomology was that the ring of regular functions on the mirror is the degree zero part of symplectic cohomology of the original variety. Following~\cite{CanonicalBasesForClusterAlgebras} for each integral point $P$ in $B$ introduce a symbol $\vartheta_P$. As a $k$-vector space the ring $QH (\check{W})$ is freely generated by the $\vartheta_P$. We take the content of~\cite{MirrorSymmetryforLogCY1} Theorem 2.34 as the definition of the product of $\vartheta_P$ and $\vartheta_Q$, so this product is equal to

\[ \sum_{R} \sum_{((l_P, m_P), (l_Q, m_Q)) \in T_{P,Q \rightarrow R}} m_P (0) m_Q (0) \vartheta_R\]

This is analogous to what the approach of ~\cite{GrossHackingSiebert}. A key result of ~\cite{MirrorSymmetryforLogCY1} is that for fixed order $J^n$ and \Lawrence{for a consistent scattering diagram in the sense of
~\cite{MirrorSymmetryforLogCY1}, Definition 2.26,} this does not depend on the choice of irrational point near $R$. \Lawrence{This consistency property will hold
in particular for the canonical scattering diagrams considered below.}
This sum need not terminate and indeed will in general only produce a power series. However, in the case that
$D$ supports an ample divisor, this power series will in fact be a polynomial.
Our goal in what follows is to limit the terms which can occur and find a generating basis of the $\vartheta$-functions.

\subsection {The combinatorics of scattering diagrams}

\Lawrence{We now tackle the final bullet point of~\ref{generalisationtable}. It is not clear that the relations between the $\vartheta$ functions should be algebraic}. Let us explain how to exploit the structure of a scattering diagram to prove \Lawrence{algebraic} convergence. To do this we require an example to work with, for simplicity the del Pezzo surface $dP_5$.

\begin{example}Figure~\ref{fig:dP5Structure} below gives a choice of boundary divisor,
where circles denote blown-up points and red lines denote components of $D$. 
\begin{figure}\label{fig:dP5Structure}
\end{figure}
From this we construct the base of the scattering diagram as in Figure~\ref{fig:dP5DualComplex}, where the grey section is glued isomorphically to the standard upper right quadrant

\begin{figure}\label{fig:dP5DualComplex}
\end{figure}

In this case one can construct the entire scattering diagram: it is finite and only non-trivial on the one-cells. We will not do this since in future examples the scattering diagram will be non-trivial on a dense set of rays.

Each point of $B$ defines a curve class as via $a \; v_i + b \; v_{i+1} \mapsto a D_i + b D_{i+1}$. Define the piecewise linear function $E$ on $B$ given by $-K_S \cdot (a D_i + b D_{i+1})$. From the Fano condition this defines a strictly convex piecewise linear function on the space $B$ which is positive everywhere
but at the origin. The function $E$ extends to a linear function on the tangent space at each point in the interior of a maximal cell. This allows us to evaluate
$E$ on tangent vectors to broken lines away from one-cells. 

In fact, given a broken line, $E(\partial l/\partial t)$ is increasing
as a function of $t$. Indeed,
at a point $t$ where $l$ is non-linear by definition the tangent vector
changes by a positive generator of the ray $\mathfrak{d}$, and $E$ is 
positive on this generator.
Similarly if $l$ crosses from one maximal cell of $B$ to another then by
convexity of $E$ the value of $E (\partial l / \partial t)$ increases. 

We use this as follows.
Suppose that two broken lines $(l_1, m_1)$ and $(l_2. m_2)$ from $v_1$ and $v_2$ combine to form a pair of pants at $P$. Then by definition we must have 
\[
(\partial l_1/\partial t)|_{t=0} + (\partial l_2/\partial t)|_{t=0} + P = 0.
\]
Since $E(P)\ge 0$, and $-E(v_i)\le E((\partial l_i/\partial t)|_{t=0})$, we
see that
$E(v_1)$ and $E(v_2)$ control how many new maximal cells $l_1$ and $l_2$ can enter, and how many times they can bend.

 Let us begin by understanding what could possibly contribute to the product $\vartheta_{(-1,0)} \vartheta_{(0,1)}$. Let us number the two-cells anti-clockwise starting at the upper right cell as in Figure~\ref{fig:dP5DualComplex}.
  
  Let $l_1$ be a broken line coming from $(-1,0)$ and $l_2$ a broken line from $(0,1)$. We have that $E((-1,0)) = E ((0,1)) = 1$. This immediately implies
that $0 \le E(P)\le 2$. Further, if $E(P)=2-n$, then the total number
of bends and crossings of one-dimensional cells of $l_1$ and $l_2$ is $n$.
First note that $n=0$ does not occur: since $v_1$ and $v_2$ are in different
cones, at least one of the broken lines must cross a one-cell.
If $n=1$, then we cross one one-cell and further bends or crossings can
occur. There is then only one possibility, 
the obvious pair of pants as described in Figure~\ref{Unbent} contributing to $\vartheta_{(-1,1)}$. 
Finally let us study the case $n=2$. In this case, $E(P)=0$, hence $P=0$,
and we can choose a point near the origin, say in the second two-cell.
If $l_2$ enters the grey region as detailed in Figure~\ref{Transition} it emerges moving in the direction $(-1,0)$ and so has $E(\partial l / \partial t) = 1$. 
Therefore in order to reach the second cell, it must cross two more one-cells,
and hence it can't contribute to a pair of pants. Thus $l_2$ must start
in cell 2 and never leaves it.

A similar argument shows that $l_1$ can only pass from cell 3 to cell 2,
and either $l_1$ or $l_2$ bends at a ray $\RR^{\ge 0}v$ with $E(v)=1$
and $v\in B(\ZZ)$. There is only one choice for such a ray, namely
the one-cell separating cells 2 and 3. See Figure~\ref{Bend}.
\label{combinatorialfiniteness}
\end{example}

In general, there will only be a finite number of rays with bounded $E$
and only a finite combination of bends. Thus in more complicated cases
this analysis can be implemented in a computer algebra package via the algorithm outlined in Algorithm~\ref{ThetaAlg}.

\begin{algorithm}
\caption{Theta relations algorithm}\label{ThetaAlg}
\begin{algorithmic}[1]
\State {\textbf{object} \textsc{AffineManifold}}
\LeftComment{Define a data structure consisting of a fan $\Sigma$, a piecewise linear function $\phi$ and a collection of integer points of $\Sigma$, $\Sigma_{\ZZ}$.}
\State {\textbf{object} \textsc{BrokenLine}}
\LeftComment {Define a data structure consisting of an initial direction $v$, a monomial, a tangent direction thought of as $\partial l/ \partial t \mid _{t=0}$ and a list of points $P_i$ generating rays along which the broken line bent}
\Function{ProductFormula} {$l_1$, $l_2$, affinemanifold}
\LeftComment {Define a function of two broken lines with tangent directions $v_1$ and $v_2$ in an affine manifold, affinemanifold which finds all possible terms appearing in the product $\vartheta_{v_1} \vartheta_{v_2}$}
\State{\textbf{int} maxE := $\phi (v_1) + \phi (v_2)$}
\State{\textbf{def} endpoints := The set of points in $\Sigma_{\ZZ}$ such that $\phi(P) \leq maxE$.}
\For {P in endpoints}
\If {$l_1$ and $l_2$ could pass through P and $v_1 + v_2 + P =0$}
\State {\textbf{print} the data of how $l_1$ and $l_2$ have bent}
\EndIf
\If {$l_1$ could pass through $P$}
\State {\textbf{def} newline := $l_1$ but now also scattering off of $\RR^{\geq 0} P$ as well}
\State {ProductFormula (newline, $l_2$)}
\EndIf
\If {$l_2$ could pass through $P$}
\State {\textbf{def} newline := $l_2$ but now also scattering off of $\RR^{\geq 0} P$ as well}
\State {ProductFormula ($l_1$, newline)}
\EndIf
\EndFor
\EndFunction
\State {\textbf{def} affinemanifold := AffineManifold( \ldots )}
\LeftComment {Define the base of the appropriate scattering diagram}
\State {\textbf{def} $L_1$ := BrokenLine ( \ldots )}
\LeftComment {Define $L_1$ to be a broken line, not scattering off of any rays with initial direction $v_1$}
\State {\textbf{def} $L_2$ := BrokenLine ( \ldots )}
\LeftComment {Define $L_2$ to be a broken line, not scattering off of any rays with initial direction $v_2$}
\State{ProductFormula ($L_1$, $L_2$, affinemanifold)}
\end{algorithmic}
\end{algorithm}

\begin{figure}\end{figure}

Doing this systematically for each pair $\vartheta_{v_{i-1}} \vartheta_{v_{i+1}}$ we can find schematic relations for the products which highlight which relative invariants we need to find. Filling out the relations for $dP_5$ we find the following relations.

\begin{align}
\vartheta_{(1,0)} \vartheta_{(-1,1)} &= g_1\vartheta_{(0,1)} + f_1 \\
\vartheta_{(0,1)} \vartheta_{(-1,0)} &= g_2\vartheta_{(-1,1)} + f_2 \\
\vartheta_{(-1,1)} \vartheta_{(0,-1)} &= g_3\vartheta_{(-1,0)} + f_3 \\
\vartheta_{(-1,0)} \vartheta_{(1,0)} &= g_4\vartheta_{(0,-1)} + f_4\\
\vartheta_{(0,-1)} \vartheta_{(0,1)} &= g_5\vartheta_{(1,0)} + f_5
\end{align}

The terms $f_i$ count curves meeting $D_i$ at a single point with tangency order one together with some monomials coming from broken lines looping around the origin.

As in the work of Hulya Arguz~\cite{HulyaThesis} the easy case is when the boundary consists of at least four components and the corresponding algebra is a quadratic algebra. The three component case is tractable but to compute the two component case $dP_2$ it is best to blow up the two intersection points of the two divisors. This refines the dual complex but keeps the interior $S \setminus D$ invariant. \Lawrence{As a consequence, this changes the mirror family
because the base $\Spec k[P]$ increases in dimension. However, the mirror
family to the original surface can be found by restricting to a subfamily,
see \cite{MirrorSymmetryforLogCY1}, Section 6.2 for details.}
In keeping with our previous notation we denote the exceptional curves over these by $E_8$ and $E_9$. The choice of dual complex now becomes:
\[\begin{tikzpicture} \fill[lightgray] (0,0) -- (-2,0) -- (-2,2); \draw (0,0) -- (2,0); \draw (0,0) -- (-1,2); \draw (0,0) -- (2,2); \draw (0,0)--(0,2); \draw [dashed] (0,0) -- (-2,2); \fill[pattern = dots] (-2,0) -- (2,0) -- (2,-2) -- (-2,-2); \end{tikzpicture}\]
\Lawrence{we re-label the theta functions by the corresponding cohomology class, so $\vartheta_{D_i}$ for $\vartheta_i$, further we write $\vartheta_{nD_i}$ for the function $\vartheta_{nv_i}$. With these labels we can apply the techniques of Example~\ref{combinatorialfiniteness} to show that the possible terms appearing in the relations between the theta functions become:
\begin{align}
\vartheta_C \vartheta_L &= r_1 \vartheta_{E_8} + r_2 \vartheta_{E_9} + r_3 + c_1(E_8) \label{formulaestart}\\
\vartheta_{E_8} \vartheta _{E_9} &= r_4 \vartheta_{3L}  + \vartheta_{2L} c_1 (L) + \vartheta_{L} c_2 (L) + c_3 (C) + r_5 \nonumber \\
& \quad + \vartheta_C c_2 (C) + \vartheta_{2C} c_1 (C) + r_6 \vartheta_{3C} \\
\vartheta_{C}^3 &= \vartheta_{3C} + c_1 (L) \vartheta_{C} + r_7 + c_1 (E_9) + r_8 + r_9 \vartheta_{E_8} + r_{10} \vartheta_{E_9}\\
\vartheta_L^3 &= \vartheta_{3L} + c_1 (C) \vartheta_L + r_{11} + c_1 (E_8) +  r_{12} + r_{13} \vartheta_{E_9} + r_{14} \vartheta_{E_8}\\
\vartheta_{C}^2 &= \vartheta_{2C} + r_{15} \vartheta_{C} + c_1 (L) \\
\vartheta_{L}^2 &= \vartheta_{2L} + r_{16} \vartheta_{L} + c_1 (C) \label{formulaeend}
  \end{align}
where the functions $r_i$ come from the combinatorics of broken lines which don't bend. The monomials which they carry come from the fact that the values of $\phi (\partial l/\partial t)$ are different at the start and end of the broken lines. The functions $c_n (D)$ are counts of broken lines bending due to curves tangent to a class $D$ at a single point to order $n$. It is this finite list of open Gromov-Witten invariants which we must calculate.} To calculate these invariants we have two options, either one could directly compute the moduli space and find the degree of the virtual fundamental class or one can use properties of the scattering diagram. Since we have not yet explained the role monodromy plays in the scattering diagram we will do the latter since it forms an important tool in the theory.

\section{Explicit Gromov-Witten counts and Monodromy}
\label{sec:explicit-GW}
One of the features required for the Mumford construction was that there was a global height function, something lost outside of the toric case. We must correct this in the example where $\phi$ is only a \Lawrence{multi}-valued piecewise linear function. In this case the graph construction of Mumford still can be applied locally. To do so if $\tau$ is a one- or two-dimensional face of $\Delta_{(S,D)}$, we write $\Delta_{(S,D)}/\tau$ for the localised fan at $\tau$\Lawrence{, i.e., the fan given by adding $\RR\tau$ to each cone containing $\tau$ and
removing cones of $\Delta_{(S,D)}$ not containing $\tau$}. Then after choosing a representative for $\phi$, $\phi_{\tau}$, we have a piecewise linear function single-valued on $\Delta_{(S,D)}/\tau$ and one can consider the graph $\Gamma (\Delta_{(S,D)}/\tau, \phi_{\tau})$. If $\tau$ lies as the one-cell separating two two-cells, $\sigma_+$ and $\sigma_-$, let $\phi_{\sigma_\pm}$ be the linear
extensions of $\phi_{\tau}$ restricted to the maximal cones $\sigma_{\pm}$.
Then we have canonical inclusions of graphs and an equality
\begin{equation}
\label{eq:Gamma-rel}
\Gamma(\Delta_{(S,D)}/\tau, \phi_{\tau}) = \Gamma(\Delta_{(S,D)}/\sigma_+, \phi_{\sigma_+}) \cap \Gamma (\Delta_{(S,D)}/\sigma_-, \phi_{\sigma_-})
\end{equation}
All \Lawrence{the monoids $\Gamma(\Delta_{(S,D)}/\rho, \phi_{\rho})$} have actions by $M$, giving rings 
$k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]$ with $k[M]$-algebra structures,
and different choices of representative for $\phi$ lead to canonically 
isomorphic $k[M]$-algebras. Choose a monomial ideal $I\subset M$ such that
$A_I:=k[M]/I$ is Artinian, and write
\[
k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]_I:=
k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]\otimes_{k[M]} A_I.
\]
Let 
\[
U_{\rho}=\Spec k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]_I.
\]
Then the relation \eqref{eq:Gamma-rel} leads to natural open inclusions
$U_{\sigma_{\pm}}\subset U_{\tau}$, which gives us gluing data for a scheme.
The paper~\cite{MirrorSymmetryforLogCY1} gives explicit formulae for the rings involved and proves that this gluing data does indeed satisfy the descent condition. This is the strict analogue of the Mumford degeneration, but does not contain any data about the instanton corrections. In particular, with the above
gluing, \Lawrence{it is not obvious that the glued scheme will carry enough
global functions to be embedded into an affine variety. For example,
a naive approach to finding global functions is to attempt to extend
monomial functions from $U_{\rho}$, but this frequently
doesn't work precisely because of monodromy.}

 The authors of~\cite{MirrorSymmetryforLogCY1} correct the gluing using 
automorphisms of the rings $k[\Gamma (\Delta_{(S,D)}/\sigma, \phi_{\sigma})]_I$ 
for the two-cells $\sigma$ and taking a limit over all $I$.

Suppose that we have a scattering diagram $ (f, \Delta_{(S,D)}, R, J)$ and fix a thickening ideal $I$. Let $\sigma$ be a two-cell with boundary one-cells
$\tau_+$, $\tau_-$. Choose points $P_+,P_-\in\sigma$ very close to $\tau_+$,
$\tau_-$ respectively. Choose a path $\gamma$ connecting $P_+$ to $P_-$.
Such a path selects a primitive normal vector $n_{\mathfrak{d}}$ to each ray through the origin in $\sigma$, pairing negatively with the tangent vector of the path. By assumption there are only finitely many rays $\mathfrak{d}$ passing through the origin such that $f(\mathfrak{d})$ is not 1 mod $I$. To each such ray take $\theta_{\mathfrak{d}}$ to be the automorphism defined on a toric monomial $z^p$, $p\in \Gamma(\Delta_{(S,D)}/\sigma,\phi_{\sigma})$ by
\[\theta_{\mathfrak{d}} (z^p) = z^p f(\mathfrak{d}) ^{\langle n_{\mathfrak{d}}, r(p)\rangle}\]
Now we define the automorphism $\theta_{\sigma}$ to be the ordered composition of the $\theta_{\mathfrak{d}}$ as one moves along $\gamma$. This yields
an automorphism of $U_{\sigma}$. In addition, the definition of $U_{\tau}$
is modified slightly using $f(\tau)$, see \cite{MirrorSymmetryforLogCY1},
Equation (2.7). Thus we obtain a modification of the $U_{\tau}$'s and 
a modification of the gluing inclusions $U_{\tau_+}\supset U_{\sigma} \subset
U_{\tau_-}$. This gives a new glued scheme which depends on the
scattering diagram and the ideal $I$. One can define the notion of a 
a \emph{consistent scattering diagram} (see \cite{MirrorSymmetryforLogCY1},
Definition 2.26). Consistent scattering diagrams then have the property
that they have a natural basis of regular functions $\{\vartheta_p\,|\,
p\in\Delta_{(S,D)}(\ZZ)\}$ which yield an embedding in an affine scheme
over $\Spec A_I$. One of the main theorems of \cite{MirrorSymmetryforLogCY1},
namely Theorem 3.8, states the canonical scattering diagram is consistent.

Importantly, however, there is an algorithmic way of computing consistent
scattering diagrams in certain situations, and a way of reducing the
calculation of the canonical scattering diagram to these situations.

So for the moment, consider a case where $\Delta_{(S,D)}$ is in fact affine isomorphic to $\RR^2$ and $\phi$ is single-valued. Then the sheaf $\mathcal{M}^+$
extends across $0$, with the stalk at $0$ being $\Gamma(\Delta_{(S,D)},\phi)$.
Then $\mathcal{R}_0$ is the completion of $k[\mathcal{M}^+_0]$ at its maximal
monomial ideal. All of the automorphisms $\theta_{\sigma}$ act on $\mathcal{R}_0$. Our construction so far has not attempted to incorporate the monoid ring $\mathcal{R}_0$. Take a \Lawrence{cyclic} ordering of the two-cells $\sigma_1, \ldots \sigma_n$ and let $\theta$ be the composite $\theta_{\sigma_n} \circ \ldots \circ \theta_{\sigma_1}$ considered as a map $\mathcal{R}_0 \rightarrow \mathcal{R}_0$. Then the 
scattering diagram is consistent if the map $\theta$ is the identity, see
\cite{ATropicalViewOfLandauGinzburgModels} and 
\cite{MirrorSymmetryforLogCY1}, Theorem 3.30.

Those scattering diagrams for which this composite $\theta$ is the identity 
are also called \emph{consistent}. Let us give some example calculations.

\begin{example}
  Any toric variety relative to its toric boundary has trivial canonical scattering diagram. Therefore we will find no interesting examples here. Let us give an example motivated instead by $dP_5$: we will see shortly the connection.

