text:
\section{Introduction}

The Kodaira dimension measures the size of the canonical model, and hence is a measure of the complexity of a projective variety $X$, while the structure of the pseudo-effective cone of divisors broadly dictates the geometry of the variety. For instance, when $X$ is $\mathbb{Q}$-factorial and the Cox ring is finitely generated, then the effective cone is not only polyhedral but further decomposes into finitely many convex chambers each representing a different birational model of the variety. Such a variety is known as a Mori dream space and when the variety is a moduli space, the chambers often have a modular meaning. If $X$ has canonical singularities then $X$ is of maximal Kodaira dimension, or of general type, when the canonical divisor lies in the interior of the pseudo-effective cone, and is of minimal Kodaira dimension and uniruled, when the canonical divisor is not pseudo-effective, cf. \cite{BDPP}. When the the Kodaira dimension of $X$ lies somewhere between these two extremes, $X$ is of intermediate type and the canonical divisor lies on the boundary of the pseudo-effective cone.\\

The computation of the Kodaira dimension of $\overline{\mathcal{M}}_{g,n}$ for $g\geq 2$ has been a central problem over recent decades. It is well established \cite{HM, Lo} that the moduli spaces $\overline{\mathcal{M}}_{g,n}$ are all of general type with only finitely many exceptions. When the genus and number of markings are low these spaces are usually unirational or uniruled, but as the genus or the number of markings grow, the problem of determining the Kodaira dimension becomes much harder with long-standing open cases such as $\overline{\mathcal{M}}_{g}$ with $17\leq g\leq21$. The transition from negative to maximal Kodaira dimension is poorly understood and only two such moduli spaces are known to be of intermediate Kodaira dimension; ${\rm{Kod}}\left(\overline{\mathcal{M}}_{11,11}\right)=19$ and ${\rm{Kod}}\left(\overline{\mathcal{M}}_{10,10}\right)=0$, cf. \cite{FV, BM}. \\

Similarly, the structure of the pseduo-effective cone and the related Mori dream space question on $\overline{\mathcal{M}}_{g,n}$ for $g\geq 2$ has presented a challenge. The cone is non-polyhedral for $n\geq2$ and they are also not Mori dream spaces for $g\geq 3$ and $n\geq1$, cf. \cite{Mu, Keel}. A full chamber decomposition of the cone is known only in the remaining cases for very small genera $g=2,3$, cf. \cite{Ru}. Both questions remain open on $\overline{\mathcal{M}}_{g}$ for $g\geq 4$ and despite many authors contributing to the known effective divisors in $\overline{\mathcal{M}}_{g}$, e.g.~\cite{HM,FarkasPopa, F}, there are no known extremal rays of the pseudo-effective cone other than the boundary divisors for $g\geq12$. \\

In this paper, we study the moduli space of marked hyperelliptic curves and their stable degenerations $\overline{\mathcal{H}}_{g,n}\subset \mathcal{\overline{M}}_{g,n}$. We have two main results seemingly pointing in opposite directions. The first might advocate for the simplicity and the second for the complexity of $\overline{\mathcal{H}}_{g,n}$. In strong contrast to $\overline{\mathcal{M}}_{g,n}$, the transition from negative to maximal Kodaira dimension is surprisingly uniform and happens exactly at $n=4g+6$ for all $g\geq 2$.

\begin{theorem}
\label{thm1}
For all $g\geq 2$, the moduli space $\overline{\mathcal{H}}_{g,4g+6}$ has Kodaira dimension zero.
\end{theorem}

One might interpret Theorem \ref{thm1} as implying $\overline{\mathcal{H}}_{g,n}$ is a much simpler variety than $\overline{\mathcal{M}}_{g,n}$. In our second result we provide a full classification of the structure of the cone of effective divisors of $\overline{\mathcal{H}}_{g,n}$ which is as complicated as that of $\overline{\mathcal{M}}_{g,n}$ in the cases where the structure of the latter is known.

\begin{theorem}
\label{thm2}
The pseudoeffective cone of $\overline{\mathcal{H}}_{g,n}$ is
\begin{enumerate}
\item
generated by the irreducible components of the boundary for $n=0$,
\item
generated by the Weierstrass divisor and the irreducible components of the boundary for $n=1$,
\item
non-polyhedral and hence generated by infinitely many extremal rays for $n\geq 2$.
\end{enumerate}
\end{theorem}

In particular, as the Cox ring cannot be finitely generated when the pseudo-effective cone is non-polyhedral, we obtain the following corollary.

\begin{restatable}{corollary}{MDS}
\label{cor:MDS}
For all $g\geq 2$, the moduli space $\Hbars{g}{n}$ is not a Mori dream space for $n\geq2$.
\end{restatable}

Keel~\cite{Keel} shows that $\overline{\mathcal{M}}_{g,n}$ is not a Mori dream space for $g\geq3$ and $n\geq 1$ via a method that utilises surfaces in the boundary that arise from gluing morphisms not available in $\Hbars{g}{n}$ and hence the method cannot be used in our situation. \\