  Our initial scattering diagram is $\RR^2$ with two incoming rays $(1 + z^{(1,0)})$ and $(1+z^{(0,1)})$:
  \[\begin{tikzpicture} \draw[-<] (0,0) -- (1,0); \draw[-<] (0,0) -- (0,1); \draw (1,0)--(2,0); \draw (0,1) -- (0,2);\end{tikzpicture}\]
  Let us calculate the composition $\theta$ along a circle anti-clockwise around the origin starting at $(-1,-2)$ acting on $z^{(1,0)}$ and $z^{(0,1)}$. The first wall we meet is $\mathfrak{d}_1=\RR_{\ge 0}(1,0)$. Applying the above definition of the automorphism
$\theta$:
  \[\theta_{\mathfrak{d}_1} (z^{(1,0)}) = z^{(1,0)} (1 + z^{(1,0)}) ^{\langle (1,0),(0,-1) \rangle} = z^{(1,0)}\]
  \[\theta_{\mathfrak{d}_1} (z^{(0,1)}) = z^{(0,1)} (1 + z^{(1,0)}) ^{\langle (0,1),(0,-1) \rangle} = z^{(0,1)}(1 + z^{(1,0)})^{-1}\]
  Continuing along our path we meet the wall $\mathfrak{d}_2=\RR_{\ge 0}(0,1)$. Here $\theta_{\mathfrak{d}_2}$ is given by:
  \[\theta_{\mathfrak{d}_2} (z^{(1,0)}) = z^{(1,0)} (1 + z^{(0,1)}) ^{\langle(1,0),(1,0) \rangle} = z^{(1,0)}(1+z^{(0,1)})\]
  \[\theta_{\mathfrak{d}_2} (z^{(0,1)}) = z^{(0,1)} (1 + z^{(0,1)}) ^{\langle(0,1),(1,0) \rangle} = z^{(0,1)}\]
  Therefore the composition $\theta$ sends $z^{(1,0)}$ to $z^{(1,0)}(1 + z^{(0,1)})$ and $z^{(0,1)}$ to $z^{(0,1)} (1 + z^{(1,0)} + z^{(1,1)})^{-1}$. To first order this looks as if we are missing outgoing rays with monomials $(1+z^{(1,0)})$ and $(1+z^{(0,1)})$. Let us add those in and then repeat this calculation.
  \[\begin{tikzpicture} \draw[-<] (0,0) -- (1,0); \draw[-<] (0,0) -- (0,1); \draw (1,0)--(2,0); \draw (0,1) -- (0,2);\draw[->] (0,0) -- (-1,0); \draw[->] (0,0) -- (0,-1); \draw (-1,0)--(-2,0); \draw (0,-1) -- (0,-2);\end{tikzpicture}\]
  Taking again a path anti-clockwise starting at $(-1,-2)$ we first meet the wall $\mathfrak{d}_3=\RR_{\ge 0}(0,-1)$. The associated $\theta_{\mathfrak{d}_3}$ acts by:
    \[\theta_{\mathfrak{d}_3} (z^{(1,0)}) = z^{(1,0)} (1 + z^{(0,1)}) ^{\langle(1,0),(-1,0) \rangle} = z^{(1,0)}(1+z^{(0,1)})^{-1}\]
  \[\theta_{\mathfrak{d}_3} (z^{(0,1)}) = z^{(0,1)} (1 + z^{(0,1)}) ^{\langle(0,1),(-1,0) \rangle} = z^{(0,1)}\]
  The next wall is $\mathfrak{d}_1$, which again acts as before. The third wall we meet is $\mathfrak{d}_2$, again acting as before. The final wall we encounter is $\mathfrak{d}_4=\RR_{\ge 0}(-1,0)$, where the associated $\theta$ is given by:
  \[\theta_{\mathfrak{d}_4} (z^{(1,0)}) = z^{(1,0)} (1 + z^{(1,0)}) ^{\langle (1,0),(0,1) \rangle} = z^{(1,0)}\]
  \[\theta_{\mathfrak{d}_4} (z^{(0,1)}) = z^{(0,1)} (1 + z^{(1,0)}) ^{\langle (0,1),(0,1) \rangle} = z^{(0,1)}(1 + z^{(1,0)})\]
  One easily calculates the composite of all of these and one finds that to second order the composite acts by $z^{(1,0)} \mapsto z^{(1,0)} (1 + z^{(1,1)} + \ldots)$ so it appears that we are missing an outgoing ray carrying monomial $(1 + z^{(1,1)})$. Introducing such a ray to obtain a new scattering diagram
  \[\begin{tikzpicture} \draw[-<] (0,0) -- (1,0); \draw[-<] (0,0) -- (0,1); \draw (1,0)--(2,0); \draw (0,1) -- (0,2);\draw[->] (0,0) -- (-1,0); \draw[->] (0,0) -- (0,-1); \draw (-1,0)--(-2,0); \draw (0,-1) -- (0,-2);\draw[->] (0,0) -- (-1,-1); \draw (-1,-1) -- (-2,-2);\end{tikzpicture}\]
 Now one calculates $\theta$ is the identity, so we have finally constructed a consistent scattering diagram.
  
\end{example}

The term-by-term nature of the above construction seems to suggest that in the background lurks a general procedure. The Kontsevich-Soibelman lemma of~\cite{AffineStructuresAndNonArchimedeanAnalyticSpaces} proves that from any scattering diagram embedded in $\RR^2$ (i.e,  one with no singularities at the origin) one can construct a consistent scattering diagram by adding in only outgoing rays. In particular one can perform this correction order-by-order as in our example making it computable. This is explicitly described in the following lemma.

\begin{lemma}[The Kontsevich-Soibelman lemma]

  Let $(B,f,R,J)$ be a scattering diagram with $B \cong \RR^2$ as affine manifolds. Let $P$ be a point of $B$ contained inside a two-cell $\sigma$. \Lawrence{The ordered composite of the functions $\theta$ along a path $\gamma$ around the origin with $\gamma(0) = \gamma(1) = P$ gives rise to an automorphism $\theta_{\gamma}$ of the stalk of sheaf $\mathcal{R}$ at zero}. There exists another scattering diagram $(B, f', R, J)$ such that the corresponding automorphism $\theta_{\gamma}$ is the identity and such that for each ray $\mathfrak{d}$ one has
\[ f' (\mathfrak{d}) = f(\mathfrak{d}) (1 + \sum a_i z^{m_i})\]
where the $z^{m_i}$ are all outgoing.

\end{lemma}

A key result of Gross, Hacking and Keel, Theorem 3.25 of~\cite{MirrorSymmetryforLogCY1} explores how to flatten the base of the scattering diagram by using \emph{toric models}. A toric model of $(S,D)$ is a choice of birational morphism $\phi:(S, D) \rightarrow (S',D')$ obtained by blowing down curves in the interior of $S$ meeting $D$ in precisely one point such that $S'$ is toric and $D'$ the toric boundary. This restricts to an isomorphism $D \cong D'$ and so induces a map of dual complexes which ``flattens'' the singularities in the sense that the ensuing dual intersection complex is isomorphic to $\RR^2$ as an affine manifold. In the philosophy of Gross Hacking and Keel it pushes the singularities out to infinity. We reincorporate the singularities by instead including them into the scattering diagram as rays carrying expressions $1+z^{[E_i]+\phi(v_i)}$ 
where $E_i$ is an exceptional curve meeting $D_i$ and $v_i$ is the primitive
generator of the ray in $\Delta_{(S',D')}$ corresponding to $D_i$. \Lawrence{We make this construction explicit in the example in the next section}. There is a canonical morphism $\nu: \Delta_{(S,D)} \rightarrow \Delta _{(S',D)}$ and in Lemmas 3.27 and 3.28 of~\cite{MirrorSymmetryforLogCY1} the authors prove that this induces an isomorphism of sheaves $\mathcal{R}(S,D) \cong \mathcal{R} (S',D)$ together with a bijection of broken lines between $\Delta_{(S,D)}$ and $\Delta _{(S',D)}$. To be precise we have the following theorem of~\cite{MirrorSymmetryforLogCY1}:

\begin{theorem}
  Let $(S,D)$ be a Looijenga pair with $(S',D)$ a toric model. Then the complex $\Delta_{S',D}$ is isomorphic as an affine manifold to $\RR^2$. Let $(\RR^2, f_{S'}, \underline{M}, \underline{M} \setminus 0)$ be the scattering diagram with values in $A_1(S)$ on $\Delta_{(S',D')}$ whose value along $\RR_{\ge 0}v_i$ is the product of incoming rays $\prod_j( 1 + z^{[E_{i,j}]+\phi(v_i)})$ and outgoing rays $\prod_j(1 + z^{-[E_{i,j}]+\phi(v_i)})^{-1}$ where the $E_{i,j}$ are the blown-down curves meeting $D_i$. Since this satisfies the assumptions of the Kontsevich-Soibelmann lemma, we let $\bar{f}_{S'}$ be the associated consistent diagram.

  Let $\bar{f}_{S}$ be the canonical scattering diagram on $\Delta_{(S,D)}$. Locally $\Delta_{(S,D)}$ and $\Delta_{(S',D')}$ are isomorphic and we can evaluate $f_{S'}$, $\bar f_S$ and $\bar f_{S'}$ on the same line $\mathfrak{d}$. 
Then one has an equality
\[ \bar{f}_{S'} (\mathfrak{d}) = \bar{f}_S (\mathfrak{d}) f_{S'} (\mathfrak{d}).\]
this has the effect of replacing \Lawrence{the base of the scattering diagram by an affine manifold isomorphic to $\RR^2$} and swapping the directions of rays $( 1 + z^{[E_{i,j}]+\phi(v_i)})$ from outgoing to incoming. This is seen explicitly in the example in the next section.
\end{theorem}

  After possibly blowing \Lawrence{up} some double points of $D$, we can then blow down appropriate curves to construct toric models, see \cite{MirrorSymmetryforLogCY1}, Proposition 1.3. This allows one to calculate the original scattering diagram by applying the Kontsevich-Soibelman lemma to $f_{S'}$. An implementation of this algorithm in Python is available from me on request, and hopefully will be incorporated into a Sage library in future work.

\section{Explicit formulae}

Let us practice constructing toric models in the cases we are interested in.

\begin{example}
  Recall the dual intersection complex for $dP_5$:

  \[\begin{tikzpicture} \fill[lightgray] (0,0) -- (2,0) -- (2,-2); \draw (0,0) -- (2,0); \draw (0,0)--(0,2); \draw (0,0) -- (-2,2); \draw (0,0) -- (-2,0); \draw (0,0) -- (0,-2); \draw[dashed] (0,0) -- (2,-2); \end{tikzpicture}\]
  
  This is certainly not affine isomorphic to $\RR^2$. The boundary has five components, and this suggests that we should attempt to find a toric
model isomorphic to $dP_7$, since there is a choice of boundary divisor there also consisting of five components. A model for $dP_7$ together with its toric boundary is contained below. The red lines are a triangle in $\PP^2$, the
red circles are blown up to obtain $dP_7$, and the black points are further
blown up to obtain $dP_5$ with $dP_7$ as its toric model.

  \[\begin{tikzpicture} \draw [red] (0,0) circle (2pt); \draw [red] (2,0) circle (2pt); \fill (0,2) circle (1pt); \node at (0.15,2) {1}; \fill (0.4,0.4) circle (1pt); \node at (0.55,0.55) {2}; \draw [red] (-1,0) -- (3,0); \draw [red] (0,-1)--(0,3); \draw [red] (3, -0.25) -- (-1, 0.75); \end{tikzpicture}\]

The corresponding dual complex is given by
  \[\begin{tikzpicture} \draw (0,0) -- (2,0); \draw (0,0)--(0,2); \draw (0,0) -- (-2,2); \draw (0,0) -- (-2,0); \draw (0,0) -- (0,-2); \end{tikzpicture}\]
In particular the self intersection of every component of the boundary for $dP_5$ was -1, whereas two of the components of the boundary of $dP_7$ have self intersection 0.

  We now incorporate this into the construction of the scattering diagram. Initially after making a choice of ordering we have have the scattering diagram:
\[\begin{tikzpicture} \draw[-<] (0,0) -- (1,0); \draw[-<] (0,0) -- (0,-1); \draw (1,0)--(2,0); \draw (0,-1) -- (0,-2); \node at (3.2, 0) {$1 + z^{E_1+\phi(v_1)}$}; \node at (0, -2.4) {$1 + z^{E_2+\phi(v_2)}$};\end{tikzpicture}\]
and we saw in our motivation for the Kontsevich-Soibelman lemma that this produces a consistent diagram
\[\begin{tikzpicture} \draw[-<] (0,0) -- (1,0); \draw[-<] (0,0) -- (0,-1); \draw (1,0)--(2,0); \draw (0,-1) -- (0,-2);\draw[->] (0,0) -- (-1,0); \draw[->] (0,0) -- (0,1); \draw (-1,0)--(-2,0); \draw (0,1) -- (0,2);\draw[->] (0,0) -- (-1,1); \draw (-1,1) -- (-2,2); \node at (2.2,0.3) {$1 + z^{E_1 + \phi(v_1)}$}; \node at (-2.2,-0.3) {$1 + z^{E_1 + \phi(v_1)}$}; \node at (-2.2,2.7) {$1 + z^{E_1 + \phi(v_1)}$}; \node at (-2.2,2.2) {$+z^{E_2 + \phi(v_2)}$}; \node at (-0,-2.4) {$1 + z^{E_2 + \phi(v_2)}$};\node at (0.2,2.4) {$1 + z^{E_2 + \phi(v_2)}$};\end{tikzpicture}\]
which agrees up to changing the direction of the incoming rays and pushing forwards with the scattering diagram found in~\cite{MirrorSymmetryforLogCY1} Figure 1.1. This is exactly as predicted by the Kontsevich-Soibelman Lemma.
\label{ex:monodromycalculation}
\end{example}

This suggests Algorithm~\ref{KSAlg} as a general algorithm for calculating consistent scattering diagrams.

Since by the previous section only a finite collection of classes need be calculated we may terminate the calculation once those coefficients have been obtained and substitute them into the formulae \Lawrence{\eqref{formulaestart}-\eqref{formulaeend} we found using the techniques in section 2.2.2}.

Recall that the Picard group of $dP_d$ is generated by a hyperplane class $H$ and the class of the exceptional curves $E_1, \ldots E_{9-d}$. Implementing \Lawrence{the procedure of Example~\ref{ex:monodromycalculation}} in a computer algebra package one reads off the coefficients and the resulting families are contained in Table~\ref{explicitequations}.

 \thispagestyle{empty}

\begin{algorithm}[H] 
\caption{Kontsevich-Soibelmann algorithm}\label{KSAlg}
\begin{algorithmic}[1] 
\State {\textbf{int} maxn}
\LeftComment{A maximum bound on the value of $-[C].K_S$ amongst curves $[C]$ we wish to calculate.}
\State {\textbf{def} R} := $\mathbb{Q}[P]/ \langle P \setminus P^* \rangle ^{maxn+1}$ 
\LeftComment{Define $R$ to be the rational monoid ring associated to the monoid $P$ modulo terms of order $maxn+1$. We take a basis of $P$ such that $r$ is simply projection to the first two coordinates and write $z^{(a,b)}$ for the monomial $z^{\phi((a,b))}$}
\State{\textbf{int} \emph {n:=1}}
\State{\textbf{object} \textsc{Ray}}
\LeftComment{Define a data structure consisting of a primitive generator for a ray $m_{\mathfrak{d}}$, an oriented primitive normal to that ray $n_{\mathfrak{d}}$ and a (finite) power series on that ray $f_{\mathfrak{d}}$ considered as an element of $R$}
\State {\textbf{object} \textsc{ScatteringDiagram}}
\LeftComment{Define a data structure consisting of an ordered collection of Ray objects.}
\Function{ThetaTransformation} {ray, f, n}
\LeftComment {Define a function of an element of $R$, $f$, a Ray $ray$ and an integer $n$. For each term $c_p z^p$ appearing with non-zero coefficient in $f$ it calculates \[f_p = z^p f_{\mathfrak{d}}^{\langle n_{\mathfrak{d}}, r(p) \rangle}\] correct to terms of order $n$}
\State \textbf{return} $\sum_p c_pf_p$
\EndFunction
\State{ \textbf{def} scatteringdiagram := ScatteringDiagram (\ldots)}
\LeftComment {Create a scattering diagram with rays of the form $(1+z^{E_{i,j}}, v_i)$ as detailed above}
\While {n < maxn}:
\State{x := $z^{(1,0)}$}
\State{y := $z^{(0,1)}$}
\For {ray in ScatteringDiagram}
\State x $\leftarrow$ ThetaTransformation (ray, x)
\State y $\leftarrow$ ThetaTransformation (ray, y)
\For {monomial $c_p z^p$ in x}
\If {$z^p$ is order $n$}
\State add $(r(p), 1-\frac{c_p}{\langle (1,0), r(p) \rangle} z^{p-(1,0)})$ to scatterdiagram
\EndIf
\EndFor
\For {monomial $c_p z^p$ in y \textbf{not} in x}
\If {$z^p$ is order $n$}
\State add $(r(p), 1-\frac{c_p}{\langle (0,1), r(p) \rangle} z^{p-(0,1)})$ to scatterdiagram
\EndIf
\EndFor
\EndFor
\State{ n $\leftarrow$ n+1}
\EndWhile
\State {\textbf{return} scatteringdiagram}
\end{algorithmic}
\end{algorithm}

\begin{center}
  \begin{table}
\begin{tabular}{l|ll}
  Surface & Equations & Super-potential \\
  $dP_5$ & $\vartheta_{1} \vartheta_3 = z^{[D_2]}\vartheta_2 + z^{[D_4] +[D_5]}$ & $\sum \vartheta_i$ \\
  & $\vartheta_2 \vartheta_4 = z^{[D_3]}\vartheta_3 + z^{[D_1]+[D_4]}$  & \\
  & $\vartheta_3 \vartheta_5 = z^{[D_4]}\vartheta_4 + z^{[D_1]+[D_2]}$  & \\
  & $\vartheta_4 \vartheta_1 = z^{[D_5]}\vartheta_5 + z^{[D_2]+[D_3]}$  & \\
  & $\vartheta_5 \vartheta_2 = z^{[D_1]}\vartheta_1 + z^{[D_3]+[D_4]}$  & \\
   & & \\
  $dP_4$ & $\vartheta_1 \vartheta_3 = z^{[D_2]}\vartheta_{2} + z^{[D_4]} \vartheta_4 +$  & $\sum \vartheta_i$\\
   & \quad $z^{H-E_1} + z^{2H - E_1 - E_2 - E_3 - E_5} + z^{2H - E_1 - E_2 - E_4 - E_5}$ & \\
  & $\vartheta_2 \vartheta_4 = z^{[D_1]} \vartheta_1 + z^{[D_3]} \vartheta_3 +$ & \\
  & \quad $z^{H-E_3} + z^{H-E_4} + z^{2H- E_2 - E_3 - E_4 - E_5}$ & \\
  & & \\
  $dP_3$ & $\vartheta_1 \vartheta_2 \vartheta_3 = z^{[D_1]}\vartheta_1^2 + z^{[D_2]} \vartheta_2^2 + z^{[D_3]}\vartheta_{3}^2 + $ & $\sum \vartheta_i$ \\
  & \quad $z^{[D_1]} \vartheta_1 (z^{E_1} + z^{E_2} + z^{H- E_3 - E_5} + z^{H - E_3 - E_6} + $ & \\
  & \quad $z^{H- E_4 - E_5} + z^{H- E_4 - E_6}) + z^{[D_2]} \vartheta_2 (z^{E_3} + z^{E_4} +$ & \\
  & \quad $z^{H- E_1 - E_5} + z^{H- E_1 - E_6} + z^{H- E_2 - E_5} + z^{H- E_2 - E_6})+$ & \\
  & \quad $z^{[D_3]} \vartheta_3 (z^{E_5} + z^{E_6} + z^{H- E_1 - E_3} + z^{H - E_1 - E_4}$ & \\
  & \quad $z^{H - E_2 - E_3} + z^{H - E_2 - E_4}) + z^H + $ & \\
  & \quad $(z^{2H}  + z^{4H - \sum E_i}) \alpha +$ & \\
  & \quad $z^{3H-\sum E_i}(z^{E_2-E_1} + z^{E_1-E_2} + z^{E_4-E_3} + z^{E_3-E_4} +$ & \\
  & \quad $z^{E_6-E_5} + z^{E_6-E_5})+ z^{5H-2 \sum E_i} - 4z^{[D]}$ & \\
  & $ \alpha = (z^{-E_1} + z^{-E_2})(z^{-E_3} + z^{-E_4})(z^{-E_5} + z^{-E_6})$ & \\
  & &
\end{tabular}
  \caption{}
  \label{explicitequations}
\end{table}
\end{center}

The expression for the mirror family to $dP_3$ agrees with that found in~\cite{MirrorSymmetryforLogCY1}, Example 6.13. To express the formulae for $dP_2$ requires some additional notation. In this case we had to blow up two additional points which lay on the intersection of the choosen conic $C$ and line $L$. Let $E_8$, $E_9$ be the exceptional divisors lying over the intersections points of $C$ and $L$. Let us relabel the $\vartheta_i$ by the cohomology class of the corresponding \Lawrence{boundary divisor, e.g., writing $\vartheta_{D_i}$ for $\vartheta_i$}. There is an action of $C_2$ exchanging $E_1$ and $E_2$, an action of $S_5$ permuting $E_3$ through $E_7$ and an action of $C_2$ exchanging $E_8$ and $E_9$. When we write $\sum z^{h H + \sum a_i E_i}$ we mean to take the sum of $z^{h[H] + \sum a_i [E_i]}$ over elements of the orbit under the action of $C_2 \times S_5 \times C_2$. In particular we choose here representatives with $a_1 \geq a_2$ and $a_3$ through $a_7$ decreasing and $a_8 \geq a_9$. For instance $\sum z^{H+E_1}$ would be the sum $z^{H+E_3} + z^{H+E_4} + z^{H+E_5} + z^{H+E_6} + z^{H+E_7}$.
\Lawrence{Now we apply our algorithm outlined in Example~\ref{ex:monodromycalculation} to find the coefficients we were missing in the equations \eqref{formulaestart} to \eqref{formulaeend} for the mirror family to $dP_2$ relative to our choice of boundary. After calculating these we find that the mirror family is given by}
\begin{align*}
\vartheta_C \vartheta_L &= z^{[E_8]} \vartheta_{E_8} + z^{[E_9]} \vartheta_{E_9} + 3 z^{[C] + [L] + 2[E_8] + 2[E_9]} + B_1(E_8)z^{[E_8]}\\
\vartheta_{E_8} \vartheta _{E_9} &= \vartheta_{3L} z^L + \vartheta_{2L} B_1 (L) z^{C+L+2E_8+E_9} + \vartheta_{L} B_2 (L) z^{2C +L+4E_8 + 2E_9} \\
 & \quad + z^{C+3E_8} B_3 (C) + 3 z^{2C + 2L + 3E_8 + 3E_9} + \vartheta_C B_2 (C) z^{C + 2E_8} \\
& \quad + \vartheta_{2C} B_1 (C) z^{C+E_8} + \vartheta_{3C} z^C\\
\vartheta_{C}^3 &= \vartheta_{3C} + 3 B_1 (L) z^{C+L+3E_8+2E_9} \vartheta_{C} + 6z^{C+2L+3E_8+3E_9} + 6 B_1 (E_9)\\
 & \quad z^{C+L+4E_8+2E_9} + 3 z^{L + 2E_8 + E_9}\vartheta_{E_8} + 3 z^{L+ E_8 + 2E_9} \vartheta_{E_9}\\
\vartheta_L^3 &= \vartheta_{3L} + 3 B_1 (C) z^{C+L+2E_8+E_9} \vartheta_L + 6z^{2C+L+3E_8+3E_9} + 6 B_1 (E_8)\\
 & \quad z^{2C+4E_8+2E_9} + 3 z^{C + E_8 + 2E_9} \vartheta_{E_9} + 3 z^{C + 2E_8 + E_9} \vartheta_{E_8}\\
\vartheta_{C}^2 &= \vartheta_{2C} + 2 \vartheta_{C} z^{L+E_8+E_9} + 2 B_1 (L) z^{C+L+3E_8+2E_9} \\
\vartheta_{L}^2 &= \vartheta_{2L} + 2 \vartheta_{L} z^{C+2E_8+E_9} + 2 B_1 (C) z^{C+2E_8+E_9}\\
\end{align*}
where the coefficients $B_k(E)$ are the relative invariants of curves meeting the curve $E$ in one point with tangency order $k$. The superpotential is given by the sum $\vartheta_C + \vartheta_L + \vartheta_{E_7} + \vartheta_{E_9}$. Explicit formulae for these are given by:
\begin{align*}
  B_3 (C) &= 9 (\sum z^{2H - E_1 - E_2 - E_3}  + \sum z^{3H - 2E_1 - E_2 - E_3 - E_4} + \sum z^{4H - 2 E_1 - 2 E_2 - 2 E_3 - E_4 - E_5 - E_6} \\ & + \sum z^{4H - 3 E_1 - E_2 - E_3 - E_4 - E_5 - E_6 - E_7} + 8 \sum z^{4H - 2E_1 - 2 E_2 - E_3 - E_4 - E_5 - E_6 - E_7} \\ & + \sum z^{5H - 3E_1 - 2E_2 - 2E_3 - 2E_4 - E_5 - E_6 - E_7})\\
  B_2 (C) &= 4 (\sum z^{H-E_1} + \sum z^{2H - E_1 - E_2 - E_3 - E_4} + \sum z^{3H - 2E_1 - E_2 - E_3 - E_4 - E_5 - E_6}\\ & + \sum z^{4H - 2E_1 - 2E_2 - 2E_3 - E_4 - E_5 - E_6 - E_7})\\
  B_1 (C) &= \sum z^{E_3} + \sum z^{H-E_1 - E_3} + \sum z^{2H - E_1 - E_2 - E_3 - E_4 - E_5} \\  & + \sum z^{3H - 2E_1 - 2E_2 - 2E_3 - E_4 - E_5 - E_6 }\\
  B_3 (L) &= 9 (\sum z^{3H - 2E_3 - E_4 - E_5 - E_6 - E_7} + \sum z^{4H - E_1 - 2E_3 - 2E_4 - 2E_5 - E_6 - E_7}\\ & + \sum z^{5H - E_1 - E_2 - 3 E_3 - 2 E_4 - 2E_5 - 2E_6 - E_7} + 8\sum z^{5H - E_1 - E_2 - 2E_3 - 2E_4 - 2E_5 - 2E_6 - 2E_7} \\ & + \sum z^{5H - 2 E_1 - 2E_3 - 2E_4 - 2E_5 - 2E_6 - 2E_7} + \sum z^{6H - 2E_1 - E_2 - 3E_3 - 3E_4 - 2E_5 - 2E_6 - 2E_7}\\ & + \sum z^{7H - 2E_1 - 2E_2 - 3E_3 - 3E_4 - 3E_5 - 3E_6 - 2E_7})\\
  B_2 (L) &= 4 (\sum z^{2H - E_3 - E_4 - E_5 - E_6} + \sum z^{3H - E_1 - 2E_3 - E_4 - E_5 -E_6 - E_7}\\ & + \sum z^{4H - E_1 - E_2 - 2E_3 - 2E_4 - 2E_5 - E_6 - E_7} + \sum z^{5H - 2E_1 - E_2 - 2E_3 - 2E_4 - 2E_5 - 2E_6 - 2E_7})\\
  B_1 (L) &= \sum z^{E_1} + \sum z^{H - E_3 - E_4} + \sum z^{2H - E_1 - E_3 - E_4 - E_5 - E_6} \\ & + \sum z^{3H - E_1 - E_2 - 2E_3 - E_4 - E_5 - E_6 - E_7}\\
  B_1 (E_8) &=\sum z^{H-E_3-E_8} + \sum z^{2H - E_1 - E_3 - E_4 - E_5 - E_8} + 9\sum z^{3H - E_1 - E_2 - E_3 - E_4 - E_5  - E_6 - E_7 - E_8} \\ & + \sum z^{3 H - E_1 - E_2 - 2E_3 - E_4 - E_5 - E_6 - E_8} + \sum z^{3H - 2 E_1 - E_3 - E_4 - E_5 - E_6 - E_7 - E_8} \\ & + \sum z^{4H - 2E_1 - E_2 - 2 E_3 - 2 E_4 - E_5 - E_6 - E_7 - E_8} + \sum z^{5H - 2E_1 - 2 E_2 - 2 E_3 - 2E_4 - 2E_5 - 2E_6 - E_7 - E_8}\\
  B_1 (E_9) &=\sum z^{H-E_3-E_9} + \sum z^{2H - E_1 - E_3 - E_4 - E_5 - E_9} + 9\sum z^{3H - E_1 - E_2 - E_3 - E_4 - E_5  - E_6 - E_7 - E_9} \\ & + \sum z^{3 H - E_1 - E_2 - 2E_3 - E_4 - E_5 - E_6 - E_9} + \sum z^{3H - 2 E_1 - E_3 - E_4 - E_5 - E_6 - E_7 - E_9} \\ & + \sum z^{4H - 2E_1 - E_2 - 2 E_3 - 2 E_4 - E_5 - E_6 - E_7 - E_9} + \sum z^{5H - 2E_1 - 2 E_2 - 2 E_3 - 2E_4 - 2E_5 - 2E_6 - E_7 - E_9}\\
\end{align*}

In cases of $dP_5$ and $dP_4$ the canonical scattering diagram is finite and can be computed. By an observation of Gross, Hacking and Keel the entire scattering diagram for $dP_3$ is generated by the series associated to a single ray under the birational automorphism group of this surface. This breaks down in lower degrees, as far as I am aware there is no known closed form expression for the scattering diagram associated to the final surface.
\Lawrence{
Now to finish we compare this to the predictions of ~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves} for the mirrors. These predictions state the mirror to $dP_n$ ought to be a compactification of a fibration of the two-torus $W:\mathbb{G}_m^2 \rightarrow \CC$ with an $I_{n}$ fibre at infinity. Here $W$ is the so-called Landau-Ginzburg potential. For example,
in the case of $dP_5$ we can take equations:
\[x_{i+1} x_{i-1} = x_i + 1\]
defining a surface inside $\AA^5$ with $W=\sum x_i$. It turns out this
is already a partial compactification of the mirror to $dP_5$.
To further compactify, we take the projective closure of the above
surface, taking coordinates $Z_1,\ldots,Z_5$ and $T$ to obtain equations
\[
Z_{i+1}Z_{i-1}=Z_i T + T^2.
\]
Then $T$, $W$ can be viewed as sections of ${\mathcal O}(1)$ on this
surface, hence giving a pencil. After resolving the base-points of this
pencil, one obtains the desired compactification. 
 The fibre at infinity therefore has equations
\[Z_{i-1} Z_{i+1} = 0\]
 for $i \in \{1, 2, 3,4,5\}$  inside $\PP^4$, hence defining a cycle of five planes. This is exactly as predicted by~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves}. For the other del Pezzo surfaces
$dP_n$ with $n\ge 4$, the same shape of the mirror as defined by
quadratic equations confirms that the fibre at infinity will be an $I_n$ fibre.
More careful considerations with the equations show the same for $n=3$ and $2$.}

\chapter{The mirror family to a rational elliptic surfaces}

Having built our intuition by studying the del Pezzo surfaces, we now turn to the main problem, constructing the mirror to a rational elliptic surface. We will begin by explaining their relevance to the general project of Gross, Hacking and Keel. They wished to study the moduli space of $K3$ surfaces and the mirror families for these surfaces. To do this one would hope to deform to the boundary where one can understand the Gross-Siebert program, prove convergence in a neighbourhood of the boundary and then smear this out to the entire moduli. Recall the following theorem on the structure of degenerations of $K3$ surfaces

\begin{theorem}[Structure of $K3$ degenerations]

Let $\pi:X \rightarrow \AA^1$ be a semi-stable family such that the fibre $X_t$ for $t \neq 0$ is a smooth $K3$ surface and the canonical bundle of the family is trivial. Then the central fibre $X_0$ is isomorphic to one of the following types

  \begin{itemize}
  \item[I] A smooth $K3$ surface.
  \item[II] A chain $S_1, \ldots S_n$ of surfaces such that $S_1$ and $S_n$ are rational, the other components are ruled surfaces over elliptic
curves and $S_i \cap S_{i+1}$ are smooth elliptic curves.
    \item[III] A complex of rational surfaces $S_1 \ldots S_n$ such that $V_i \cap V_j$ are smooth rational curves. The dual complex gives a triangulation of $S^2$.
  \end{itemize}

In the second case one can perform a birational modification of the family to produce a type II degeneration where the central fibre is the union of only two components. In this case the class $S_1 \cap S_2$ is an anti-canonical curve on each component.

\begin{proof}

  See~\cite{DegenerationsOfK3SurfacesAndEnriquesSurfaces}.
  
\end{proof}  

\end{theorem}

They therefore asked if their construction converges for a type II degeneration by first studying the construction for each half of the central fibre $S_1$ and $S_2$. Therefore let $S$ be such a surface. In an initial version of~\cite{MirrorSymmetryforLogCY1} the authors prove that if the intersection matrix for $S$ is negative definite then \Lawrence{the construction converges over an analytically closed set, which is the union of strata, and then further formally over a formal neighbourhood of this set}. On page 6 of the first version of~\cite{MirrorSymmetryforLogCY1} they however claim that their construction converges analytically over a disk when the intersection matrix for $S$ is negative semi-definite. However,
their proposed proof was not correct and this thesis corrects that omission; we answer this question positively at least in the case where the surface $S$ \Lawrence{is a rational elliptic surface with} an $I_4$ fibre $F_0$.

Rational elliptic surfaces hold a special position in the theory of algebraic surfaces in many respects. In particular they appear as limiting surfaces in the classification of Fano surfaces: they are not Fano but $-K_S$ is nef. We begin by introducing these surfaces and \Lawrence{recalling} a structure theorem for them.

\begin{definition}

A \emph{rational elliptic surface} (RES) is a non-singular proper rational surface $S$ admitting a relatively minimal elliptic fibration $\pi: S \rightarrow \mathbb{P}^1$ and a section $\sigma_0: \mathbb{P}^1 \rightarrow S$.

\end{definition}

The theory of Weierstrass models allows us to classify all such surfaces. Recall that a Weierstrass model for an elliptic curve $(E,0)$ is an isomorphism to an elliptic curve of the form $V(Y^2Z + a_1 XYZ + a_3 Y Z^2 = X^3 + a_2 X^2 Z +  a_4 X Z^2 + a_6 Z^3) \subset \mathbb{P}^2$ with identity the point at infinity. Following~\cite{GeometryofEllipticSurfaces} one can form Weierstrass models for rational elliptic surfaces. These describe the surface as being birational
to a $(3,1)$ hypersurface in $\mathbb{P}^2 \times \mathbb{P}^1$. We prefer to
view the surface by resolving the singularities of the hypersurface and
projecting to the first factor, hence describing $S$ as a blow-up of
$\mathbb{P}^2$.

\begin{lemma}[Weierstrass models of rational elliptic surfaces]

Let $\pi: S \rightarrow \mathbb{P}^1$ be a rational elliptic surface. Then it is isomorphic to the blow up of $\mathbb{P}^2$ in nine points lying on the intersection of two cubics. This isomorphism can be chosen to map the given section $\sigma_0$ to the exceptional curve over any one of the blown up points.

\begin{proof}

  This is Lemma IV.1.2 of~\cite{GeometryofEllipticSurfaces}.
  
\end{proof}

\end{lemma}

Using Proposition 13.12 of~\cite{3264AndAllThat} the Chow theory of the surface can be described very explicitly. From now on $H$ will denote the pullback of a hyperplane class from $\mathbb{P}^2$ and $E_1, \ldots , E_9$ the exceptional curves over the nine blown up points. Together these classes span the Chow group $A_1 (S)$. We write $1$ for the fundamental class of $S$ generating $A_2 (S)$ and $H^2$ for the unique class in $A_0 (S)$. By construction the fibre class is $F = 3H - \sum_i E_i$, and the class of a section $\sigma$ satisfies $\sigma.\sigma = -1$, hence any section is rigid.

\Lawrence{Following the techniques of~\cite{MirrorSymmetryforLogCY1} we can construct the base of the SYZ fibration as being the dual intersection complex. Figure~\ref{fig:DualIntersectionRES} depicts the universal cover of this space.

\begin{figure}
\caption{The universal cover of the dual intersection complex for $S^\dagger$.}
\label{fig:DualIntersectionRES}
\end{figure}

We now study \Lawrence{the algebra structure on} the scattering diagram\Lawrence{~\ref{fig:DualIntersectionRES}} associated to the pair $(S, F_0)$. Following the philosophy built up by studying the del Pezzo surfaces we study the products $\vartheta_{D_1}\vartheta_{D_3}$ and $\vartheta_{D_2} \vartheta_{D_4}$. Here the function $E$, analogous to the one defined in section 2.2.2, is just the $y$-component. \Lawrence{Summing the value of $E$ evaluated on the two incoming rays we get $-2$} and every point in $B(\ZZ)$ has positive $y$-component. Therefore the broken lines can bend off of rays representing curves tangent to the boundary to order one, order two or meeting the common intersection points $D_i \cap D_{i+1}$ transverse to $D_i$ and $D_{i+1}$. These classes are the classes whose log Gromov-Witten theory we must calculate. The first collection, those which meet the boundary transversely at a single point, were studied in~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms}

Expanding out the products according to the multiplication law for theta functions we see that the only non-trivial non-primitive theta functions that can occur are of the form $\vartheta_{(2k,2)}$, but they may come in infinite families as described in Figure~\ref{fig:TopLevelBrokenLines}. Schematically this will induce six formulae
\begin{align}
  \vartheta_{D_1} \vartheta_{D_3} & = f_{(2,2)} \vartheta_{2D_2} + f_{(6,2)} \vartheta_{2D_4} + \sum f_{(i,1)} \vartheta_{D_i} + f_0   \label{eqn:MirrorToRES1} \\
  \vartheta_{D_2} \vartheta_{D_4} & = g_{(0,2)} \vartheta_{2D_1} + g_{(4,2)} \vartheta_{2D_3} + \sum g_{(i,1)} \vartheta_{D_i} + g_0 \label{eqn:MirrorToRES2}\\
  \vartheta_{D_i}^2 & = (1+r^{i}_{(i,2)}) \vartheta_{2D_i} + r^{i}_{(i+2,2)} \vartheta_{2D_{i+2}} + \sum r^{i}_{(i,1)} \vartheta_{D_i} + r^i_0\label{eqn:MirrorToRES3}
\end{align}

\begin{figure}
\caption{Two pairs of broken lines, dotted and densely dotted, contributing to $\vartheta_{D_1}
\label{fig:TopLevelBrokenLines}
\end{figure}with $r^{i}_{(i,2)}$ and $r^{i}_{(i+2,2)}$ vanishing on the central fibre. \Lawrence{The functions $f_{(2,2)}$, $f_{(6,2)}$, $g_{(0,2)}$, $g_{(4,2)}$, $r^{i}_{(i,2)}$ and $r^{i}_{(i+2,2)}$ correspond to pairs of broken lines not bending. The terms $f_{(i,1)}$, $g_{(i,1)}$ and $r_{i,1}^i$ come from pairs of broken lines where one of the lines bends once off of a ray with primitive $(k,1)$. The remaining terms correspond to pairs of broken lines bending in three different ways, either one broken line bends off of a ray with primitive $(k,2)$, both broken lines bends off of rays with primitives $(k,1)$ and $(k',1)$ or one of the broken lines bends twice, off of rays with primitives $(k,1)$ and $(k',1)$}. Therefore from these we can extract two equations for the primitive $\vartheta$ functions and thus an embedding of the mirror family into affine space.}

  \section{Counting sections of rational elliptic and $K3$ surfaces}

  
\section{The unravelled threefold}

Let $\rho: S \rightarrow \mathbb{P}^1$ be a rational elliptic surface with fibre class $[F]$. Here we will describe a novel approach to calculating the virtual class \Lawrence{of the moduli space of} rational unmarked curves in a class $\Lawrence{\beta =} [C]$ with $[C].K_S = -2$. \Lawrence{From now on $\beta$ will always denote such a class.}

\subsection{The ramified fibres}

Our goal will be to study $\StableModuli_{0,0} (S, [C])$ via its image under $ram$ and therefore understanding the fibres of $ram \circ \pi$ will be very useful. Explicitly the map $ram$ takes a stable map, contracts out any contracted components of the curve and any components mapping into fibres and then classifies the remaining cover of $\mathbb{P}^1$ based on where it ramifies. Let $f:C \rightarrow S$ be a stable map stable map \Lawrence{corresponding to a closed point of} $\StableModuli_{0,0} (S, [C])$. There are three types of components of $C$: contracted components, components covering fibres of $\rho$ and components which cover the base $\mathbb{P}^1$ of the fibration. We will assign one of the two labels $\fib$ and $\tr$ to each component. Components covering fibres will be labelled with $\fib$ whilst components covering the base will be labelled $\tr$. Any contracted component which is part of a chain linking two components covering the base will be labelled $\tr$ and all the remaining contracted components will be labelled $\fib$. This divides $C$ into two pieces based on these labels, which we will denote by $C_{\tr}$ and $C_{\fib}$. The component $C_{\tr}$ consists either of a single component mapping as a double cover to the base $\mathbb{P}^1$ or as a chain of contracted components joining two components mapping isomorphically to the base or is the union of two disconnected components. Note that $C_{\fib}$ may very well be disconnected. We write $C_{\contr}$ for the curve obtained from $C_{\tr}$ by contracting out the contracted components. With this notation we can attempt to understand the structures of the fibres of $ram\circ \pi$. The following lemma shows that inside such a fibre $C_{\tr}$ is rigid, only the component $C_{\fib}$ deforms.

\begin{theorem}[Fibres of $ram\circ \pi$ vary]
\label{thm:varying-fibres}
The component $f(C_{\tr})$ is locally constant as a subscheme of
$S$ along fibres of $ram\circ\pi$. Hence 
the image under $ram\circ \pi$ of a family of curves in $\StableModuli_{0,0} (S, [C])$ with $f(C_{\tr})$ varying as a subscheme of $S$ is positive dimensional.