When a variety is of intermediate type the computation of the Kodaira dimension is usually a difficult undertaking. Our novel idea is to study the cone of moving or nef curves that is dual to the cone of pseudo-effective divisors. Our investigation of the curve geometry of $\overline{\mathcal{H}}_{g,n}$, leads us to the existence of an extraordinary nef curve $M$ in $\overline{\mathcal{H}}_{g,n}$ with the property that any effective divisor $D$ satisfying $M\cdot[D]=0$ must be rigid. This means that for any $k\geq 1$, the linear system $\left|kD\right|$ consist on a single divisor. The curve $M$ intersects the canonical divisor of $\overline{\mathcal{H}}_{g,n}$ negatively when $n\leq 4g+5$ and zero when $n=4g+6$. This shows that $\overline{\mathcal{H}}_{g,n}$ has negative Kodaira dimension when $n\leq 4g+5$, reproving a result of Benzo and Agostini--Barros \cite{Be, AB}, and as Schwarz~\cite{Sch} has shown the canonical divisor is effective for $n=4g+6$, this shows that $\overline{\mathcal{H}}_{g,n}$ is of Calabi-Yau type when $n=4g+6$.\\

As pointed out in \cite{Sca} and more recently in \cite{EH}, the smoothness of $\overline{\mathcal{H}}_{g,n}$ is unknown for $n\geq1$. Recently, Schwarz \cite{Sch} showed assuming the smoothness of $\overline{\mathcal{H}}_{g,n}$ that the Kodaira dimension is non-negative when $n\geq 4g+6$ and maximal when $n\geq 4g+7$. We also show the following theorem.

\begin{theorem}
The stack $\overline{\mathcal{H}}_{g,n}$ is smooth.
\end{theorem}

Together with \cite{Be, AB, Sch} we obtain the complete Kodaira classification.

\begin{corollary}
The Kodaira dimension of  the moduli space $\overline{\mathcal{H}}_{g,n}$ is given by
$${\rm{Kod}}\left(\overline{\mathcal{H}}_{g,n}\right)=\left\{\begin{array}{lcl}-\infty&\hbox{if}&n\leq 4g+5,\\
0&\hbox{if}& n=4g+6,\\
2g-1+n&\hbox{otherwise.}&\end{array}\right.$$
\end{corollary}

Throughout the paper we make use of the classical Hurwitz correspondence

\begin{equation}
\label{intro:eq1}
\begin{tikzcd}
&{\rm{Hur}}_{g,2}^n\arrow[dr, "\pi"]\arrow[dl, "\phi"']&\\
\overline{\mathcal{H}}_{g,n}&&\overline{\mathcal{M}}_{0,[2g+2]+n}.
\end{tikzcd}
\end{equation}
Where ${\rm{Hur}}_{g,2}^n$ be the Hurwitz space that parameterises admissible double covers
$$\left[f:C\to R, p_1,\ldots,p_n\right]\in {\rm{Hur}}_{g,2}^n,$$
with $R$ a rational curve and $p_i$ are marked points on the source curve where the image of all $p_i$ and the branch points of $f$ are distinct on the interior of the moduli space. See Section \ref{Section:Hur} for definitions. The map $\phi$ forgets the cover, and $\pi$ forgets the source curve while remembering branch points and the images under $f$ of the markings.\\

The proof of Theorem \ref{thm1} relies on the construction of a curve $M$ and its many splittings, enough such that combining with the natural rational tail versions of the curves provide a full rank set of conditions on the coefficients of any effective decomposition of the canonical class of $\overline{\mathcal{H}}_{g,n}$. More precisely (Corollary \ref{corollary:splittingM}), for any $1\leq i\leq g$ and $S\subset \{1,\ldots,n\}$,
$$M=B_{i,S}^\eta+B_{g-i-1, S^c}^{\delta},$$
where the summands are covering curves for the divisors $\eta_{i:S}$ and $\delta_{g-i-1:S^c}$ respectively. Any irreducible effective divisor $D$ not supported on  $\eta_{i:S}$ or $\delta_{g-i-1:S^c}$ such that $M\cdot[D]=0$ is forced to have trivial intersection with both $B_{i,S}^\eta$ and $B_{g-i-1, S^c}^{\delta}$ for all $i,S$. These conditions, along with those from the analogous rational tails curves are enough to ensure rigidity.\\

The curves $B^\delta$ and $B^\eta$ are obtained by pulling back a curve and different effective decompositions of this curve in $\overline{\mathcal{M}}_{0,[2g+2]+n}$ to $\Hbars{g}{n}$ via the Hurwitz correspondence \eqref{intro:eq1}. \\

Along the way we compute the class two geometric divisors independently obtained by Edidin and Hu in \cite{EH}. The first is the closure of the Weierstrass locus $W\subset\mathcal{H}_{g,1}$, that is the locus of $1$-pointed hyperelliptic curves $[C,p]$, where $p$ is a Weierstrass point. The second is the closure of the conjugate divisor $D\subset \mathcal{H}_{g,2}$, defined as the locus of $2$-pointed hyperelliptic curves $[C,p,q]$, such that $p$ and $q$ are conjugate points.\\