\end{theorem}

\begin{proof}

Let $B \xleftarrow{\pi} C \xrightarrow{f} S$ be a family of genus zero unmarked stable maps contained inside a fibre of $ram$. First we \Lawrence{treat} the case where $C_{\tr}$ is irreducible.

Fix a choice of section $E$ for the rational elliptic surface which we take to be the identity for the group law. Choose an \'etale $U\rightarrow B$ with dense
image such that $(C|_U)_{\tr}$ is isomorphic to $U \times \mathbb{P}^1$. 
Thus we may identify fibres over each $u \in U$ as follows, using the assumption
that $C_{tr}$ is irreducible. After passing to another \'etale cover we may identify $(C|U)_{tr}$ with the pull-back of the universal curve over $\MTwoOne$. By assumption the image in $\MTwoOne$ is a point $P$. Therefore for each $u \in U$ we may identify the curve $C_{u,tr}$ with the same double cover of $\PP^1$.

\begin{figure}\label{identificationoffibres}
\end{figure}

\Lawrence{Fix a choice of closed point $u \in U$ and so a stable map to $S$ and for each $v \in U$ we identify the two covers via the above isomorphism as described in Figure~\ref{identificationoffibres}} and consider $f(C_{tr,v}) - f(C_{tr, u})$. For $u = v$ this defines a double cover of the section $E$, while if $f(C_{\tr})$ is varying then for $u \neq v$ this is not a double cover of that section. By rigidity of sections this does not occur.

Now if $C_{\tr}$ is reducible, $C_{\contr} = C_1 \cup C_2$ then both $C_1$ and $C_2$ are sections and admit no deformations. In particular this shows that $f$ is constant on these fibres.

\end{proof}

The virtual dimension of the moduli space of stable maps
$\StableModuli_{g,n}(S,\beta)$ is given by the formula $(1-g)(dim \: S - 3) +n - K_S.\beta$,
which gives a virtual dimension of $1$ for $\StableModuli_{0,0}(S,[C])$. \Lawrence{Let us assume that $C_{tr}$ is connected. We write $C_{contr}$ for the curve obtained by contracting out contracted components of $C_{tr}$ and note that this curve carries a natural map to $S$, making it into a stable map. Applying the above formula we see that $C_{contr}$ also moves in an at least one dimensional family.}

  Thus there are three types of irreducible components of $\StableModuli_{0,0} (S, \beta)$, \emph{point components} whose image is dimension zero, \emph{curve components} whose image is dimension one and \emph{bubble components} which surject onto $\Hilb \cong \mathbb{P}^2$. Schematically the types of curves which can occur in each component is contained in Figure~\ref{fig:animals}. We can draw representative curves of each type in the class 
$E_1 + E_2 + 3F$ for sections $E_1, E_2$ satisfying $E_1=E_2$, $E_1\cdot E_2=1$
or $E_1\cdot E_2=0$ in the three cases depicted. 
The dashed line in the first figure shows that this is a double cover. We will later prove that these are representative of the general case.

\begin{figure}
\caption{Bubble}
\label{fig:gull}

\caption{Curve}
\label{fig:tiger}

\caption{Point}
\label{fig:mouse}
\end{figure}

\begin{lemma}[The structure curves over point components]

Let $\mathcal{M}$ be a point component. Then the restriction of the universal curve to this component decomposes as two constant components each mapping to potentially distinct sections joined by a tree of curves containing at least one component mapping to a fibre.

\end{lemma}

\begin{proof}

  If the transverse component $C_{\tr}$ is smooth and connected then \Lawrence{we have seen that we can deform it as} a double cover of $\mathbb{P}^1$. This leaves how to glue the component $C_{\fib}$ to this deformation. Suppose that $C_{\fib}$ meets $C_{\tr}$ in $P_1, \ldots , P_n$. \Lawrence{If none of these are ramification points of the map $C_{\tr} \rightarrow \PP^1$ then this deformation gives unique deformations of the $P_i$. If not then take the double cover $\AA^1 \rightarrow \AA^1$ given by $z \mapsto z^2$. Over this family there is a unique deformation given by choosing a branch to move along}. This allows us to lift the glueing data to the entire deformation contradicting Lemma \ref{thm:varying-fibres} and the definition of a point component. Therefore the universal curve must be generically reducible containing two distinct sections. The same argument applies if $C_{\tr}$ is the union of two sections meeting at a point.

Now suppose that $C_{\tr}$ is connected but not smooth hence of the form two sections joined by a chain of contracted components. Then by stability each contracted component in the chain must have an attached component mapping to a rational fibre.

\end{proof}

The point components will not really appear in our calculations, as they are not generic enough to appear in our main construction and do not support the correct log structure to count towards log Gromov Witten invariants. The bubble components however will certainly contribute and controlling this contribution will be key to our construction.

\begin{theorem}[Curve and bubble components are disjoint]

Let $\mathcal{M}$ be a bubble component. Then $\mathcal{M}$ does not meet any curve components of the moduli space. 

\end{theorem}

\begin{proof}

We will begin by showing $f(C_{\tr})$ does not vary for any stable map in this space, a double cover of some section $\sigma$. We do this by first studying the restriction of the family to the boundary of the moduli space. Let $\partial \Hilb$ denote the conic of degenerate double covers inside $\mathcal{H}ilb^2 (\mathbb{P}^1) \cong \mathbb{P}^2/C_2$. Take a point $P \in \partial \Hilb$ such that the ramification point of the corresponding double cover $C_{\tr}\rightarrow \mathbb{P}^1$ is distinct from the images of any of the rational fibres of $S$. In this case, $C_{\tr}$ is reducible, the union of two sections meeting at a point which does not lie on one of the singular fibres and admits deformations which remain reducible. Sections however are rigid and so this can only happen if one has a double cover of a single section, $\sigma$.

Now suppose a curve component intersects $\mathcal{M}$. In this case, one
can find\Lawrence{ the spectrum of a DVR $B = \Spec R$} with closed point $b_{0}$ and generic point $b_{gen}$ and
a morphism $cl:B\rightarrow \StableModuli_{0,0}(S,[C])$ such that
$cl(b_0)\in \mathcal{M}$, $cl(b_{gen})\not\in\mathcal{M}$ and $cl(b_{gen})$
lies in a curve
component. Let
$\pi: C \rightarrow B$ be the pullback of the universal curve and $f: C \rightarrow S$ the corresponding stable 
map.

In fact, it is enough to show the following.
First, replace $C$ by removing the closures of irreducible components
of $C_{gen}$ mapping into the rational fibres of $S$ \Lawrence{and taking the closure inside $C$}. Thus we may assume
that $(C_{gen})_{\tr}=C_{gen}$. If we can show that $(C_0)_{\tr}=C_0$, then
as necessarily $f_*([C_0])=2[\sigma]$ and $\sigma^2=-1$, we see
that $f(C_{gen})\subseteq \sigma$ also. Thus $f$ yields a double cover
of $\sigma$, and hence $cl(b_{gen})$ in fact lies in a bubble component.

Our goal will be to decorate $C_{gen}$ with the structure of a log stable map. Then since the moduli space of log curves is proper the family $B$ admits a unique completion, and this completion must be a base-change of the classical stable map. But we will now see that results of~\cite{TheTropicalVertex} lead to
a contradiction. To this end we take the divisorial log structure on $S$ coming from the union 
$\cup F_i$ of all singular fibres.

To give a log structure on $C_{gen}$ \Lawrence{we need to control the location of the singularities. Therefore} we first exclude the case
that $C_{gen}$ has any nodes on the transverse component. Indeed, \Lawrence{if $C_{gen}$ possesses a node then} it is
reducible and the two components map to sections of $S\rightarrow \mathbb{P}^1$.
But then the family is constant since sections admit no deformations.
Thus we assume $f_{gen}^{-1}(\cup F_i)$ is a set of smooth points of $C_{gen}$
and we give $C_{gen}$ the divisorial log structure given by this divisor.
We then have a log morphism $C_{gen}\rightarrow S$.

Now we know that the generic fibre has a log structure and hence there is a unique choice of limit of stable log maps, possibly after passing to a branched
cover of $B$, see \cite{LogarithmicGromovWittenInvariants}, Theorem 4.1. We now claim that there are no log stable curves with components mapping into the fibre $F_0$. This is a consequence \Lawrence{of the proof of} Proposition 4.3 of~\cite{TheTropicalVertex}. This shows that $C_{gen}$ cannot degenerate to one of these curves. Thus we conclude all curve components are disjoint from $\mathcal{M}$.

\end{proof}

\begin{theorem}[A fibration of bubble components]

Let $\mathcal{M}$ be a bubble component. The map $\mathcal{M} \rightarrow \Hilb$ fibres $\mathcal{M}$ by moduli spaces of stable maps to (potentially singular) $K3$ surfaces. This fibration is trivial away from a codimension one set.
  
\end{theorem}

\begin{proof}

One might think that the fibration should be trivial as they correspond to different parametrisations of the rational fibres. This is only true generically but fortunately we can describe the locus where it does not hold. Let $R$ denote the locus inside $\Hilb$ of stable maps ramifying over the images of the $F_i$. Away from $R$ the fibration will be trivial since the only deformations are deformations of the rational tails.

Let $P \in \Hilb \setminus R \cup \partial \Hilb$ be a smooth double cover not ramifying over one of the rational fibres. Let $f: C \rightarrow S$ and $\pi: C \rightarrow B$ be a curve in $ram^{-1} P$ with $B$ an infinitesimal extension over $k[t]/t^2$. Let $B_0, C_0$ denote the reduced structure. We wish to show that this deformation of $C_0$ does not smooth any of the nodes between $C_{0,tr}$ and the other components. To do this we construct a moduli space of stable maps to a $K3$ surface where none of these smoothings exists and compare the obstruction theories. Take the following diagram
\[
\xymatrix@C=30pt
{
K\ar[r]^d\ar[d]&S\ar[d]^{\rho}\\
C_{0,tr}\ar[r]_{\rho\circ f}&\mathbb{P}^1
}
\]

This defines a smooth elliptically fibred $K3$ surface $K$ (here we are using the assumption that the curve does not ramify over the images of any of the rational fibres). $C_0$ maps to both $C_{0, tr}$ and $S$ and hence maps to $K$ as a section together with a collection of covers of the rational fibres. This shows that the fibres of $ram$ are isomorphic to moduli spaces of stable maps to a $K3$ surface. \Lawrence{In particular} by the beginning of section 5 ~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} there are no infinitesimal deformations smoothing the nodes lying on $C_{0,tr}$ since there are none on $K$. In particular this shows that over this set the fibration is trivial.

We now turn our attention to the boundary $\partial \Hilb$. We will show
that if $P\in\partial\Hilb$, then no map $\Spec k[\epsilon]/(\epsilon^2)
\rightarrow ram^{-1}(P)$ produces an infinitesimal deformation of stable
maps which
smooths any nodes between $C_{\tr}$ and $C_{\fib}$. Since $\partial\Hilb$ meets every codimension one locus in $\Hilb$ this would show that the set of points on which at least one of the deformations smooths these nodes is codimension at least two. Take $P \in \partial \Hilb$ and $f: C \rightarrow S$, $\pi:C \rightarrow B$ an infinitesimal deformation of a point $\pi: C_0 \rightarrow B_0$, $f_0: C_0 \rightarrow S$ in $ram^{-1}P$. Then $C_{0}$ contains a chain of rational curves connecting two components $D_1$ and $D_2$ which are mapped to sections of $S$. Since the deformation remains within the fibre it fails to smooth at least one node in this chain. Removing this node disconnects the domain curve $C_0$ into two components $C_a$ and $C_b$ and the deformation restricts to each of these. But $C_a$ and $C_b$ are sections together with rational tails, hence deformations of them fail to smooth any of the nodes connecting the rational tails to the section by the discussion of Section 5 of ~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms}. Pulling back to the curve $C_0$ we obtain the desired statement.

Finally we claim that the fibres over any point in $\Hilb \setminus R$ are isomorphic. This is trivial from the above description. All these spaces are just moduli spaces of covers of the rational tails. It is worth saying that it is not true that this fibration is globally trivial. We can construct examples where the fibres of $ram$ over $R \cap \partial \Hilb$ are much higher dimension than the surrounding fibres since there could be more moduli in the way we connect the two transverse components together and such examples exist for three sheeted covers of one of the rational tails.

\end{proof}

\subsection{The unravelled threefold}

We have seen that the map $ram$ from \Lawrence{$\StableModuli_{0,0} (\beta, S)$} to $\MTwoOne\cong[\mathcal{H}ilb^2 (\mathbb{P}^1)/C_2]$ which sends a stable map to the two ramification points \Lawrence{classifies where the double cover ramifies, in other words it classifies those fibres to which $C_{tr}$ is tangent at a single point}, and we now apply it to count the desired curves. Classically one can construct a $K3$ surface by taking the base change of a rational elliptic surface by a double cover of the base $\mathbb{P}^1$. Fortunately we have a large number of double covers of the base at hand in the corresponding double covers $C_{\contr}\rightarrow \PP^1$ given
by the curves we wish to count. Take such a curve $C$ and consider the following fibre product

\centerline{\xymatrix{K \ar[r]^{d} \ar[d] & S \ar[d]^{\rho} \\ C_{\contr} \ar[r]^{\rho \circ f} & \mathbb{P}^1}}

This defines a (possibly singular) $K3$ surface $K$. The curve $C$ maps to $S$ by the map $f$ and to $C_{\contr}$ by contracting out the components contained inside fibres. Thus there is a canonical map $\tilde{f}: C \rightarrow K$, the ``unravelling'' of the curve $C$. On the other hand any stable map in the class $\tilde{f}_* ([C])$ pushes forward to one in the class $f_*[C]$ on $S$. We will attempt to construct a threefold which detects these possible liftings.

\begin{construction}

  The moduli space we are considering provides us with a collection of components mapping to $\Hilb$. Of these we will be interested in those whose image is at least dimension one so we should search for a curve inside $\Hilb$. We construct this as follows. Let $D$ be a generic tri-degree $(1,1,2)$ surface inside $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ with projection maps $\pi_L$, $\pi _M$ and $\pi_R$ to the respective factors. We view this as a family of curves over $\mathbb{P}^1$ via the projection \Lawrence{$\pi_L$}. Taken this way each fibre is a bi-degree $(1,2)$ hypersurface inside $\mathbb{P}^1 \times\mathbb{P}^1$, so via $\pi_L$ the graph of a double cover of $\mathbb{P}^1$ by $\mathbb{P}^1$. This produces a morphism $\PP^1 \rightarrow \MTwoOne$ such that the family $\pi_L : D \rightarrow \PP^1$ is the pullback of the universal family of double covers. We wish to know that  \Lawrence{the trivial $C_2$ action on the Hilbert scheme $\Hilb$} is \Lawrence{intertwined with the involution of $D$} swapping the two sheets of the double cover.

  \begin{lemma}

    With the above notation the family $[D/C_2] \rightarrow [\PP^1/C_2]$ is the pull-back of the universal family of double covers of $\PP^1$ over the base.

  \end{lemma}

  \begin{proof}

    By construction the family $D$ is the pull-back of the universal family of
double covers to $\PP^1$. Therefore we have a diagram

    \[\begin{tikzpicture}
    \node (P1) at (0,0) {$\PP^1$};
    \node (M21) at (4,0) {$\MTwoOne$};
    \node (P1/2) at (2,1) {$[\PP^1/C_2]$};
    \node (D) at (0,2) {$D$};
    \node (C) at (4,2) {$\mathcal{C}$};
    \node (D/2) at (2,3) {$[D/C_2]$};
    \draw [->] (P1) -- (M21);
    \draw [->] (P1) -- (P1/2);
    \draw [->] (P1/2) -- (M21);
    \draw [->] (D) -- (P1);
    \draw [->] (C) -- (M21);
    \draw [->] (D/2) -- (P1/2);
    \draw [->] (D) -- (C);
    \draw [->] (D) -- (D/2);
    \draw [->] (D/2) -- (C);
    \end{tikzpicture}\]
    where the front face is a pullback square. Now $\PP^1$ is simply connected, hence there is only the trivial $C_2$ torsor over $\PP^1$. This shows that $\PP^1 \rightarrow [\PP^1/C_2]$ has a chart given by $\PP^1 \coprod \PP^1 \rightarrow \PP^1$. Taking the pull-back of $[D/C_2]$ to this and using that $D$ is the universal curve over $\PP^1$ we obtain the desired result.
  \end{proof}
\end{construction}

Let us explore some general features of this via an example.

\begin{example}

  Let us study the family with defining equation
  \[ x_1x_2x_3^2 + x_1y_2x_3y_3 + 2y_1x_2x_3y_3 + y_1y_2y_3^2 \]
where the $i^{th}$ factor has coordinates $x_i$ and $y_i$. The fibre over $(x_2, y_2)$ is the graph of a double cover of $\mathbb{P}^1$, ramifying where the discriminant $(x_1y_2+2y_1x_2)^2 - 4 x_1x_2y_1y_2 = x_1^2y_2^2 + 4y_1^2x_2^2$ vanishes. This gives two values for $(x_1:y_1)$ for each value of $(x_2:y_2)$. The fibre is singular just when these two values coincide, so when the discriminant of this \Lawrence{quadratic} vanishes, $-16 x_2^2y_2^2$. Therefore there are two singular fibres each with multiplicity two.
  
\end{example}

In general this shows that there will be four singular fibres and it is easy to construct examples where these occur as distinct fibres. We have constructed here a collection of double covers of $\mathbb{P}^1$, so the base $\mathbb{P}^1$ should map to $\Hilb$. The degenerate fibres inside $\Hilb$ form a conic inside $\mathbb{P}^2$, so by counting intersection points we know that the image of this family is also a conic. Therefore to search for curve and bubble components of the moduli space we are lead naturally to study the following fibre product:
\[\centerline{\xymatrix{X \ar[r]^{d} \ar[d] & S \ar[d]^{\rho} \\ D \ar[r]^{\pi_L} & \mathbb{P}^1}}\]
We call the constructed threefold $X$ an \emph{unravelled threefold}. Such a \Lawrence{threefold} is the intersection of a $(3,1,0,0)$ hypersurface and a $(0,1,1,2)$ hypersurface inside $\mathbb{P}^2 \times (\mathbb{P}^1)^3$. The $(0,1,1,2)$ hypersurface we may take to be generic whilst the $(3,1,0,0)$ hypersurface depends on the choice of rational elliptic surface $S$. The first hypersurface gives rise to a choice of line $\iota: [\PP^1/C_2] \rightarrow \MTwoOne \cong [\PP^2/C_2]$. We will restrict our range of choices later in the discussion to reflect the surfaces we choose to work with. Let $\mathcal{M}$ be a component of the moduli space $\StableModuli_{0,0} (S, \Lawrence{f_*[C]})$ and consider the image of $\mathcal{M}$ under the morphism $ram:\StableModuli_{0,0}(S,\Lawrence{f_*[C]})\rightarrow \mathcal{H}ilb^2 (\mathbb{P}^1) \cong \mathbb{P}^2/C_2$. Whenever the image intersects the image of the family of double covers of $\PP^1$ one can lift the corresponding stable map to $X$. The existence of a lift suggests that $\StableModuli_{0,n} (X, \tilde{f}_* ([C])) \cong \StableModuli_{0,n} (S, \Lawrence{f_*[C]}) \times_{\mathcal{H}ilb ^2 \mathbb{P}^1} \mathbb{P}^1$ along the map $ram$. Soon we will see that this description induces a relation between the Gromov-Witten invariants of these two moduli spaces.