To prove Theorem \ref{thm2} we follow the strategy of~\cite{Mu} and identify an extremal ray of the dual cone of nef curves $\overline{\text{Nef}}_1(\Hbars{g}{n})$  where the cone is not polyhedral. As in $\overline{\mathcal{M}}_{g,n}$, the general fibre of the forgetful morphism
$$\pi:\Hbars{g}{n}\longrightarrow\Hbars{g}{n-1}$$
forms a nef curve class we denote $[F]$. For any curve class $[B]$ we define the pseudo-effective dual space by
$$[B]^\vee:=\{[D]\in \overline{\text{Eff}}^1(\Hbars{g}{n})\hspace{0.3cm}\big|\hspace{0.3cm} [B]\cdot[D]=0\}.$$
By analysing the fibres of the forgetful morphisms we bound the rank of the strictly effective divisors appearing in $[F]^\vee$. Then by considering the intersection of any divisor in $[F]^\vee$ with the surfaces obtained from
$$i:C\times C\longrightarrow \Hbars{g}{2}$$
for fixed general genus $g$ curve $C$, we obtain in Proposition~\ref{prop:corank}, bounds on the rank of $[F]^\vee$. For $g\geq 2$ and $n\geq2$,
$$\rho(\Hbars{g}{n})-n\leq\text{rank}([F]^{\vee}\otimes\mathbb{R})\leq \rho(\Hbars{g}{n})-2,$$
where $\rho(\Hbars{g}{n})$ denotes the Picard number of $\Hbars{g}{n}$.\\

The remaining task is to show that $[F]$ is indeed an extremal nef curve. This is achieved by identifying a series of effective divisors that contradict the nefness of curve classes appearing in any non-trivial nef decomposition of the curve $[F]$. The required divisors come from Hurwitz space constructions that Lemma~\ref{lem:eff} shows provides codimension one conditions. The classes for the corresponding divisors in $\Mbar{g}{n}$ were computed in~\cite{Mu} and pullback to give the required classes.\\

Finally, the isomorphism $\overline{{\bf{H}}}_{g}\overset{\sim}{\to} \overline{\mathcal{M}}_{0,[2g+2]}$ and the morphism $ \overline{\mathcal{M}}_{0,[2g+2]+1}{\to}\overline{{\bf{H}}}_{g,1}$ give the finite generation of the Cox rings in these cases and complete the proof of Corollary~\ref{cor:MDS}.\\

The outline of the paper is as follows. In Section \ref{section:can_class} we recall some basic facts about the Picard group of $\overline{\mathcal{H}}_{g,n}$ and the computation of the canonical class. The end of Section \ref{section:can_class} is devoted to the proof of the smoothness of $\overline{\mathcal{H}}_{g,n}$. In Section \ref{sec:int_type} we show that $\overline{\mathcal{H}}_{g,4g+6}$ is of intermediate type. In Section \ref{Section:Hur} we study in detail the Hurwitz correspondence and compute the classes of the Weierstrass and conjugate divisors. In Section \ref{section:M} we compute the numerics of a nef curve $M$ that cuts out a rigid face of the cone of divisors. In Section \ref{section:KodDim} we prove Theorem \ref{thm1} by showing that it lies on the aforementioned rigid face. Finally in Section \ref{section:eff} we prove Theorem \ref{thm2}.\\

We will denote by $\overline{\mathcal{H}}_{g,n}$ the stack defined as the inverse image of the closure of the moduli of hyperelliptic curve $\mathcal{H}_{g}\hookrightarrow\overline{\mathcal{M}}_g$ via the forgetful morphism $\overline{\mathcal{M}}_{g,n}\to \overline{\mathcal{M}}_g$. The stack $\overline{\mathcal{H}}_{g,n}$ parameterises families of stable models of $n$-pointed admissible double covers. We denote by $\overline{{\bf{H}}}_{g,n}$ the coarse space. As it is standard in the literature, by Kodaira dimension of the moduli space $\overline{\mathcal{H}}_{g,n}$ we mean the Kodaira dimension of the variety $\overline{{\bf{H}}}_{g,n}$.

\subsection*{Acknowledgments}

This paper benefited from correspondence and conversations with Daniele Agostini, Johannes Schmitt, and Irene Schwarz. Special thanks go to Ana-Maria Castravet and Jenia Tevelev for pointing out an issue with an earlier version of this paper. The first author was supported by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK. The second author was supported by the Alexander von Humboldt Foundation.

\section{The canonical class of $\overline{\mathcal{H}}_{g,n}$}
\label{section:can_class}

Recall that ${\rm{Pic}}_{\mathbb{Q}}(\overline{\mathcal{M}}_{g, n})$ is generated by $\lambda$, the first Chern class of the Hodge bundle, $\psi_i$ the first Chern class of the cotangent bundle on $\overline{\mathcal{M}}_{g,n}$ associated with the $i$-th marking for $i=1,\dots,n$ and the classes of the irreducible components of the boundary. We denote by $\delta_{\rm{irr}}$ the class of the locus $\Delta_{\rm{irr}}$ of curves with a non-separating node and $\delta_{i:S}$ for $0\leq i\leq g$, $S\subset\{1,\dots,n\}$ the class of the locus $\Delta_{i:S}$ of curves with a separating node that separates the curve such that one component has genus $i$ and contains precisely the markings of $S$. Hence $\delta_{i:S}=\delta_{g-i:S^c}$ and we require $|S|\geq 2$ for $i=0$ and $|S|\leq n-2$ for $i=g$. For $g\geq 3$ these divisors freely generate ${\rm{Pic}}_{\mathbb{Q}}(\overline{\mathcal{M}}_{g, n})$. Recall that $\delta_{1:\varnothing}$ is defined as $\frac{1}{2}\Delta_{1:\varnothing}$ accounting for the fact that the general element in $\Delta_{1:\varnothing}$ has automorphism group of order $2$.\\