\section{Our approach}

In~\cite{AMirrorTheoremforToricCompleteIntersection} Givental gave an explicit method to calculate the Gromov-Witten invariants for classes on nef complete intersections in toric varieties. \Lawrence{This approach constructs two different cohomology valued functions, the $I$ and $J$ functions, whose coefficients are linearly independent solutions to two differential equations. These two equations are related by an invertible change of basis and the coefficients of each function span the entire solution space to their respective equations}. Hence there is a change of basis relating the two. Further Givental showed that this change of basis can be explicitly calculated by comparing coefficients. This change of basis is referred to in the literature as the \Lawrence{\emph{mirror map}}.

One could hope to use this to count the number of families of curves on $S$ in a class $[C]$. We will soon see such approaches require very delicate analysis of the virtual number of curves of given degree meeting certain tangency conditions. Without this one cannot reconstruct the log invariants. Our approach is to count the number of curves in an unravelled threefold $X$. The problem will be to relate the new moduli space of stable maps to $X$ to the moduli space of stable maps to $S$ and compare the obstruction theories. We will construct the series described by Givental and use them to bound the relative Gromov-Witten invariants on $S$. This final step will involve some hard analysis of the growth of certain series.

\subsection{Givental's techniques}

Let us begin with a toric variety $\mathbb{T}$ with codimension one strata $D_\rho$. Take a basis for the \Lawrence{rational} Chow ring $A^* (\mathbb{T})$, $\{H_i\}_1^n$, and a dual basis $\{H^i\}_1^n$ under cup product.

\begin{definition}

The $I_X$ and $J_X$ functions are defined for a smooth complete intersection $X \subset \mathbb{T}$. Suppose that $X$ is the common vanishing of sections of line bundles $\mathcal{L}_i$ with $-K_\mathbb{T} - \sum c_1(\mathcal{L}_i)$ nef. Then the $I_X$ function is defined by

\begin{align*}I_X (t_0, \ldots t_n) &= \\ e^{(t_0+\sum t_i H_i) / \hbar} Eul& (\oplus \mathcal{L}_i) \sum_\beta q^\beta \frac{\prod_i (\prod_{m=1} ^{\mathcal{L}_i (\beta)} (c_1 (\mathcal{L}_i) + m\hbar) \prod_\rho \prod_{m=-\infty}^{0} (D_\rho + m\hbar))}{\prod_\rho \prod_{m=-\infty}^{D_\rho \cap \beta} (D_\rho + m\hbar)} \end{align*}
where $q^{\beta} = e^{\sum t_i H_i \cap \beta}$ and the sum is over $\beta$ effective. This function is valued in $A^* (\mathbb{T})[\![t_0, \ldots t_n, \hbar\Lawrence{^{-1}}]\!]$. The function $J_X$ is a generating function for the gravitational correlators of $X$ and is valued in $A^* (X)[ \! [t_0, \ldots t_n, \hbar\Lawrence{^{-1}} ] \! ]$. It is defined by the formula
\[J_X (t_0 \ldots t_n) = e^{(t_0+\sum t_i H_i)/ \hbar} \left(1 + \sum_{\beta, i, k} \hbar^{-(k+1)} q^\beta \langle \tau_k \Lawrence{H}_a, 1 \rangle_{0,\beta} \Lawrence{H}^a \right)\]
where $\langle \tau_k \Lawrence{H}_a, 1 \rangle_{0,\beta}$ are the gravitational descendants, the sum is over positive $i, k$ and effective curve classes $\beta$.

\end{definition}

These two functions are not valued in the same ring so cannot be equal. Let $\iota: X \rightarrow \mathbb{T}$ be the inclusion. The main result of~\cite{AMirrorTheoremforToricCompleteIntersection} is an equality of generating functions $I_X(t_i) = \iota_* J_X (s_i)$  after some change of basis $t_i \mapsto s_i \in A^* (\mathbb{T})[\! [t_0, \ldots t_n ]\!]$.

\begin{theorem}[Mirror symmetry for Givental's $I$ and $J$ functions]

The functions $I_X$ and $\iota _* J_X$ are equal up to a homogeneous change of variables of the form $t_0 \mapsto t_0 + f_0 (z^{\beta}) \hbar + h (z^\beta)$, $t_i \mapsto t_i + f_i (z^\beta)$ where the $f_i$ and $h$ are homogeneous power series of weights $\deg f_i = 0$ and $\deg h = 1$ with degrees of
the variables being given by $c_1 (\mathbb{T}) - c_1 (\sum \mathcal{L}_i) = \sum deg (z^{H_i}) H_i$, $\deg \hbar = 1$, $\deg t_0=1$ and $\deg t_i=0$ for
$i>0$.

\begin{proof}

  See~\cite{AMirrorTheoremforToricCompleteIntersection} Theorem 0.1.
  
\end{proof}

\end{theorem}

To explore these techniques we will begin by applying them to the rational elliptic surface itself. The surface $S$ is a $(3,1)$ hypersurface in $\mathbb{T} = \mathbb{P}^2 \times \mathbb{P}^1$ and we take generators $H_1, H_2$ of $A^1 (\mathbb{T})$ with dual coordinates $t_1, t_2$. 

Writing out the formula for $I_S$ we have

\begin{align*}
I_S &= e ^{\sum t_i H_i / \hbar} (3 H_1 + H_2) \sum _{a,b \in \ZZ^2} z^{a t_1 + b t_2} \frac{ \prod\Lawrence{_{m=1}}^{3a + b} (3H_1 + H_2 + m \hbar)}{\prod\Lawrence{_{m=1}}^a (H_1 + m \hbar)^3 \prod\Lawrence{_{m=1}} ^b (H_2 + m \hbar)^2}\\
& = 1 + 6 z^{t_1} + 90 z^{2t_1} + z^{t_2} \hbar^{-1} + 1680 z^{3t_1} + 24 z^{t_1} z^{t_2} \hbar^{-1} + 15 z^{t_1} H_1 \hbar^{-1} + \\
& \quad 11 z^{t_1} H_2 \hbar^{-1} + 34650 z^{4t_1} + 630 z^{2t_1} z^{t_2} \hbar^{-1} + 513/2 z^{2t_1} H_1 \hbar^{-1} + \ldots
\end{align*}

Then the change of coordinates can be found by comparing coefficients. A good explanation of this process may be found in~\cite{MirrorSymmetryAndAlgebraicGeometry}. Performing this the new coordinates are given by

\begin{align*}
z^{t_0'} &= z^{t_2} + 18 z^{t_1 + t_2} + 432 z^{2t_1 + t_2} + 10908 z^{3t_1 + t_2} + 281232 z^{4t_1 + t_2} \\
z^{t_1'} &= z^{t_1} + 15 z^{2t_1} + 279 z^{3t_1} + 5729 z^{4t_1} + 124554 z^{5t_1} + 1799811 z^{6t_1} + 56647267/2 z^{7t_1} + \ldots \\
z^{t_2'} &= z^{t_2} + 11 z^{t_1 + t_2} + 215 z^{2t_1 + t_2} + 4757 z^{3t_1 + t_2} + 111212 z^{4t_1 + t_2} + 7079849/5 z^{5t_1 + t_2} + \ldots
\end{align*}

Inverting this change of basis we obtain the following expression for the $J_S$ function

\begin{align*}
J_S = {} & z^{t'_0 + \sum t'_i H_i / \hbar} (3H_1 + H_2) (1 + 9 z^{t'_2} H_1^2 \hbar^{-2} + 27 z^{t'_1 + 2t'_2} H_1 \hbar^{-2}\\
& + 9 z^{t'_1 + 2t'_2} H_2 \hbar^{-2} + 36 z^{t'_1 + t'_2} H_1^2 \hbar^{-2} + 
 36 z^{t'_1 + t'_2} H_1 H_2 \hbar^{-2}\\
{} & + 378 z^{2t'_1 + 2t'_2} H_1 \hbar^{-2} + 126 z^{2t'_1 + 2 t'_2} H_2 \hbar^{-2} + 126 z^{2t'_1 + t'_2} H_1^2 \hbar^{-2} + \cdots)
\end{align*}

\Lawrence{Let us try to understand heuristic geometric interpretations for these coefficients.} If we view $S$ as the blow up of $\mathbb{P}^2$ in the nine intersection points of two cubics we immediately see the nine base points in this series as the coefficient of $H_1^2 \hbar ^{-2}$. The term $9z^{t_1' + 2t_2'} H_2 \hbar^{-2}$ 
corresponds to the nine families of bisections in $|H - E_i|$ and in each such family precisely one curve passes through a point. However, we can't see how
many of these curves are tangent to a given fibre directly from the above
expression, but it is easy to see that four of the curves in each family 
are indeed tangent to a fibre, giving $36$ such $\AA^1$ curves.
Similarly the term $126z^{2t_1'+t_2'}H_1^2\hbar^{-2}$ arises from 
the 126 bisections of the form $2H - E_i - E_j - E_k - E_l$. In each family precisely four curves meet the desired tangency conditions and so we expect to see a factor of 504 appearing in the $\AA^1$ invariants.

As the degree grows the presumed correspondence with the log invariants becomes intractable. To find the next term we would have to answer the question ``given a rational cubic $E$ passing through seven points $P_i$ what is the virtual contribution from rational cubics pass through those points meeting $E$ in one other point of tangency order two?''.

\subsection{Formulae for the invariants}

We will calculate using the above techniques generating functions for Gromov-Witten invariants on an unravelled three-fold which we expect to compare favourably with the invariants of the original surface $S$. We begin by relating the underlying moduli spaces.
The following lemma is the entire motivation behind defining this threefold. 

\begin{lemma}[The unravelled threefold lifts curves]
Let $X \subset \mathbb{P}^2 \times (\mathbb{P}^1)^3$ be an unravelled three-fold, $\pi_M$ the projection to the third factor, a copy of $\mathbb{P}^1$,
and let
\[
d^{-1} \beta = \{ \alpha \in A_1 (X) \mid \pi _{M, *} \alpha = 0, d_* \alpha = \beta\}
\]
be the set of effective classes on $X$ contained inside a fibre of $\pi_M$  
and such that $d_* \alpha = \beta$. \Lawrence{We write $\StableModuli(X,d^{-1}(\beta))$ for the union of moduli spaces $\bigcup _{\alpha \in d^{-1} \beta} \StableModuli_{0,0} (X, \alpha)$}. Then consider the following diagram
\[
\xymatrix{\StableModuli(X,d^{-1}(\beta)) \ar@/^2pc/[rrrd]^{d_*} \ar@/_2pc/[rdd]^{ pr _X} \ar[rd]^{\psi} & & & \\
 & \mathbb{P}^1 \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)  \ar[rr]^{\iota} \ar[d] ^{pr _S } & & \StableModuli_{0,0} (S, \beta) \ar[d]^{ram} \\
 & \mathbb{P}^1 \ar[rr]^{\iota} & & \MTwoOne}
\]
where $pr_X$ associates to a stable map $f:C/B\rightarrow X$ the morphism
$B\rightarrow \mathbb{P}^1$ given by the ramification of the composed
map $\rho\circ d\circ f$, where $\rho:S\rightarrow \mathbb{P}^1$ is the
elliptic fibration.
Then the map $\psi$ is an isomorphism, and thus a two to one cover of $[\mathbb{P}^1/C_2] \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta) \subset \StableModuli_{0,0} (S, \beta)$.
\end{lemma}

\begin{proof}
By genericity we may assume that the choice of map $\mathbb{P}^1 \rightarrow \Hilb$ avoids all point components and is transverse to the images of all curve components. We use the universal property of products to construct an inverse.

Let $S \xleftarrow{f} C \xrightarrow{\pi} B$ be a family of stable maps inside $\mathbb{P}^1 \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)$. Since $D$ is the restriction of the universal curve on $\MTwoOne$ to $\PP^1$ there is an induced morphism $C \rightarrow D$, and hence by properties of fibre products a unique morphism $C \rightarrow X$. This constructs an inverse to $\psi$, which we call it $\phi$.

We now need to prove that the two compositions $\phi \psi$ and $\psi \phi$ are in fact the identity maps. To show that $\psi \cdot \phi$ is the identity we show that $\iota \cdot \psi	 \cdot \phi = \iota$ and $pr_S \cdot \psi \cdot \phi = pr_S$. These are both automatic from the construction of $\phi$. Indeed, $\iota \cdot \psi \cdot \phi = d_* \cdot \phi = \iota$ and $pr_S \cdot \psi \cdot \phi = pr_X \cdot \phi = pr_S$.

Finally we show the composition $\phi \cdot \psi$ is the identity. Let $f: C \rightarrow X$ be a stable map in $\bigcup _{\alpha \in d^{-1} \beta} \bar{\mathcal{M}}_{0,0} (X, \alpha)$. The image $\psi (f:C \rightarrow X)$ is given by pushforward to $S$ and has a unique lift. But $f: C \rightarrow X$ is such a lift and so this composition too is the identity map.

\end{proof}

Since any degree two map from $[\mathbb{P}^1/C_2]$ to $[\mathbb{P}^2/C_2]$ is a regular embedding
there exists a Gysin map for the inclusion $\iota: \StableModuli (X, d^{-1}\beta) \rightarrow \StableModuli (S, \beta)$ covering $\mathbb{P}^1 \rightarrow \Hilb \cong [\mathbb{P}^2/C_2]$. We follow~\cite{VirtualPushforward} in constructing a compatibility datum between the two spaces.

\begin{theorem}[Compatibility of Gromov-Witten invariants betwen $S$ and $X$]

  For a generic choice of unravelling $d: X \rightarrow S$ there is an equality between the virtual classes associated to the obstruction theories on $\StableModuli_{0,0}(X, d^{-1} \beta)$ and $\StableModuli_{0,0} (S, \beta)$ under the Gysin map $\iota ^!$:
\[[\StableModuli_{0,0}(X, d^{-1} \beta)]^{vir} = 2 \iota^! [\StableModuli_{0,0} (S, \beta)] ^{vir}\]

\label{thm:SandXcomparisons}

\end{theorem}

\begin{proof}

We choose a generic unravelling which does not meet any point components and is transverse to any curve components. We have the following commutative diagram
\[
\xymatrix{X \ar[r]^{d} & S \\
\mathcal{C}_{0,0} (X,d^{-1} \beta) \ar [r]^{\iota} \ar[d]^{\pi_X} \ar [u]^{f_X} & \mathcal{C}_{0,0} (S, \beta) \ar[d]^{\pi_S} \ar [u]^{f_S} \\
\StableModuli_{0,0} (X,d^{-1} \beta) \ar [r]^{\iota} \ar[d]^{ram}  & \StableModuli_{0,0} (S, \beta) \ar[d]^{ram}\\
\mathbb{P}^1 \ar [r]^{\iota}& \MTwoOne
}
\]

The desired compatibility datum is a morphism of distinguished triangles
\[
\xymatrix{\cdots \ar[r] & (\iota^* R \pi_{S, *} f_S^* \Theta_{S/k})^{\vee} \ar[r]^{\phi} \ar[d]^{\gamma_1} & (R \pi_{X, *} f_X^* \Theta_{X/k})^{\vee} \ar[r] \ar[d]^{\gamma_2} & ram^* L_{\mathbb{P}^1 / \mathbb{P}^2} \ar[r] \ar[d]^{\gamma_3} & \cdots\\
\cdots\ar[r] & \iota^* L_{\StableModuli_{0,0}(S, \beta)/\mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)} \ar[r] & \cdots}
\]

To prove this we must show that the cone over $(\iota^* R \pi_{S, *} f_S^* \Theta_{S/k})^{\vee} \rightarrow (R \pi_{X, *} f_X^* \Theta_{X/k})^{\vee}$ is quasi-isomorphic to $ram^* L_{\mathbb{P}^1 / \mathbb{P}^2}$. First let us show that the map $\gamma_3 : ram ^* L_{\mathbb{P}^1/\Hilb} \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)}$ is an isomorphism. The morphism $\mathbb{P}^1 \rightarrow \Hilb$ is a local complete intersection morphism, defined by a single equation $F$ of degree two in $k[X,Y,Z]$. So in particular, $F$ forms a length one regular sequence
in $k[X,Y,Z]$. By genericity we may take the images of $\mathbb{P}^1$ and the curve components of $\StableModuli_{0,0}(S, \beta)$ to be transverse inside $\Hilb$. Therefore the sequence $F$ remains regular when pulled back to $\StableModuli_{0,0}(S, \beta)$ and $\StableModuli_{0,0}(X, d^{-1} \beta)$ is just the vanishing locus of the pullback of $F$. In the case of a regular embedding
there is an explicit formula for the cotangent complex, namely it is $\mathcal{I}/ \mathcal{I}^2[1]$ where $\mathcal{I}$ is the defining sheaf of ideals. Both source and target of $\gamma_3$ is $ram^*(\langle F \rangle / \langle F \rangle^2[1])$ and $\gamma_3$ is the identity.

Now taking the cone \Lawrence{over $\phi$} there is a morphism of triangles of the following form
\[
\centering
\xymatrix{... \ar[r] &(\iota^* R \pi_{S, *} f_S^* \Theta_{S/k})^{\vee} \ar[r] \ar[d] &(R \pi_{X, *} f_X^* \Theta_{X/k})^{\vee} \ar[r] \ar[d] & \Cone \ar[r] \ar[d]^{\delta} & ...\\
  ... \ar[r] & \iota^* L_{\StableModuli_{0,0}(S, \beta)/\mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \mathfrak{M}} \ar[r] & L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)} \ar[r] & ...}
\]
\begin{equation}
\label{diag:bigone}
\end{equation}
To begin with we need a result allowing us to commute pull-back and push-forward. We claim that there is an isomorphism
\[\iota^* R \pi _{S \; *} f_S^* \Theta_{S/k} \cong R \pi_{X \; *} \iota ^* f_S^* \Theta_{S/k} \cong R \pi_{X \; *} f_X^* d^* \Theta_{S/k}\]
The second isomorphism is automatic, so we focus on the first. By~\cite{EGA4}, III.2, Corollary 6.9.9 and the fact that families of curves are flat, cohomology and base change commute. Applying~\cite{GromovWittenInvariantsInAlgebraicGeometry}, Proposition 5, $f^* _S \Theta_{S/k}$ admits a resolution by a two term complex of vector bundles $b: F^{-1} \rightarrow F^0$ such that $\pi_* F^i = 0$ and $R^1 \pi_* F^i$ is a vector bundle. 
Now $\pi_* f^* \Theta_{S/k}$ is just the cokernel of $R^1 \pi _* (F^0) \rightarrow R^1 \pi _* (F^1)$. But we have seen that cohomology and base change commute, thus we have an isomorphism $\iota^* R^1 \pi _* (F^i) \cong R^1 \pi_* (\iota ^* F^i)$ and $\iota^* R^0 \pi_* (F^i) = 0$. Now this is enough to show that there is the desired equality.

This allows us to calculate $\mathcal{C}one$ from the distinguished triangle on $X$
\begin{equation}
\label{exact-sequence-upstairs} \rightarrow \Theta _{X/k} \rightarrow d^* \Theta_{S/k} \rightarrow \MacBeth \rightarrow 
\end{equation}
The complex $\MacBeth$ is a line bundle in degree minus one and supported only on the critical locus of $X$ over $S$ in degree zero. The fibres of the map $d: X \rightarrow S$ are isomorphic to the fibres of $d:D \rightarrow \PP^1$. By an elementary calculation this is only singular over two points of $\PP^1$ where the corresponding double cover degenerates. The choice of $D$ was arbitrary so we can arrange that these two singular points do not occur over the critical values of the map $S \rightarrow \PP^1$, so over the images of the singular fibres. This ensures that $H^0 (\MacBeth)$ is supported only on two elliptic fibres of the fibration $\rho: S \rightarrow \PP^1$.