Let $\overline{{\bf{H}}}_{g,n}$ denote the coarse moduli space for the stack $\overline{\mathcal{H}}_{g,n}$. Recall that the morphism $\overline{\mathcal{H}}_{g,n}\longrightarrow \overline{\bf{H}}_{g,n}$ induces a natural isomorphism on codimension one cycles
\begin{equation}
\label{sec2:eq1}
{\rm{CH}}^1\left(\overline{\mathcal{H}}_{g,n}\right)\otimes{\mathbb Q}\cong{\rm{CH}}^1\left(\overline{\bf{H}}_{g,n}\right)\otimes{\mathbb Q}.
\end{equation}
When $n=0$, the stack $\overline{\mathcal{H}}_g$ is smooth \cite[Ch. XI, Lemma 6.15]{ACG} and equation \eqref{sec2:eq1} is an isomorphism of Picard groups with rational coefficients. As pointed out in \cite{Sca} and more recently in \cite{EH}, the smoothness of stack $\overline{\mathcal{H}}_{g,n}$ was unknown for higher $n$. We settle this question at the end of the section.

\begin{theorem}
\label{thm:smHgn}
The stack $\overline{\mathcal{H}}_{g,n}$ is smooth.
\end{theorem}

In particular, $\overline{\bf{H}}_{g,n}$ is ${\mathbb Q}$-factorial and equation \eqref{sec2:eq1} holds also for rational Picard groups. In what follows we recall the main result in \cite{Sca} about the structure of ${\rm{CH}}^1(\overline{\mathcal{H}}_{g,n})\otimes {\mathbb Q}$ generalising \cite{Co} in the unpointed case.\\

The divisors $\Delta_{i:S}\subset \overline{\mathcal{M}}_{g,n}$ for $i\geq 1$ intersect $\overline{\mathcal{H}}_{g,n}$ transversally on an irreducible divisor whose class on $\overline{\mathcal{H}}_{g,n}$ we also denote $\delta_{i:S}$.\footnote{When $n=0$, $\delta_{i}$ is defined as $\frac{1}{2}$ times the pull back of $\delta_{i}$ on $\overline{\mathcal{M}}_g$. This is again to take into account the fact that a general curve in $\delta_i$ has twice as many automorphisms as the general hyperelliptic curve. Similarly, $\delta_{i:\varnothing}$ is also $\frac{1}{2}$ times the pull back of $\delta_{i:\varnothing}$ on $\overline{\mathcal{M}}_{g,n}$.} A general point in $\delta_{i:S}$ consist on two hyperelliptic curves of genus $g-i$ and $i$ meeting transversally at a common Weierstrass point. The divisor $\Delta_{\rm{irr}}$ intersects $\overline{\mathcal{H}}_{g}$ on
$$\lfloor\frac{g-1}{2}\rfloor+1$$
many components denoted $\eta_0,\eta_1,\ldots, \eta_{\lfloor\frac{g-1}{2}\rfloor}$, where the multiplicity of the intersection along $\eta_{i}$ for $i>0$ is $2$. Hence
\begin{equation}
\label{sec2:eq2}
{\delta_{\rm{irr}}}=\eta_0+2\sum_{i=1}^{\lfloor\frac{g-1}{2}\rfloor}\eta_i.
\end{equation}
The element $\eta_0\in{\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g}\right)$ is the class of the locus $\Xi_0$ of curves whose general element is an hyperelliptic curve with two conjugate points identified. Similarly, $\eta_i\in{\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g}\right)$ is the class of the locus $\Xi_i$ of curves whose general element consist of two hyperelliptic curves $C_1, C_2$ and two pairs of conjugate points $p,\overline{p}\in C_1$ and $q,\overline{q}\in C_2$, where the point $p$ is identified with $q$ and $\overline{p}$ with $\overline{q}$. When $n\geq 1$, we denote by $\eta_{i:S}\in{\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g, n}\right)$ the class of the locus $\Xi_{i:S}$ whose general element is a curve on $\Xi_i$, and the marked points indexed by $S$ are in the component of genus $i$.

\begin{theorem}[Thm. 1.1 in \cite{Sca}]
For $g\geq 2$, the classes $\psi_i$, $\eta_{irr}$, $\eta_{i:S}$, and $\delta_{i:S}$ freely generate ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,n}\right)$.
\end{theorem}