Let us describe the pullback of \Lawrence{$\mathcal{D}$} to fibres of the universal curve of stable maps. Let $f: C \rightarrow X$ be a stable map. By necessity such a stable map factorises through a (possibly singular) $K3$ double cover of $S$. This shows that $L^0 f^* \MacBeth$ is supported only on a finite set since the universal curve can have no components mapping into an elliptic fibre. By applying the Grothendieck spectral sequence we therefore see that both $H^1 (R\pi_*Lf^*\MacBeth)$ and $H^{-1}((R\pi_*Lf^*\MacBeth)^\vee)$ vanish.

By pushing forward \eqref{exact-sequence-upstairs} and dualizing in the
derived category, we obtain a long exact sequence

\begin{align*}
0\rightarrow & H^{-1} ((R\pi_*f^* d^* \Theta _{S/k})^\vee)
 \rightarrow  H^{-1}((R\pi_* f^* \Theta_{X/k})^\vee)
 \rightarrow   H^{0}((R\pi_*Lf^*\MacBeth)^\vee)\\
 \rightarrow & H^{0}((R\pi_*f^* d^* \Theta _{S/k})^\vee) \rightarrow H^{0}((R\pi_* f^* \Theta_{X/k})^\vee)
 \rightarrow  H^{1}((R\pi_*Lf^*\MacBeth)^\vee) \rightarrow 0
\end{align*}
we see that $H^{1}((R\pi_*Lf^*\MacBeth)^\vee)$ vanishes by mapping via the canonical isomorphisms to the non-negative part of the cotangent exact sequence:
\[  \iota^* L_{\StableModuli_{0,0}(S, \beta)/\mathfrak{M}} \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \mathfrak{M}} \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta) } \rightarrow \]
and applying the 5-lemma. We now have an exact sequence
\[0\rightarrow H^{-1} ((R\pi_*f^*d^* \Theta _{S/k})^\vee) \rightarrow H^{-1}((R\pi_* f^* \Theta_{X/k})^\vee) \rightarrow H^{0}((R\pi_*Lf^*\MacBeth)^\vee) \rightarrow \]
\[ \rightarrow H^{0}((R\pi_*f^* d^* \Theta _{S/k})^\vee) \rightarrow H^{0}((R\pi_* f^* \Theta_{X/k})^\vee) \rightarrow 0\]
and by the 5-lemma $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)$ maps surjectively to $ram^* L_{\PP^1/\PP^2}$. 
Taking stalks at a generic point $\eta=\Spec K$ of the moduli space,
we may apply the Riemann-Roch theorem to the two tangent sheaf terms and using additivity of Euler characteristics we find that $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)_{\eta}$ has dimension one as a vector space over $K$. Now $(R\pi_*Lf^*\MacBeth)^\vee$ has cohomology supported in degrees minus one and zero. Let $E^\bullet$ be a resolution of $R\pi_*Lf^*\MacBeth$, which is quasi-isomorphic to the truncation $\tau_{\leq 0} E^\bullet$ defined in~\cite{StacksProject} Tag 0118. Taking a projective Cartan Eilenberg double resolution of this we obtain a resolution of $R\pi_*Lf^*\MacBeth$ by projectives concentrated in non-positive degree. Therefore after dualising $R\pi_*Lf^*\MacBeth^\vee$ has a resolution concentrated in non-negative degree. This shows that $H^0(R\pi_*Lf^*\MacBeth^\vee)$ embeds into a vector bundle, and hence is torsion free. \Lawrence{We prove an analogous result to the statement that a surjective map of line bundles is an isomorphism}. Take the morphism
\[\delta: H^{0}((R\pi_*Lf^*\MacBeth)^\vee) \rightarrow L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta)} \]
of the diagram \eqref{diag:bigone}.
After localising at a point $x$ we have an isomorphism 
\[ 
L_{\StableModuli_{0,0}(X, d^{-1} \beta) / \StableModuli_{0,0}(S, \beta), x}
\cong \mathcal{O}_{\StableModuli_{0,0}(X, d^{-1} \beta), x}
\]
 and we know that $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)$ is a rank one torsion free sheaf with $\delta$ is surjective. Therefore we have the following situation: a local ring $R$, a rank one torsion free $R$-module $M$ and a surjection $\delta:M \rightarrow R$. We claim that this implies that $M$ is isomorphic to $R$. Since $\gamma$ is surjective we can choose $m \in M$ mapping to $1 \in R$. Suppose that there were $n \in M$ not in the submodule of $M$ generated by $m$. The element $n$ maps to $f \in R$ and so at every generic point of $R$ one has $n = fm$. But $M$ is torsion free and so $n = fm$. Therefore $m$ generates $M$ freely, i.e., $R \cong M$ as $R$-modules. Therefore $H^{0}((R\pi_*Lf^*\MacBeth)^\vee)$ is isomorphic to $ram^* L_{\PP^1/\PP^2}$.

\Lawrence{The Gysin map $\iota^!$ produces a class on $[\mathbb{P}^1/C_2] \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)$. However the moduli space $\StableModuli_{0,0} (X, d^{-1} \beta)$ is a two to one \'etale cover of $[\mathbb{P}^1/C_2] \times_{\MTwoOne} \StableModuli_{0,0} (S, \beta)$. Therefore by ~\cite{VirtualPushforward} we obtain the formula
\[ 2 \iota^! [\StableModuli_{0,0} (S, \beta)]^{virt} = \iota_* [\StableModuli_{0,0} (X, d^{-1} \beta)]^{virt}.\]
We call this \emph{the comparison formula for bisections on $S$ and sections on $X$}.}

\end{proof}

\subsection{Log Geometry and Algebraic Geometry}

We will require a specific case of what is a general rule of thumb in log geometry: one can construct log invariants by manually restricting to the tangency conditions one desires and twisting the virtual class down by an appropriate line bundle. Let us begin with an easy case, relating the invariants of sections.

Here, we fix a rational elliptic surface $\rho:S\rightarrow \PP^1$ which
has an $I_n$ fibre $F_0$ for some $n$ over $0\in \PP^1$. 
Let $S^{\dagger}$ denote $S$ with the divisorial log structure coming
from $F_0\subset S$. \Lawrence{Recall that we write $\LogModuli (S^\dagger, \beta)$ for the moduli of log stable maps to $S^\dagger$ in a class $\beta$ tangent to the boundary at a single point with maximal tangency.}

\begin{theorem}[Log invariants of sections are restrictions of Gromov-Witten invariants]

Let \Lawrence{$E$} be a section of a rational elliptic surface $S$, and let $\beta = E + nF$. Then there is an inclusion of components $\iota:  \LogModuli (S^\dagger, \beta) \rightarrow \StableModuli_{0,0} (S, \beta) $ and this induces an equality of virtual classes $\iota^* [\StableModuli_{0,0} (S, \beta)] =  [ \LogModuli (S^\dagger, \beta) ] $.
\label{thm:LogEqualityForSections}

\begin{proof}

In ~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} the authors constructed the moduli space $\StableModuli_{0,0} (S, \beta)$: any curve is the union of the section $E$ together with different covers of the rational fibres. Since any curve in $\LogModuli(S, \beta)$ has a unique choice of marked point $\sigma$ the forgetful map is injective and since different components correspond only to different covers of the rational fibres this map is an inclusion of components.
Indeed by Proposition 4.3 of~\cite{TheTropicalVertex} the components of
$\LogModuli(S,\beta)$ are precisely those components of $\StableModuli_{0,0}
(S,\beta)$ where no component of the universal curve maps into the boundary.

Let us denote by $\mathfrak{M}_{0,0}$ the moduli Artin stack of prestable
genus zero curves and by $\mathfrak{M}^{\dagger}_{0,1}$ the moduli Artin
stack of all pre-stable genus one-pointed log curves. Recall that
the virtual fundamental class of $\StableModuli_{0,0}(S,\beta)$ is
defined via an obstruction theory relative to $\mathfrak{M}_{0,0}$
and recall
from \cite{LogarithmicGromovWittenInvariants} that the virtual
fundamental class of the moduli space $\LogModuli(S,\beta)$ is defined
by an obstruction theory relative to $\mathfrak{M}^{\dagger}_{0,1}$.

We consider the product $\StableModuli (S, \beta)_{0,1} \times_S F_0$ and restrict to those components of
$\StableModuli (S, \beta)_{0,1}$ where no component of the domain curve maps into $F_0$. We have inclusions $\LogModuli(S, \beta) \subset \StableModuli (S, \beta)_{0,1} \times_S F_0 \cong \StableModuli (S, \beta)_{0,0}$. Furthermore there is a virtual class on $\StableModuli (S, \beta)_{0,1} \times_S F_0$ relative to $\mathfrak{M}_{0,1}$ defined by the obstruction bundle induced by the pull-back
of the kernel of the natural morphism $\Theta_{S/k} \rightarrow{\mathcal N}_{F_0/S}$. Note that this kernel is rank two bundle away from the singular
points of $F_0$, and no curve in the moduli space passes through these
singular points.
Sections of the pull-back of this bundle then correspond to infinitesimal
deformations that deform the marked point in a direction along $F_0$.
It is a folklore result exposited in remark A.5 of the appendix of~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} that the virtual class this defines is equal to the pullback of the virtual class on $\StableModuli (S, \beta)_{0,0}$. \Lawrence{Their result holds so long as $F_0$ and $S$ are both smooth. This is not the case and we instead consider $\Spec k \in \PP^1$ which indeed does have smooth source and target. Then the Gysin map of $F_0$ in $S$ is compatible with the Gysin map of a point in $\PP^1$ for which we do have the desired equality.}

It remains to compare the virtual classes on $\StableModuli (S, \beta)_{0,1} \times_S F_0$ and on $\LogModuli(S, \beta)$. Write $D_i$ for the components of $F_0$. There is a standard exact sequence of sheaves on $S$ given by
\[0 \rightarrow \Theta_{S^\dagger/k}^\dagger \rightarrow \Theta_{S/k} \rightarrow \bigoplus \Lawrence{i_*} \mathcal{N}_{D_i/S} \rightarrow 0.\]
By definition the curves we wish to count do not map into the intersections $D_i \cap D_j$ and so the pull-backs of $\Theta^{\dagger}_{S^{\dagger}/k}$ or
$\ker(\Theta_{S/k}\rightarrow {\mathcal N}_{F_0/S})$ to any stable curve in $\LogModuli(S, \beta) \cong \StableModuli (S, \beta)_{0,1}$ are isomorphic. We consider the two cotangent complexes $L_{\LogModuli (S^{\dagger}, \beta) / \mathfrak{M}^{\dagger}_{0,1}}$ and $L_{\StableModuli_{0,1} (S, \beta) \times_S F_0 / \mathfrak{M}_{0,1}}$. Note the log structure on $\LogModuli (S^{\dagger}, \beta)$
is the pull-back of the basic log structure on $\mathfrak{M}_{0,1}$ under
the forgetful map because there are no components or nodes of the domain
curve mapping into $F_0$. Thus we have a strict factorisation 
$\LogModuli(S^{\dagger},\beta)\rightarrow \mathfrak{M}_{0,1}
\rightarrow \mathfrak{M}_{0,1}^{\dagger}$, with the second morphism
\'etale. Thus the two cotangent complexes are isomorphic.

Since we have an isomorphism of obstruction theories and an isomorphism of underlying schemes the corresponding virtual classes are equal.

\end{proof}

\end{theorem}

We move on to the harder case.

\begin{theorem}[Log invariants of bisections are restrictions of Gromov-Witten invariants]

Let $ i : \mathbb{P}^1 \rightarrow \Hilb \cong \mathbb{P}^2/C_2$ be the hyperplane of double covers ramifying over $0 \in \mathbb{P}^1$ and $\beta$ be a curve class with $\beta . K_S = -2$. We denote the product $\StableModuli_{0,0} (S,\beta) \times_{\Lawrence{[\mathbb{P}^2/C_2]}} \mathbb{P}^1$ by $\StableModuli^{ram}(S, \beta)$ with projection maps $pr$ to $\StableModuli_{0,0} (S, \beta)$ and $ram$ to $\mathbb{P}^1$. Then the moduli space of log stable maps to $S^\dagger$, $\LogModuli (S^\dagger,\beta)$ admits a morphism $j$ to $\StableModuli^{ram} (S,\beta)$ which is an inclusion of components and there is an equality of virtual classes

\begin{equation}
j^* i^! [\StableModuli_{0,0}(S,\beta)]^{virt} = [\LogModuli(S^\dagger,\beta)]^{virt}
\end{equation}

\label{thm:logtocclassicalcomparison}

\begin{proof}

The map $j$ is the forgetful map, forgetting the choice of marking. A log curve in this space only meets the boundary in a single point, so maps to the desired space. By construction of the space $\overline{\mathcal M}^{ram} (S, \beta)$ there is a canonical choice of marked point given by intersecting with the boundary $F_0$ so long as no component of the curve maps into $F_0$. Viewing this
point as a logarithmic marked point with contact order $2$ with one of the
boundary divisors $D_i$ or contact orders $1$ with adjacent divisors
$D_i$, $D_j$, 
we obtain a family of stable log maps. Thus $\LogModuli (S^\dagger, \beta)$ is open inside some components of $\StableModuli^{ram}$ but is also proper and hence closed. Thus $j$ is an inclusion of components. To be precise it is those components which do not cover the marked rational fibre and have precisely one point mapping to the boundary, which excludes any point components. 

As in the single section case we relate potentially distinct obstruction theories on $\LogModuli (S^\dagger, \beta)$ and $\StableModuli_{0,0}(S,\beta)$. Recall that the obstruction theories on $\LogModuli (S^\dagger, \beta)$ and $\StableModuli_{0,0}(S,\beta)$ are given by $R \pi_{S, *} f_S^* \Theta_{S^\dagger/k} ^{\dagger \: \vee}$ and $R \pi_{S, *} f_S^* \Theta_{S/k} ^{\vee}$ respectively. There is an exact sequence
\[ 0 \rightarrow  \Theta_{S^\dagger/k}^\dagger \rightarrow \Theta_{S/k} \rightarrow \bigoplus \mathcal{O}_{D_i}(D_i) \rightarrow 0 \]
where $F_0 = \bigcup D_i$. Pulling back to the universal curve over $\LogModuli (S^\dagger, \beta)$, pushing forward along $\pi$ and dualising we therefore obtain a distinguished triangle
\[ (R \pi_* f^* \bigoplus \mathcal{O}_{D_i}(D_i))^\vee \rightarrow (R \pi_*f^*\Theta_{S/k})^\vee \rightarrow (R \pi_*f^*\Theta_{S^\dagger/k}^\dagger)^\vee \rightarrow\]
Now $f^* \bigoplus \mathcal{O}_{D_i}(D_i)$ is supported only at a single point of each fibre of $\pi$, hence it pushes forwards to a line bundle on $\LogModuli (S^\dagger, \beta)$. Consider the cotangent triangle associated to the maps $\LogModuli (S^\dagger, \beta) \hookrightarrow \StableModuli (S, \beta) \rightarrow \mathfrak{M}_{0,0}$, noting that as before the map $\LogModuli (S^\dagger, \beta) \rightarrow \mathfrak{M}_{0,0}^\dagger$ factors through the \'etale map $\mathfrak{M}_{0,0} \rightarrow \mathfrak{M}_{0,0}^\dagger$
\[ i^*L_{\StableModuli (S, \beta)/\mathfrak{M}_{0,0}} \rightarrow L_{\LogModuli (S^\dagger, \beta)/\mathfrak{M}_{0,0}^\dagger} \rightarrow L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)} \rightarrow\]
Since $(R \pi_*f^*\Theta_{S/k})^\vee$ and $(R \pi_*f^*\Theta_{S^\dagger/k}^\dagger)^\vee$ are obstruction theories we obtain by~\cite{VirtualPullbacks} that $(R \pi_* f^* \bigoplus \mathcal{O}_{D_i}(D_i))^\vee[1]$ is an obstruction theory for $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$. But $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$ is a regular immersion defined by one equation, namely that defining $\PP^1 \subset \PP^2$. We therefore deduce that $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$ is a line bundle supported in degree minus one. But $(R \pi_* f^* \bigoplus \mathcal{O}_{D_i}(D_i))^\vee[1]$ is also a line bundle supported in degree minus \Lawrence{one} and maps surjectively and hence isomorphically to $L_{\LogModuli (S^\dagger, \beta)/ \StableModuli (S, \beta)}$. This produces an equality of virtual classes on $\LogModuli(S^\dagger, \beta)$
\[\iota^! ([\StableModuli_{0,0}(S,\beta)]^{vir}) = [\LogModuli(S,\beta)]^{vir}\]
as desired.
\end{proof}

\end{theorem}

The following follows from Theorems ~\ref{thm:SandXcomparisons} and ~\ref{thm:logtocclassicalcomparison}.

\begin{corollary}[A formula for bisections on $S$]

There are equalities
\[  deg \: (ram^* \Lawrence{(2H)} \cdot[\StableModuli_{0,0} (S, \beta)]^{virt}) = deg \:([\StableModuli_{0,0} (X, d^{-1}\beta)]^{virt}) \]
and
\[deg \: (j^* (ram^* \Lawrence{(H)} \cdot [\StableModuli_{0,0} (S, \beta)]^{virt})) = deg \: ([\LogModuli (S^\dagger, \beta)]^{virt}) \]
\end{corollary}

Thus we see that the point components do not appear in either side of the calculation since they are trivial with respect to both Gysin maps. We know that the moduli space $\LogModuli (S^\dagger, \beta)$ is not the entirety of $\StableModuli^{ram} (S, \beta)$. Rather it consists of those curves none of whose components map into the boundary. To prove convergence we will have to bound the modulus of all the components contributing to this class. We claim the following:

\begin{lemma}[Point components are trivial]

The restriction of $ram^* H \cdot [\StableModuli_{0,0} (S, \beta)]^{virt}$ to point components is zero.

\end{lemma}

\begin{proof}

This follows from conservation of number and choosing a different hyperplane to intersect with.

\end{proof}

\begin{theorem}[Curve components are positive]

The restriction of $ram^* H \cdot [\StableModuli_{0,0} (S, \beta)]^{virt}$ to the preimage of a curve component $\mathcal{M}$ is positive.

\label{thm:CurvesArePositive}

\begin{proof}
  \Lawrence{By our compatibility result relating Gromov-Witten invariants on $S$ and $X$ it is enough to prove this positivity on the threefold $X$}. This problem was studied by Maulik and Pandharipande in~\cite{GromovWittenTheoryAndNoetherLefschetzTheory} but we can prove the equality here via a direct calculation. Recall that the reduced invariants on a $K3$ surface $K$ are given by an obstruction theory $T^\circ$ fitting into an exact triangle
  \[  \tau_{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K) \rightarrow R \pi_* f^* \Theta_{K/k} \rightarrow T^\circ \rightarrow\]
    where $\tau_{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K)$ is a line bundle supported in degree minus one. Suppose now that $i:K \hookrightarrow X$ is a fibre of the $K3$ fibration of $X$ into which all the stable curves in $\mathcal{M}$ map. There is an exact sequence on $K$
    \[ 0 \rightarrow \Theta_{K/k} \rightarrow \Lawrence{i^*} \Theta_{X/k} \rightarrow \mathcal{N}_{K/X} \rightarrow 0 \]
    where the normal bundle $\mathcal{N}_{K/X}$ is trivial since $K$ is a fibre. Pulling this back to the universal curve, pushing forwards to the moduli space and dualising we obtain a distinguished triangle on $\StableModuli_{0,0} (X, \beta)$
    \[ \mathcal{L} \rightarrow (R \pi_* f^* i^* \Theta_{X/k})^\vee \rightarrow (R \pi_* f^* \Theta_{K/k})^\vee  \rightarrow \]
    where $\mathcal{L}$ is a line bundle supported in degree zero. By composition we find a morphism $(R \pi_* f^* i^* \Theta_{X/k})^\vee \rightarrow T^\circ$ and let $\Cone$ be the cone over this. Applying the octahedral axiom we obtain a diagram
 \[ \begin{tikzpicture}
      \node (T) at (0,0) {$T^\bullet$};
      \node (L) at (0,2) {$\mathcal{L}[1]$};
      \node (C) at (3,0.5) {$\Cone[1]$};
      \node (K) at (-3,0.5) {$R \pi_* f^* \Theta_{K/k}^\vee$};
      \node (X) at (-6,-1) {$R \pi_* f^* \Theta_{X/k}^\vee$};
      \node (R) at (6,-1) {$\tau _{\geq -1} R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K)[1]$};
      \node (E0) at (1.5,2.75) {};
      \node (E1) at (4.8,0.8) {};
      \node (E2) at (9,-1.5) {};
      \node (E3) at (9,-2.5) {};
      
      \draw [->] (L) -- (E0);
      \draw [->] (C) -- (E1);
      \draw [->] (R) -- (E2);
      \draw [->] (R) -- (E3);      
      \draw [->] (L) -- (C);
      \draw [->] (K) -- (L);
      \draw [->] (K) -- (T);
      \draw [->] (T) -- (C);
      \draw [->] (X) -- (K);
      \draw [->] (X) -- (T);
      \draw [->] (T) -- (R);
      \draw [->] (C) -- (R);
      \end{tikzpicture}\]
    showing that the following triangle is distinguished
    \[  \mathcal{L} \rightarrow \Cone \rightarrow  \tau _{\geq -1}R \pi _* \omega_{\pi} \otimes H^0 (K, \omega_K) \rightarrow \]
    Now the special property of the preimage of a curve components is that each component is the moduli space of stable maps to $K$. In particular the degree zero cohomology of $(R \pi_* f^* i^* \Theta_{X/k})^\vee$ and $T^\circ$ are isomorphic since both are isomorphic to the tangent space of $\mathcal{M}$. This is enough to show that in fact $\Cone$ vanishes since $\Cone$ has a two term resolution by line bundles, \Lawrence{the differential is a surjection of line bundles since the degree zero cohomology vanishes and hence this surjection is an isomorphism.}

    Now $\mathcal{M}$ is a component of the moduli space of stable maps both to $K$ and $X$ and the two induced obstruction theories are isomorphic. The result is known from~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} that each component of the moduli space contributes either one or zero to the total count.