In particular, the restriction of $\lambda$ to $\overline{\mathcal{H}}_{g,n}$ can be expressed as an explicit linear combination of boundary classes, cf. \cite[Prop. 4.7]{CH}. The coarse map $c:\overline{\mathcal{H}}_g\to\overline{{\bf{H}}}_g$ is simply ramified at the divisor $\delta_i$ this accounts for the fact that a general hyperelliptic curve in $\delta_i$ has automorphism group of order four as opposed to the general one in the interior or in any $\eta$-boundary divisor whose only non-trivial automorphism is the hyperelliptic involution. The branch isomorphism induces a map $\alpha$ fitting in the following fiber diagram
\begin{equation}
\label{intro:diag_alpha}
\begin{tikzcd}
&{\rm{Hur}}_{g,2}\arrow[dl]\arrow[dr]&\\
\overline{\mathcal{H}}_g\arrow[rr,"\alpha"]&&\overline{\mathcal{M}}_{0,[2g+2]}.
\end{tikzcd}
\end{equation}
Here ${\rm{Hur}}_{g,2}$ stands for the moduli space of admissible double covers of genus $g$, see \cite{HM} and Section \ref{Section:Hur} for definitions. The map $\alpha$ is induced by the isomorphism between $\overline{\mathcal{H}}_g$ and ${\rm{Hur}}_{g,2}$, cf. \cite[Prop. 2.2]{EH} and is factors through an isomorphism $\overline{\bf{H}}_{g}\overset{\sim}{\to} \overline{\mathcal{M}}_{0,[2g+2]}$.\\

Let $\mathcal{P}\to B$ be a family of rational curves over a smooth one-dimensional base $B$, and $\Sigma_{2g+2}\subset \mathcal{P}$ the divisor associated to the $2g+2$ unordered sections. The pull-back via $\alpha$ of the family is given by the stable reduction $\widetilde{\mathcal{C}}\to\mathcal{C}$ of the unique family of admissible degree two covers $\widetilde{\mathcal{C}}\to \mathcal{P}$ branched over $\Sigma_{2g+2}$. Recall \cite[Ch. XIII, Sec. 8]{ACG} that if $\Gamma$ if a fiber of $\mathcal{P}\to B$ and $q$ is a node of $\Gamma$, then in suitable coordinates the family $\mathcal{P}$ has equation $xy=t^k$ around $q$ with $t\in B$. The integer $k$ is called the \textit{multiplicity} of the node $q$. If fibers of $\mathcal{P}$ around $t=0$ are smooth for $t\neq 0$ and the fiber at $t=0$ has only one node, then the intersection multiplicity of $B$ and the boundary divisor $\delta\subset \overline{\mathcal{M}}_{0,[2g+2]}$ is exactly $k$ at $t=0$.\\

For a general rational curve in the divisor classes $\delta_{0:[j]}$, the parity of $j$ determines whether the admissible cover is of $\delta$-type, that is, two hyperelliptic curves glued at a Weierstrass point, or of $\eta$-type; two hyperelliptic curves glued at pairs of hyperelliptic conjugate points. Assume $B$ intersects $\delta_{0:[j]}$ at $t=0$. If $j$ is odd, the admissible cover $C\subset \mathcal{C}$ over $\Gamma$ is ramified at the node forcing the multiplicity of $q$ to be even and the equation of $\mathcal{C}$ around the node over $q$ to be of the form $x'\cdot y'=t^{k/2}$, where $x'=x^2$ and $y'=y^2$. If $j$ is odd, then $C$ has two nodes both with local equations of the form $x'\cdot y'=t^k$. In both these cases no stable reduction is needed and the map $\widetilde{\mathcal{C}}\to\mathcal{C}$ is an isomorphism around $t=0$. Finally, if $j=2$, then $C$ consists on an hyperelliptic curve of genus $g-1$ with a rational bridge attached to two conjugate points. After stable reduction, the resulting curve $C'$ has only one node and the equation of $\mathcal{C}$ around the node is $x'\cdot y'=t^{2k}$. This together with \eqref{sec2:eq2} and the commutativity of \eqref{intro:diag_alpha} give rise to the following identities \cite[Ch. XIII, eq. 8.7]{ACG}:
\begin{equation}
\label{eqn:pull-back}
\alpha^*\delta_{0:[2]}=\frac{1}{2}\eta_{irr},\quad\alpha^* \delta_{0:\left[2i+2\right]}=\eta_i\quad\hbox{and}\quad\alpha^*\delta_{0:\left[2i+1\right]}=2\delta_{i}.
\end{equation}
From the formula \cite[Lemma 3.5]{KM} for the canonical class of $\overline{\mathcal{M}}_{0,[2g+2]}$ and taking into account the ramification of the coarse map $\overline{\mathcal{H}}_g\to\overline{\bf{H}}_g$ one computes\footnote{Recall that a general element in $\delta_{0:[2]}$ has automorphism of order two. So in the notation of \cite{KM} one has $\delta_{0:[2]}=\frac{1}{2}\widetilde{B}_2$.}
\begin{align}
K_{\overline{\mathcal{H}}_g}=&-\left(\frac{1}{2}+\frac{1}{2g+1}\right)\eta_{irr}+2\sum_{i=1}^{\lfloor\frac{g-1}{2}\rfloor}\left(\frac{(i+1)(2g-2i)}{2g+1}-1\right)\eta_i\\
&+\sum_{i=1}^{\lfloor\frac{g}{2}\rfloor}\left(\frac{2(2i+1)(2g-2j+1)}{2g+1}-3\right)\delta_i.
\end{align}
Moreover, if $p:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_g$ is the forgetful morphism and $q:\overline{\mathcal{H}}_{g,n}\to\overline{\mathcal{H}}_g$ the pull-back via the restriction $i:\overline{\mathcal{H}}_{g,n}\hookrightarrow\overline{\mathcal{M}}_{g,n}$, then from the identity
$$c_1\left(\Omega_q^1\right)=c_1\left(i^*\Omega_p^1\right)=i^*\left(K_{\overline{\mathcal{M}}_{g,n}}-p^*K_{\overline{\mathcal{M}}_g}\right)=\psi-2\sum_{\left|S\right|\geq 2}\delta_{0:S}$$
together with the fact that when $n\geq 2$ the coarse map $\overline{\mathcal{H}}_{g,n}\to \overline{\bf{H}}_{g,n}$ is ramified at $\delta_{i:\varnothing}$ one obtains

\begin{align}
\label{eq:can_class}
K_{\overline{\bf{H}}_{g,n}}=&\psi-\left(\frac{1}{2}+\frac{1}{2g+1}\right)\eta_{irr}+2\sum_{i\geq 1,S}\left(\frac{(i+1)(2g-2i)}{2g+1}-1\right)\eta_{i:S}\\
&+\sum_{i\geq1, S}\left(\frac{2(2i+1)(2g-2j+1)}{2g+1}-3\right)\delta_{i:S}-2\sum_{\left|S\right|\geq 2}\delta_{0:S}-\sum_{i\geq 1}\delta_{i:\varnothing}.
\end{align}
Logan's divisor $\overline{\rm{D}}_g$ is defined as the closure in $\overline{\mathcal{M}}_{g,g}$ of the locus
$${\rm{D}}_g=\left\{\left[C,p_1,\ldots,p_g\right]\in \mathcal{M}_{g,g}\left| h^0\left(C,\mathcal{O}_C(p_1+\ldots+p_g)\right)\geq 2\right.\right\}.$$
The formula in terms of generators of ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{M}}_{g,g}\right)$ was computed in \cite[Thm 5.4]{Lo}. When $n\geq g$ one can consider the {\it{symmetric pull-back}} to $\overline{\mathcal{M}}_{g,n}$ given by
$$\overline{\rm{D}}_n=\frac{1}{\binom{n}{g}}\sum_{\substack{T\subset\{1,\ldots,n\}\\ \left|T\right|=g}}\pi_T^*\overline{\rm{D}}_g,$$
where $\pi_T:\overline{\mathcal{M}}_{g,n}\to \overline{\mathcal{M}}_{g,g}$ is the forgetful map that remembers only the markings indexed by $T$. By pulling back $\overline{\rm{D}}_g$ to $\overline{\mathcal{H}}_{g,n}$ and using the relation between $\lambda$ and the generators of ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,n}\right)$, Schwarz \cite{Sch} shows that when $n>4g+6$, there exist $a,b>0$ such that
$$K_{\overline{\bf{H}}_{g,n}}=a\left(\psi_1+\ldots+\psi_n\right)+b\overline{\rm{D}}_{n}+E,$$
where $E$ is an effective sum of boundary divisors. Then bigness of the sum of $\psi$-classes is used to conclude that the canonical class of $\overline{\bf{H}}_{g,n}$ is big. Moreover, when $n=4g+6$, there exist a constant $b>0$ such that

\begin{equation}
\label{eff.decomposition}
K_{\overline{\bf{H}}_{g,n}}=b\overline{\rm{D}}_{n}+\sum_{i\geq 1, S}e_{i:S}\eta_{i:S}+\sum_{i\geq 0, S}d_{i:S}\delta_{i:S},
\end{equation}
where $e_{i:S}$ and $d_{i:S}$ are explicit non-negative coefficients. In particular $K_{\overline{\bf{H}}_{g,4g+6}}$ is ${\mathbb Q}$-effective.\\

\begin{theorem}[Thm. 1.1 in \cite{Sch}]
The canonical divisor $K_{\overline{\bf{H}}_{g,n}}$ in the Neron-Severi group $N_{{\mathbb Q}}^1\left(\overline{\bf{H}}_{g,n}\right)$ is effective when $n\geq 4g+6$ and big when $n\geq 4g+7$.
\end{theorem}

Let $\left[C\right]\in \mathcal{H}_g$ be a general hyperelliptic curve, $A$ a general $g_{g+1}^1$ and $B$ the unique $g_2^1$. Then $A$ and $B$ induce an embedding
$$C\hookrightarrow {\mathbb P}^1\times{\mathbb P}^1,$$
where $A$ is given by projection onto the second component and $B$ onto the first. The curve $C$ lies in the linear system $\left|\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}\left(g+1, 2\right)\right|$. Benzo \cite{Be} observes that $\dim\left|\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}\left(g+1, 2\right)\right|=4g+5$ and that general points on $C$ impose independent conditions on the linear system, in particular if $p_1,\ldots,p_{4g+4}$ are general points of $C$, then
$$\left|\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}\left(C\right)\otimes\mathcal{I}_{p_1,\ldots,p_{4g+4}}\right|\cong {\mathbb P}^1.$$
This induces a rational curve ${\mathbb P}^1\dashrightarrow\mathcal{H}_{g,4g+4}$ passing through a general point. The argument was extended in \cite[Prop. 2.1]{AB} where it is shown that there exists a rational map ${\mathbb P}^2\dashrightarrow\mathcal{H}_{g,4g+5}$ passing through a general point.