\end{proof}
\end{theorem}

On the bubble components we claim the following

\begin{theorem}[Bubble components are bounded]
  
Let $X$ be a generic unravelling of $S$, $\StableModuli^{bub} \subset \StableModuli_{0,0} (X, d^{-1}\beta)$ the preimage of a bubble component, a family over $\iota : \mathbb{P}^1 \rightarrow \Hilb \cong \mathbb{P}^2/C_2$. Choosing a point $P: Spec \: k/C_2 \rightarrow \mathbb{P}^2/C_2$ in the image of $\iota$ defines a $K3$ surface $cov:K \rightarrow S$. This induces a map $P: \StableModuli_{0,0} (K, cov ^* \beta) \rightarrow \StableModuli^{bub}$ and generically fibres $\StableModuli^{bub}$ by moduli spaces of stable maps to $K3$ surfaces, $Kl: \StableModuli^{bub} \rightarrow \mathbb{P}^1$. We claim that there is a constant $q \in \mathbb{Q}$ with $P_* [\StableModuli_{0,0} (K, cov ^* \beta)] = q [_{0,0} (X, d^{-1} \beta)]$ and that $q$ is dependent only on the choice of $X$.\label{thm:BubblesArentTooNegative}
\begin{proof}

  This question is related to the comparison theory of Maulik and Pandharipande in~\cite{GromovWittenTheoryAndNoetherLefschetzTheory}. In~\cite{GromovWittenTheoryAndNoetherLefschetzTheory}, the authors consider a family of nonsingular $K3$ over a non-singular curve $C$, $p: X \rightarrow C$. The Gromov-Witten invariants of $X$ are related to those of a fibre $K$ by multiplication by the degree of the Hodge bundle of $X \rightarrow C$. The details of this calculation are contained in~\cite{HodgeIntegralsAndGromovWittenTheory}.

  In our situation however there are some singular fibres of the fibration $p: X \rightarrow \PP^1$. Twenty four of these fibres are ordinary double points, occuring where the double cover $\PP^1 \rightarrow \PP^1$ ramifies over the image of a rational fibre. One can take a resolution of this family as described in Example 5.1 of~\cite{GromovWittenTheoryAndNoetherLefschetzTheory} to obtain a new family of non-singular $K3$ surfaces with the same Gromov-Witten invariants. In their paper Maulik and Pandharipande move to an analytic space to construct the small resolution. \Lawrence{This is a strictly analytic construction and to apply techniques of log geometry we must remain in the algebraic world. We appeal to Theorem 7.3 of~\cite{AlgebraizationOfFormalModuliII} which says that any Moishezon manifold is the analytification of an algebraic space. This resolution is indeed a Moishezon manifold since it is a small resolution of another Moishezon manifold. Therefore we consider a small resolution $\pi: \tilde{X} \rightarrow X$ with $\tilde{X}$ a smooth algebraic space. The moduli stack of stable maps to a Deligne Mumford stack was constructed in~\cite{GromovWittenTheoryOfDeligneMumfordStacks}. In that paper the authors also proved that the usual technology for constructing virtual classes can be applied to such objects and so the techniques of~\cite{HodgeIntegralsAndGromovWittenTheory} continue to apply.} However there are also two singular fibres where the $K3$ surface degenerates to the union of two rational elliptic surfaces meeting along a smooth elliptic fibre. We augment this singular family to a smooth log family. Let $K_1$ and $K_2$ denote these two singular fibres and we write $X^\dagger$ for $X$ with the divisorial log structure induced by the $K_i$. Since we have not imposed any tangency conditions \Lawrence{and a generic curve is transverse} the classical and log Gromov-Witten invariants coincide. Now by construction the map $p: X^\dagger \rightarrow \PP^1$ is log smooth and we can define a relative theory by taking the cone over the natual morphism
    \[\tau_{\geq-1} R \pi_* (\omega_\pi^\dagger) ^\vee \otimes H^0 (X, \omega_{X/\PP^1}) \rightarrow R \pi_* f^* (\Theta_{X/\PP^1})\]
    This gives a relative obstruction theory which may play the role of the relative theory defined in section 2.2 of~\cite{GromovWittenTheoryAndNoetherLefschetzTheory}. In particular their equation
    \[ [\StableModuli _{0} (\pi, \epsilon)]^{vir} = c_1 (K^*) \cap [\StableModuli_{0}(\pi, \epsilon)]^{red}\]
from page 22 continues to hold, but where $K$ is the log canonical line bundle with fibre $H^0 (X_{\xi}, K^\dagger_{X_\xi})$ over a point $\xi$. This proves the desired result since we may take the Chern class to be supported away from the singular fibres.

\end{proof}

\end{theorem}

\subsection{A geometric interpretation}

We have seen that this produces predictions for the Gromov-Witten invariants of the rational elliptic surface. In low degree we can calculate everything explicitly and compare them to the numbers of curves predicted directly by the geometry. Since we are practising numerology we will not attempt to prove our claims.

The invariant of a class on $X$ can be determined by its intersection with $H_2$, that is to say the class which restricts on each $K3$ fibre to a fibre of the fibration of that $K3$. This appears as the coefficient of $H_2$ in the corresponding term in the $J_X$ function. On $S$ we should calculate the invariants by their intersection with $[F]$. This suggests that each curve on $S$ should be counted with multiplicity four compared to the corresponding curve on $X$.

The first three coefficients we calculate, the coefficients of $z^{(0,2,0,1)}, z^{(1,2,0,1)}$ and $z^{(2,2,0,1)}$, are $-9, 144$ and $1980$. The first of these we can easily explain, there are nine sections each admitting a bubble component, by the double cover formula the relative invariant should be $-9/4$. The 144 reflect the fact that for each family class $H - E_i$ there are four lines which are tangent to the boundary to order two, the coefficient therefore is $9 \times 4 \times 2 \times 2 = 144$.

For the term 1980 we begin to see an interaction between the bubble components and the curve components. Each of the lines $H - E_i - E_j$ admits a bubble component, there are 36 choices of such classes so we expect a contribution of $-36 = -36/4 \times 2 \times 2$. The classes $2H - E_i - E_j - E_k - E_l$ give a curve component of which four elements are tangent to the boundary. This gives a contribution of $126 \times 4 \times 2 \times 2 = 2016$\footnote{Making this into an IMO problem}, combining these we obtain the predicted contribution of $1980$.

\section{Some painful analysis}

We have identified that we must count the number of curves meeting the boundary to tangency order one and two. A curve tangent to order one consists of a section and a collection of components covering rational fibre. We call such a curve a \emph{section with rational tails}. The Gromov-Witten theory of sections of rational elliptic surfaces was studied in~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms}. They constructed the moduli space of stable maps of genus zero in the class $S + nF$ and found that all such curves are reducible being precisely curves with rational tails. They calculate for a fixed curve class satisfiying $C.F = 1$. We bound the number of such classes of a given degree.

Whilst the number of classes is bounded their Gromov-Witten invariants may still grow rapidly, the content of~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} is to calculate explicit formulae for these. We will combine these two results to produce a bound on the Gromov-Witten invariants of curves of degree $d$. Now let us begin bounding the number of curves tangent to order two.

\subsection{Multivariate holomorphic functions}

In a short while we will see that the function $I_X$ is a holomorphic function of several variables. Our goal will be to prove that this implies that the function $J_X$ is also a holomorphic function and therefore bound the coefficients appearing in a power series centred at the origin. Before we do this we collect here some results about the growth of coefficients for power series of holomorphic functions.

Given a vector $\underline{v} = (v_1, \ldots v_n) \in \NN^n$ we write $x^{\underline{v}}$ for the monomial $\prod x_i^{v_i}$. Recall that a holomorphic function in variables $x_1, \ldots x_n$ is given locally by an absolutely convergent power series $\sum a_{\underline{v}} x^{v}$ converging on some disc and that if such a power series exists then operations such as differentiation and so on may be performed term by term. Let $diag (\underline{v})$ be the function sending $v$ to $\sum (v_i + 1)$. Suppose now that $a_{\underline{v}}$ are real numbers satisfying $|a_{underline{v}}| < c r^{diag (\underline{v})}$ for some fixed choice of real constants $c$ and $r$. Then by construction the power series
\[ f (x_1, \ldots, x_n) = \sum a_{\underline{v}} x^{\underline{v}}\]
converges on the polydisc around the origin in $\CC^n$ of radius $1/r$. We call such a family of $a_{\underline{v}}$ \emph{exponentially bounded}. We provide a converse to this statement.

\begin{proposition}

  Let $f: D^n (0,r) \rightarrow \CC$ be a holomorphic function on a polydisc around $0$ of radius $r$. Then $f (x_1, \ldots x_n) = \sum a_{\underline{v}} x^{\underline{v}}$ with the $a_{\underline{a}}$ exponentially bounded.
  
\end{proposition}

\begin{proof}

  The existence of the power series expansion is standard. We apply the Cauchy Integral formula for many variables to find $a_{\underline{v}}$.
  \[a_{\underline{v}} = \frac{1}{2\pi i}^n\int_{\partial D^n(0,r/2)} \frac{f(x_1, \ldots x_n)}{x^{\underline{v}} x_1 \ldots x_n} dx_1 \ldots dx_n.\]
  Therefore we have
  \[|a_{\underline{v}}| \leq \frac{\max_{z \in D^n (0,r/2)}|f(x_1, \ldots x_n)|}{(r/2)^{diag(\underline{v})}} \]
  where $f(x_1, \ldots x_n)$ is bounded on $D^n (0,r/2)$ . Therefore we have the result.
\end{proof}

\begin{definition}

  We work in slightly greater generality than is stadard. Let $R = \CC[t_1, \ldots t_m] / \langle t_1^{k_1}, \ldots t_m^{k_m} \rangle$ be a finite dimensional $\CC$ algebra. We say that an element of $R[\! x_1, \ldots x_n \! ]$ is holomorphic if the coefficients of each monomial in the $t_{i}$'s are all holomorphic in the $x_i$.
  \label{def:exp-bounded}
\end{definition}
  
  The class of holomorphic functions is closed under many operations. This is a list of properties we need of them.

\begin{lemma}[A holomorphic toolkit]

The set of germs of holomorphic functions around $0$ is closed under the following operations:

\begin{enumerate}
  \item If an element $f \in \mathbb{C} [ \! [ t_1, \ldots t_n, x_1, \ldots x_m ] \! ]$  is holomorphic then its image in the quotient ring $R [ \! [ x_1, \ldots x_m ] \! ]$ obtained by dividing out by the ideal $\langle t_1^{m_1},\ldots,t_n^{m_n}\rangle$ is holomorphic.
\item If $f = \sum a_{\underline{i}, \underline{j}} t^{\underline{i}} x^{\underline{j}}$ is holomorphic and $g = \sum b_{\underline{i}, \underline{j}} t^{\underline{i}} x^{\underline{j}}$ is such that $| a_{\underline{i}, \underline{j}} | \geq | b_{\underline{i}, \underline{j}} |$ then $g$ is holomorphic.
\item Taking products of elements.
\item Taking inverses of invertible elements.
\item Taking exponentials of elements.
\item Substitutions of the form $x_i \mapsto x_i f_i$ where the $f_i$ are also holomorphic around $0$.
\item If $s_i = x_i f_i (x_0,  \ldots , x_m)$ is a change of basis with $f_i$ holomorphic and non-vanishing at zero then this change of basis locally has an inverse where the inverse map is given by $x_i = s_i g_i (s_0, \ldots, s_m)$ with the $g_i$ holomorphic.
\end{enumerate}

\end{lemma}

\begin{proof}

  The proof of i) is trivial.
  
  The proofs of ii) through vi) are standard for genuinely holomorphic functions of multiple variables. However the first result allows us to lift holomorphic elements of $R [ \! [ x_1, \ldots x_m ] \! ]$ to genuinely holomorphic functions of more variables by treating the $t_i$ as being free under the natural inclusion and then take the quotient.

  The second result says that if a collection $a_{\underline{v}}$ is exponentially bounded then any subcollection of the $a_{\underline{v}}$ are exponentially bounded too.
  
  To see the final result note that the derivative $\partial s_i / \partial x_j = \delta_{ij}$. Therefore the change of basis $\CC^n \rightarrow \CC^n$ is locally invertible by the inverse function theorem with holomorphic inverses. Since this maps $(0, \ldots 0)$ to $(0, \ldots 0)$ it has the required form.
  
\end{proof}

\subsection{Bounding Gromov-Witten invariants}

We have formulae for the Gromov-Witten invariants but we saw in our section on their geometric interpretation that calculating them is a tricky business. In particular the mirror map required in Givental's theory has no nice expression. The Gromov-Witten invariants appear as the coefficients of monomials $z^{(n,2,0,1)}$ in the power series expansion for $J_X$. When we analyse the scattering diagram we will see that sums of exponentially bounded elements will converge, hence we wish to deduce that the function $J_X$ is at least holomorphic. To show this we will show that the function $I_X$ is holomorphic and use our toolkit to show that the set of holomorphic functions are closed under the operations performed in transforming the $I_X$ function into the $J_X$ function.

\begin{theorem}[The $I$ function is holomorphic]

The function $I_X$ is holomorphic for $X$ an unravelled threefold,
where the coordinates $x_i$ in Definition \ref{def:exp-bounded} coincide
with $e^{t_i}$ in $I_X$, and the coordinates $t_i$ in
Definition \ref{def:exp-bounded} coincide with the $H_i$ in $I_X$.

\end{theorem}

\begin{proof}

To do this we prove the the coefficients of $I_X$ form an exponentially bounded set. We write out the definition of the $I_X$ function

\begin{equation}
I_X (t_0, \ldots t_n) = e^{\sum t_i H_i / \hbar} Eul (\oplus \mathcal{L}_i) \sum_\beta q^\beta \frac{\prod_i (\prod_{m=1} ^{\mathcal{L}_i (\beta)} (c_1 (\mathcal{L}_i) + m\hbar) \prod_\rho \prod_{m=-\infty}^{0} (D_\rho + m\hbar))}{\prod_\rho \prod_{m=-\infty}^{D_\rho \cap \beta} (D_\rho + m\hbar)}
\end{equation}

To prove boundedness, with $x_i=e^{t_i}$ and $x^{(a_1,\ldots,a_4)}
=\prod_i x_i^{a_i}$, we only need to prove boundedness of the sum

\begin{equation}
\sum x^{(a,b,c,d)} \frac{\prod_{m=1} ^{3a + b} (3H_1 + H_2 + m\hbar) \prod_{m=1} ^{b+c+2d} (H_2 + H_3 + 2H_4 + m\hbar)}{\prod_{m=1}^a (H_1 + m\hbar)^3\prod_{m=1}^b (H_2 + m\hbar)^2\prod_{m=1}^c (H_3 + m\hbar)^2\prod_{m=1}^d (H_4 + m\hbar)^2}
\end{equation}
Fix a coefficient $H_{\underline{i}} = \prod H_i ^{\alpha_i}$ and consider the coefficient of $H_{\underline{i}} x^{(a,b,c,d)}\hbar^{-c + \sum \alpha_i}$. Expanding out the above product we obtain the expression
\begin{align*}
  \sum x^{(a,b,c,d)} \hbar^{-c} & \frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2} \\
  & \frac{\prod_{m=1} ^{3a + b} (\frac{3H_1 + H_2}{m\hbar} + 1) \prod_{m=1} ^{b+c+2d} (\frac{H_2 + H_3 + 2H_4}{m\hbar} + 1)}{\prod_{m=1}^a (\frac{H_1}{m\hbar} + 1)^3\prod_{m=1}^b (\frac{H_2}{m\hbar} + 1)^2\prod_{m=1}^c (\frac{H_3}{m\hbar} + 1)^2\prod_{m=1}^d (\frac{H_4}{m\hbar} + 1)^2} \end{align*}
any monomial appearing in this consists of two parts, a term $\frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2}$ with no $\hbar$ term times a correction factor involving a sum of products of terms $1/i$. The correction factor is at most a polynomial in $a,b,c,d$, hence exponentially bounded since there are only finitely many choices of $\underline{i}$. We now bound the term $\frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2}$ using Stirling's formula:
\[
\frac{(3a + b)! (b + c + 2d)!}{a! ^3 b!^2 c!^2 d!^2} 

< k e^c \cdot \frac{(3a + b)^{3a+b} (b+c+2d)^{b+c+2d}}{a^{3a} b^{2b} c^{2c} d^{2d}} \cdot \frac{2 \pi \sqrt{(3a + b) (b + c + 2d)}}{\sqrt{512 \pi ^ 9 a^3 b^2 c^2 d^2 }}
\]
The middle term is the only one not obviously exponentially bounded. We rewrite it as

\begin{equation}
\frac{(3a + b)^{3a}}{a^{3a}} \cdot \frac{(3a+b)^b}{b^b} \cdot \frac{(b + c + 2d)^b}{b^{b}} \cdot \frac{(b + c+ 2d)^{c}}{c^{2c}} \cdot \frac{(b + c +2d) ^{2d}}{d^{2d}}
\end{equation}

Rearranging the first term we get $(3+b/a)^{3a}$ and we wish to compare this to $r^{a + b}$. Let $r = 27$, then $27^{a+b} = 3^{3a+3b}$ and we can rewrite the quotient $(3+b/a)^{3a} / r^{a + b}$ as
\[ \frac{(1+\frac{b}{3a})^{3a}}{3^b}.\]
Taking log and using that $\log (1+x) < x$ we find that
\[ 3a \log \left(1 + \frac{b}{3a} \right) - b \log (3) < b - b \log (3) < 0\]
Hence for $r > 27$ we see that $(3 + b/a)^{3a} < r^{a + b}$. There are similar choices of $r$ for each other term. Our choice of $r$  is the product of all of these minimal choices. Then since $r^{a+b+c+d}$ dominates all terms of the form $r^{a+b}$ etc. we see that the coefficients of $I_X$ are exponentially bounded. Thus $I_X$ defines a holomorpic function in a neighbourhood of the origin.