\begin{theorem}[\cite{AB, Be}]
The moduli space $\overline{\mathcal{H}}_{g,n}$ is uniruled for $n\leq 4g+5$.
\end{theorem}

These two results leave open only the case $n=4g+6$ as a transition case. First we complete the argument in \cite{Sch} by showing the smoothness of $\overline{\mathcal{H}}_{g,n}$.

\subsection{Smoothness of $\overline{\mathcal{H}}_{g,n}$ and Kodaira dimension}

The singularity analysis of the coarse space $\overline{\bf{H}}_{g,n}$ was done in \cite{Sch} where Schwarz observes that the argument of \cite{HM} for $\overline{\mathcal{M}}_g$ also applies in the hyperelliptic context. The extendability of $m$-canonical forms on the regular locus of $\overline{\bf{H}}_{g,n}$ to any desingularisation is obtained by means of the Reid-Tai criterion, cf. \cite{Re}, \cite[Appendix 1 to Section 1]{HM}. The criterion works under the assumption that locally around any point $\left[C,x_1,\ldots,x_n\right]$ in $\overline{\bf{H}}_{g,n}$, the coarse space is of the form $V\big/G$ where $V$ is a vector space and $G$ is a finite group acting linearly on $V$. In this case, $V$ is the space of first order deformations of $\left[C,x_1,\ldots,x_n\right]$ preserving the hyperellipticity of the stable model $\left[C'\right]\in \overline{\mathcal{H}}_g$ of $C$, and $G$ is the group of automorphisms of $C$ fixing the marked points. In particular, in order for $\overline{\bf{H}}_{g,n}$ to be locally of the form $V\big/G$, the dimension of $V$ must be equal to the dimension of $\overline{\bf{H}}_{g,n}$. If
$${\rm{Def}}_{\left[C,x_1,\ldots,x_n\right]}: {\bf{Art}}_{\mathbb C}\longrightarrow {\bf{Set}}$$
is the corresponding deformation functor for the stack $\overline{\mathcal{H}}_{g,n}$, then an equivalent formulation is the equality
$$\dim {\rm{Def}}_{\left[C,x_1,\ldots,x_n\right]}\left({\mathbb C}[{\varepsilon}]\right)=\dim T_{\left[C,x_1,\ldots,x_n\right]}\overline{\mathcal{H}}_{g,n}=2g-1+n.$$
Or, in other words, the smoothness of $\overline{\mathcal{H}}_{g,n}$. \\

We start by briefly recalling some deformation theory of nodal curves (see for instance \cite[Ch. XI]{ACG}). Let $\left[C,x_1,\ldots,x_n\right]$ be an $n$-pointed nodal curve (not necessarily stable). First order deformations are parameterised by the vector space
\begin{equation}
\label{sec2:eq3}
{\rm{Ext}}^1\left(\Omega_C, \mathcal{O}_C\left(-x_1-\ldots-x_n\right)\right)\cong H^0\left(C,\Omega_C\otimes \omega_C\left(x_1+\ldots+x_n\right)\right)^\vee,
\end{equation}
where $\Omega_C^1$ is the sheaf of $1$-forms and the isomorphism \eqref{sec2:eq3} is Serre's duality. The ${\rm{Ext}}$-group \eqref{sec2:eq3} seats in an exact sequence coming from the local-to-global spectral sequence:
\begin{equation}
\label{sec2:eq4}
0\to H^1\left(C,\mathcal{H}om\left(\Omega_C^1, \mathcal{O}_C(-\overline{x})\right)\right)\to{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C\left(-\overline{x}\right)\right)\to H^0\left(C,\mathcal{E}xt^1\left(\Omega_C^1, \mathcal{O}_C(-\overline{x})\right)\right)\to0,
\end{equation}
where $\overline{x}$ is short notation for the sum $x_1+\ldots+x_n$. The term on the left parameterises first order deformations that are locally trivial at the nodes. The term on the right parameterises first order deformations (smoothings) of the nodes. The inclusion $\mathcal{O}_C(-\overline{x})\to \mathcal{O}_C$ canonically induces a map of the corresponding exact sequences \eqref{sec2:eq4}. The middle vertical arrow
\begin{equation}
\label{sec2:eq5}
{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C\left(-\overline{x}\right)\right)\longrightarrow{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C\right)\end{equation}
is the differential at $\left[C,\overline{x}\right]$ of the forgetful map $\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_g$ only when both $\left[C,\overline{x}\right]$ and $\left[C\right]$ are stable. Using the induces map on exact sequences \eqref{sec2:eq4} one can easily see that in that case the map \eqref{sec2:eq5} is surjective, i.e., the forgetful morphism $\pi:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_g$ is smooth when restricted to the open substack consisting on $n$-pointed curves that remain stable after forgetting the markings. Since $\overline{\mathcal{H}}_g$ is smooth \cite[Ch. XI, Lemma 6.15]{ACG} and smoothness of morphisms is preserved under base change, one shows \cite[Prop. 5.4]{Sca} that $\overline{\mathcal{H}}_{g,n}$ is smooth on the locus of $n$-pointed hyperelliptic curves that remain stable after forgetting the markings. On the other hand, even for $n=1$, the proper map $\overline{\mathcal{H}}_{g,n}\to\overline{\mathcal{H}}_g$ has singular fibers so it cannot be smooth and we have to study more carefully the differentials of the morphisms in the fibered diagram:
\begin{equation}
\label{sec2:eq6}
\begin{tikzcd}
\overline{\mathcal{H}}_{g,n}\arrow[r]\arrow[d]&\overline{\mathcal{M}}_{g,n}\arrow[d, "\pi"]\\
\overline{\mathcal{H}}_g\arrow[r, "i"]&\overline{\mathcal{M}}_g.
\end{tikzcd}
\end{equation}