\end{proof}

To calculate the $J_X$ function we took subsequences, exponentials, products and took the inverse of a holomorphic substitution. Thus $J_X$ is holomorphic too. In particular the sequence of coefficients of $x^{(n,2,0,1)}$, corresponding to the desired curves tangent to the boundary to order two is exponentially bounded. Applying the results of~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} we obtain the following result

\begin{lemma}[Sections are exponentially bounded]

Let $\beta = nH - \sum a_i E_i = E + m F$ be a curve class on $S$ with $\beta.K_S = -1$ and $E$ a section. There exist constants $c,r \in \mathbb{R}$ independent of $\beta$ such that

\begin{equation}
| I_{0,0,\beta}^\dagger | < c r^n
\end{equation}

\label{thm:SectionsAreBounded}

\begin{proof}

  We know that the image of $I_{0,0,\beta}^\dagger$ inside of $I_{0,0,\beta}$ are precisely those stable maps with no component mapping into the boundary. By~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms} Lemma 5.7 the restriction of the virtual class to those components not in the image and to those components in the image is positive. Then by Theorem~\ref{thm:LogEqualityForSections} we obtain an inequality:
  \[| I_{0,0,\beta}^\dagger | \leq | I_{0,0,\beta} |\]
  
But we have an explicit formula for $I_{0,0,\beta}$ in~\cite{TheEnumerativeGeometryOfK3SurfacesAndModularForms}, Theorem 6.2:
\begin{equation}
\sum_{n=0}^\infty I_{0,0,E + mF} z^m = \prod _{m=1}^\infty (1- z^m)^{-12}
\end{equation}

Now rewriting the right hand side as $\prod (1 + z^m + z^{2m} + z^{3m} + \ldots)^{12}$ we see that it is enough to prove that the element $\prod (1 + z^{m} + z^{2m} + z^{3m} + \ldots)$ is exponentially bounded. Each term in this product with monomial $z^a$ gives a distinct unordered partition of $a$. But the number of such partitions less than the number of ordered partitions, equal to $2^a$. Since $n \geq m$ the result holds for $c = 1$ and $r=2$.

\end{proof}

\end{lemma}

Now applying this result to our compatibility relations we obtain the following important result

\begin{corollary}[Bisections are exponentially bounded]

Let $\beta = nH - \sum a_i E_i$ be a curve class on $S$ with $\beta.K_S = 2$. There exist constants $c,r \in \mathbb{R}$ independent of $\beta$ such that

\begin{equation}
| I_{0,0,\beta}^\dagger | < c r^n
\end{equation}

\begin{proof}

  To transform the $I$ function into the $J$ function we took subsequences of exponentially bounded sequences, made exponentially bounded changes of basis and took products of exponentially bounded sequences. Therefore the $J$ function is exponentially bounded. In particular the coefficients of $H_2 z^{(n,2,0,1)}$ are exponentially bounded. We write $J_n$ for these coefficients. Now as above the moduli space of stable maps breaks up into different components dependent on the combinatorics of covers of the rational fibres. By invariance of Gromov-Witten invariants under flat deformation we may assume that the surface $S$ has twelve rational fibres. In \Lawrence{our application of} the Gross Siebert program we work relative to an $I_4$ fibre  and so we want to count those stable maps which do not map to those four of the rational fibres which combine to give the $I_4$ fibre. By Theorems~\ref{thm:CurvesArePositive} and~\ref{thm:BubblesArentTooNegative} and the compatibility relation Theorem~\ref{thm:logtocclassicalcomparison} we have the inequality
\begin{equation}
| I_{0,0,\beta}^\dagger | \leq J_n + k \sum _{\beta = 2E + mF} I_{0,0,E+mF}
\end{equation}
where the sum is taken over all sections $E$ and $k$ is the universal coefficient we described earlier coming from the Hodge bundle. By Lemma~\ref{thm:BoundedlyManySections} we have seen that the number of such sections grows at worst polynomially in $n$. Combining this with the above Lemma~\ref{thm:SectionsAreBounded} we have that right hand term too is exponentially bounded.

\end{proof}

\end{corollary}

\section{Convergence of the mirror}

We now combine all of the above results on the number of sections and bisections to prove the desired convergence statement. Recall the dual intersection complex of~\ref{fig:TopLevelBrokenLines}. It should carry a multi-valued piecewise linear function $\phi$ which becomes single valued on this cover. Thus if $v$ is an element of the upper half-plane, $\phi(v)$
makes sense. In the next lemma we give an explicit formula for a choice of $\phi$.

\begin{lemma}[Bounding the function $\phi$]
  We choose the lift of $\phi$ which is zero on the cone $C$ of Figure~\ref{fig:DualIntersectionRES}. Over a one-cell $v_i$ recall that the piecewise linear function $\phi$ changes by $D_i \otimes n_i$, where $n_i$ is defined in Definition
\ref{def:sheaf-of-monoids}. Then $\phi$ is given by the following formulae on the following cones. On $\langle (4n, 1), (4n+1, 1)\rangle$ it is given by
\[nx F - (n(2n-1)D_1 + 2n^2D_2 + n(2n+1)D_3 + n(2n+2)D_4)y\]
whilst on $\langle (4n+1, 1), (4n+2, 1)\rangle$ it is given by
\[nx F + x D_1 - ((n+1)(2n+1)D_1 + 2n^2D_2 + n(2n+1)D_3 + n(2n+2)D_4)y\]
and on $\langle (4n+2, 1), (4n+3, 1)\rangle$ it is given by
\[nx F + x(D_1 + D_2) - ((n+1)(2n+1)D_1 + 2(n+1)^2D_2 + n(2n+1)D_3 + n(2n+2)D_4)y\]
and finally on $\langle (4n+3, 1), (4n+4, 1)\rangle$ it is given by
\[nx F + x(D_1 + D_2 + D_3) - ((n+1)(2n+1)D_1 + 2(n+1)^2D_2 + (n+1)(2n+3)D_3 + n(2n+2)D_4)y\]

We now find rough bounds on $\phi$. Suppose that $m > n > 0$ are integers and we wish to compute $\phi ((m,1)) - \phi ((n,1))$. Since $\phi$ is strictly convex the difference between these is bounded by $\phi((m,1)) - \phi((m-1,1))$. From the formulae above we see that $\phi ((m,1)) - \phi ((m-1,1)) \geq (2\left \lfloor{m/4}\right \rfloor -1) D_{m-1}$ where the index of $D_{m-1}$ is taken mod $4$. The inequality $\phi ((m,1)) - \phi ((m-1,1)) \geq (1- 2\left \lfloor{m/4}\right \rfloor) D_{m+1}$ similarly holds for $m < n < 0$. Now suppose that $n,k > 0$ and we consider $\phi ((m,1)) - \phi ((n,1)) - \phi ((k,1))$ with $m \ge n+k$. By convexity this value is smallest if $n = m-1$ and $k=1$. In that case we can use the above bound to see that this is bounded also by $(2\left \lfloor{m/4}\right \rfloor -1)D_{m-1}$.

\end{lemma}

Recall the structure of the equations of \ref{eqn:MirrorToRES1} through \ref{eqn:MirrorToRES3}. The functions there become progressively harder to bound as we work through, so let us work in increasing order of difficulty.

\begin{example}

The coefficients $f_{(k,2)}, g_{(k,2)}$ and $r_{(k,2)}^i$ of these equations converge in a neighbourhood of the large complex structure limit point.

\begin{proof}

  By symmetry it is enough to prove convergence only of the first equation and of one of the products $\vartheta_{D_i}^2$.

  The coefficient $f_{2,2}$ is the sum over pairs which do not bend anywhere and end close to $(2,2)$. The pairs of pants are pairs of broken lines from $(4n,1)$ and $(-4n+2,1)$. Near $(2,2)$ these lines carry the monomials $z^{\phi (4n,1)}$ and $z^{\phi (-4n+2,1)}$. Therefore we want to study the convergence of
  \[\sum |z^{\phi (4n,1) + \phi (-4n+2,1)}| < \sum |z^{n^2F}|\]
  which converges since it decays at least exponentially. Exactly the same argument applies to show the convergence of $f_{(4,2)}$, $r_i^i$ and $r_{i+2}^i$. In all these cases none of the broken lines can bend, there is a $\ZZ$ indexed family of pairs of pants and the monomials are controlled entirely by the function $\phi$. For the product $\vartheta_{D_i}^2$ there is precisely one pair of broken lines which lie in only one maximal cell of $B$. This explains the leading coefficient of $1$ appearing in the formulae.

\end{proof}

\end{example}

We now study the contribution to the product of points with $E(P) = 1$, i.e.,
the theta functions $\vartheta_{D_i}$, restricting our analysis by symmetry to $\vartheta_{D_1}\vartheta_{D_3}$ and $\vartheta_{D_4}^2$. Let us begin with $\vartheta_{D_1}^2$ where we describe all pairs of broken lines which contribute to $\vartheta_{D_4}$. All such pairs contain one bend which for the purpose of proving convergence we can assume occurs on the first broken line. We may also assume that the first broken line is from $(4m + 4n, 1)$ with $m, n$ positive and scatters off of $(4n, 1)$ with a monomial $z^{E+kF}$ for $E$ a section meeting $D_1$ and $k$ non-negative. As a consequence the second line is from $(-4m, 1)$. This pair of lines contributes $I_{0,0,E+kF}^{\dagger}z^{E + \phi ((4m+4n,1)) - \phi ((4n,1)) + \phi ((-4m,1)) + kF}$ to the coefficient. This sum is taken over all $m,n > 0$, $k \geq 0$ and sections $S$ meeting $D_1$ in one point. 

\begin{lemma}[Single bends converge]

The following sum converges
\[\sum_{m,n,k, S} I_{0,0,S+K}^{\dagger}z^{S + \phi ((4m + 4n, 1))  + \phi ((-4n,1)) - \phi ((4m,1)) + kF}\]
\begin{proof}
We rewrite this sum as
\begin{equation}
\left( \sum _{S,k} I_{0,0,S+kF}^{\dagger} z^{S + kF}  \right) \left( \sum_{m,n}  z^{\phi ((4m + 4n, 1)) + \phi ((-4n,1)) - \phi ((4m,1))}\right)
\end{equation}

We know that the term $I_{0,0,S+kF}^{\dagger}$ has modulus at most $2^k$, therefore for $| z^F | < 1/2$ and $| z^S | < 1$ and after applying Lemma~\ref{thm:BoundedlyManySections} one has convergence of the left hand sum.

To prove convergence of the right hand sum we expand out the sum using the explicit formula for $\phi$ found above and bound it
\[\sum_{m,n} | z^{\phi ((4m+4n,1)) - \phi ((4m,1)) + \phi ((-4n,1))} | \le \sum_{m,n} | z^{\phi ((-4n,1)) + ((2m+2n-1)D_{3})} |\]
And this converges since $\phi ((-4n,1))$ is positive and grows quadratically in $n$.

\end{proof}

\end{lemma}

Exactly the same calculation proves the convergence of the contributions to $\vartheta_{D_i}$. The choice of classes $E$ changes: they must be tangent to different components of the boundary. Our bounds on the log invariants though are independent of the choice of component of the boundary. Similarly all we used about $\phi(n)$ is our linear bound given above. Therefore the all the coefficients of $\vartheta_{D_i}$ in the formulae converge.

Finally we calculate the contribution to the constant monomial $z^{(0,0)}$. Here there are four families we can consider, those where the same line scatters twice off of two sections, those where one of the lines scatters off of a rational curve meeting the boundary twice, those where the lines scatter off of a curve meeting $D_i \cap D_{i+1}$ or finally each line scatters once off of sections.

We take the first of these and calculate the coefficient $r^i_{0}$, all the other coefficients in the other formulae have the same combinatorics but with different incoming directions. By symmetry we may assume that it is the first line which scatters twice and this initially has slope $(4m+n+k, 1)$, scatters off of lines $(n,1)$ with monomial $z^{S + aF}$, and $(k,1)$ with monomial $z^{S' + bF}$ for sections $E$ and $E'$ meeting $D_{n}$ and $D_k$ respectively and $n \leq k$. The second line has slope $(-4m,1)$ and we have the restriction that $n+k$ is divisible by four. This pair of broken lines carries monomial
\begin{equation}
I_{0,0,E + aF}^\dagger I_{0,0,E' + bF}^\dagger z^{E + E' + \phi ((4m+n+k, 1)) + \phi ((-4m,1)) - \phi((n,1)) - \phi((k,1)) + (a + b)F}
\end{equation}
We claim that the sum over all of these terms converge.
\begin{lemma}[Double bends converge]

The sum of the above terms converges in a small neighbourhood of the maximal degeneracy point.

\begin{proof}

As before we expand out the sum and show that each term converges
\begin{align*}
\sum & I_{0,0,E + aF}^\dagger I_{0,0,E' + bF}^\dagger z^{E + E' + \phi ((4m+n+k,1))+ \phi(-4m,1) - \phi((n,1)) - \phi((k,1)) + (a + b)F} = \\
&  \sum _{E, a} I_{0,0,E + aF}^\dagger z^{E + aF} \sum_{E', b} I_{0,0,E' + bF}^\dagger z^{E' + bF} \sum_{m,n,k} z^{\phi ((4m+n+k,1)) + \phi ((-4m,1)) - \phi ((n, 1)) - \phi ((k,1)) }
\end{align*}
The first two terms converge precisely as before so long as $z^E$ is of modulus at most one for each section class $E$ and $| z^F | < 1$. To prove convergence of the final sum we expand out the bounds we found above
\[ \sum_{m,n,k} z^{\phi ((4m+n+k,1)) + \phi ((-4m,1)) - \phi ((n, 1)) - \phi ((k,1))} < \sum _{m,k,n \in \mathbb{N}^3} z^{\phi ((-4m,1))}z^{(2\left \lfloor{(4m+n+k)/4}\right \rfloor -1)D_3} .\]
This sum clearly converges for $| F | < 1/2$ and $|D_3| < 1/2$.

We now turn to the second family, where one of the lines scatters off of a class $[C]$ meeting the boundary with tangency order two. As in the case of single sections suppose that we have lines from $(4m+4n,1)$ and $(-4m,1)$ with the first line bending off of a ray $(2n,1)$\footnote{The tangency order two means that this bends the tangent vector by $(4n,2)$}. The associated sum is then
\[ \sum_{[C], m,n,k} I^\dagger_{[C]}z^{[C]} z^{\phi ((4m+4n,1)) + \phi((-4m,1)) - \phi ((4n,2))}\]
which we bound as being less than
\[ \sum_{[C]} |I^\dagger_{[C]}z^{[C]}| \sum_{m,n} |z^{\phi ((4m+4n,1)) + \phi((-4m,1)) - \phi ((4n,2))}|.\]
We have seen that such a class $[C]$ is of the form $E + E' + kF$ for sections $E$ and $E'$. For a given degree $d$ of curve $[C] = dH + \sum a_i E_i$ there are exponentially bounded many pairs $E$ and $E'$ with $[C] = [E] + [E']$. From our calculations of the log Gromov-Witten invariants the sum $I^\dagger ([C])$ associated to $[C]$ is exponentially bounded. Therefore the first sum converges.

To deduce the convergence of the second sum we note that $\phi ((4n,2)) = \phi ((2n,1)) + \phi ((2n,1)) \leq \phi ((4n,1)) + \phi ((0,1))$. Therefore $\phi ((4m+4n,1)) - \phi ((4n,2)) \geq \phi ((4m+4n,1)) - \phi ((4n,1)) - \phi ((0,0)) \geq (2m+2n-1)D_3$. Therefore so long as all the $|z^{D_i}|$ are sufficiently small this converges.

The third case where one of the broken lines scatters off of a line corresponding to a curve meeting $D_i$ and $D_{i+1}$ at one point transversely. This is exactly the same combinatorics as in the above case an converges for the same reasons.

Finally we treat the case where both lines scatter off sections. Suppose that we have two lines from $(m,1)$ and $(n,1)$ scattering off of $(k,1)$ and $(l,1)$ respectively. By assumption $m$ and $n$ have opposite signs, so let us write $m  < k \geq l < n$ with $m+n = k+l$. Suppose that the lines on which they scatter carry monomials $z^{E + aF}$ and $z^{E' + bF}$ for sections $E$ and $E'$. This pair of broken lines carries monomial 
\[\sum  I_{0,0,E + aF}^\dagger I_{0,0,E' + bF}^\dagger z^{E + E' + \phi ((m,1))+ \phi(n,1) - \phi((k,1)) - \phi((l,1)) + (a + b)F}\]
Again this reduces to proving that the following sum converges
\[ \sum |z^{\phi ((m,1))+ \phi(n,1)  - \phi((k,1)) - \phi((l,1))}|\]
which in turn is bounded by
\[ \sum |z^{(2\left \lfloor{n/4}\right \rfloor -1)D_{n-1}} z^{(1-2\left \lfloor{m/4}\right \rfloor)D_{m-1}}|.\]
Therefore for $| z^{D_i} |$ small this certainly does converge.

\end{proof}

\end{lemma}

Thus we have shown the desired result, the functions appearing in our schematic formula do indeed converge in a neighbourhood of the origin, at least for a rational elliptic surface relative to an $I_4$ fibre. This induces an embedding of the mirror via the functions $\vartheta_i$ into affine space which we take to be our analytic family.

\section{Reflections}

Our construction here can be used to construct the mirror to a rational elliptic surface relative to an $I_n$ fibre for any $n > 3$. The proof is essentially unchanged in this case since we are studying equations of the form $\vartheta_{i-1} \vartheta_{i+1} = \vartheta_{i}^2 + \ldots$ and these have the same combinatorics. To construct the mirror to a less degenerate fibre though is hard, one would need to count the number of curves tangent to the boundary to orders three and four. We do not know of any techniques that would allow one to bound these coefficients. One possibility would be to look for modularity in the coefficients by computation and prove that these relations hold. Unfortunately we have no way to know the weight of the attached modular forms and so this would require new insight into the geometry. A second technique would be to use the degeneration formula to degenerate to a reducible variety. Unfortunately the naive degeneration of $S$ to the union of $\PP^2$ and a component $I_n \times \PP^1$ viewed as the union of $(0,1)$ amd $(3,0)$ hypersurfaces in $\PP^2 \times \PP^1$ does not produce any control on the invariants. It could be that there is a nicer choice of degeneration with more useful properties, but I certainly do not know of one.  

\section{The moduli space of surfaces}

We said originally that motivation to study the mirror to a rational elliptic surface came from attempts to understand a compactification of the moduli space of K3 surfaces. Gross Hacking and Keel wanted to understand the Gross-Siebert program when applied to a Type II degeneration. The base of the $SYZ$ fibration for a generic elliptically fibred $K3$ surface is homeomorphic as a topological manifold to $S^2$ with $24$ singularities and this degenerates to the union of two triangles glued along their boundary. A consequence of this is that we lose the ability to push our singularities together to a point. In turn this means that the scattering process and broken lines are much more complicated since broken lines can move from one rational elliptic surface and to the other. In particular we must account for far more curve classes than just the ones described in the above project. Even still the tropical picture of these broken lines should be reasonable to understand. Our problem is counting the corresponding curves.

Conceptually one can calculate the entire Gromov-Witten theory of a rational elliptic surface from the mirror, even the open strings. We are interested in the combinatorics of how open strings glue together on the central fibre of a type II degeneration, so the open invariants of a rational elliptic surface relative to a smooth ellipitic fibre.

There are two problems we need to consider. Firstly we need to understand how the mirror family to $(S,D)$ changes as we deform $D$, believing that this allows us to at least reconstruct and bound the log invariants relative to a smooth fibre. The second problem is that the invariants on the $K3$ family are reduced invariants, not actual Gromov-Witten invariants. Degeneration formulae for these reduced invariants have not been found.

\section{Understanding the mirror family}

We have constructed an analytic mirror family for the rational elliptic surface smoothing the central degenerate fibre. On the base $B$ Gross and Siebert point out that there are certain strata corresponding to smoothing only a subset of the nodes. These strata are given by choices of toric ideals inside our monoid $P$. The particular strata we are interested are those where the $D_i$ are non-zero but every other monomial is zero. Studying our equations we see that the family now becomes a cone over an elliptic curve. One could hope to find the explicit equations for this elliptic curve since they correspond to counts only of unbroken lines and relate this to the boundary at infinity of the Landau-Ginzburg model we have constructed.

\section{Mirrors to low degree del Pezzo surfaces}

In the firts project we calculated explicit formulae for the mirror to $dP_2$. Such equations are very hard to understand as presented, in particular it is not obvious that they have the expected properties of the mirror. One test for this should be that the original del Pezzo should appear as a compactification of some member of this family. We conjecture that the original del Pezzo should be a compactification of the fibre obtained by setting all the parameters to one. The only obstacle in doing this is understanding how to embed this affine piece into an appropriate projective space. 

\end{document}