Let $C$ be a stable hyperelliptic curve. Recall that \cite[Ch. XI, Lemma 6.15]{ACG} the hyperelliptic involution $\sigma$ acts linearly on the tangent space of $\left[C\right]$ in $\overline{\mathcal{M}}_g$. Moreover, the tangent space of $\overline{\mathcal{H}}_g$ at $\left[C\right]$ can be identified with the $\sigma$-invariant subspace and the differential of the inclusion $i:\overline{\mathcal{H}}_g\to\overline{\mathcal{M}}_g$ is given by the natural inclusion:
\begin{equation}
\label{sec2:eq7}
di:{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C\right)^\sigma\longrightarrow{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C\right).
\end{equation}
Assume $\left(C,x_1,\ldots,x_{n}\right)$ is an $n$-pointed stable curve and let
\begin{equation}
\label{sec2:eq8}
f: C\longrightarrow C'
\end{equation}
be the stable reduction after forgetting the markings. The map $f$ enjoys several funtorial properties \cite[Tag 0E8A]{S-P}, \cite[Ch. X, Prop. 6.7]{ACG}, \cite[Lemma 1.6]{Kn}. It is defined in families, respects base change, $f_*\mathcal{O}_{C}=\mathcal{O}_{C'}$, $f_*\omega_C=\omega_{C'}$, $\omega_C=f^*\omega_{C'}$, and $R^1f_*\mathcal{O}_{C}=0$. Moreover, if
\begin{equation}
\label{sec2:eq9}
f^*\Omega_{C'}^1\to\Omega_C^1\to\Omega_f^1\to 0
\end{equation}
is the cotangent sequence for $f$, then the Leray spectral sequence for $Rf_*$ together with the map $\Omega_{C'}\to f_*\Omega_C$ induce a natural map that corresponds \cite{Kn} to the differential of the forgetful morphism $\pi:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_g$:
\begin{equation}
\label{sec2:eq10}
d\pi:{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C(-\overline{x})\right)\longrightarrow{\rm{Ext}}^1\left(\Omega_C^1, \mathcal{O}_C\right)\overset{\sim}{\longrightarrow}{\rm{Ext}}^1\left(f_*\Omega_C^1, f_*\mathcal{O}_C\right)\longrightarrow{\rm{Ext}}^1\left(\Omega_{C'}^1, \mathcal{O}_{C'}\right).
\end{equation}
The first map is induced by the inclusion $\mathcal{O}_{C}(-\overline{x})\to\mathcal{O}_{C}$, the isomorphism in the middle is induced by taking the Serre dual of the isomorphism induced by the Leray spectral sequence
$$H^0\left(C,\Omega_{C}^1\otimes\omega_{C}\right)\overset{\sim}{\longrightarrow} H^0\left(C,f_*\Omega_{C'}^1\otimes\omega_{C'}\right),$$
and the last map is obtained by taking the dual of the map induced by taking global sections of $\Omega_{C'}^1\to f_*\Omega_C^1$ after tensoring with $\omega_{C'}$. \\

The map $f$ can be expressed as a composition of $n$ maps, where each forgets one marking and contracts either a rational tail or a rational bridge. The following lemma shows that we can reduce to the case where $f$ only contracts rational bridges.

\begin{figure}[h]
\label{figure:rt-rb}
\caption{Local picture around the two possible contractions.}
\includegraphics[scale=0.1]{rt-rb}
\end{figure}

\begin{lemma}
\label{sec2:lemma1}
Let $(C,x_1,\ldots,x_n,x_{n+1})$ be an $(n+1)$-pointed stable curve and let $f:C\to C'$ be the stable reduction after forgetting the last marking. Assume further that $f$ only contracts one rational tail. Then, the differential of $\pi:\overline{\mathcal{M}}_{g,n+1}\to \overline{\mathcal{M}}_{g,n}$ at $\left[C,x_1,\ldots,x_n,x_{n+1}\right]$:
$${\rm{Ext}}^1\left(\Omega_{C}^1,\mathcal{O}_{C}(-x_1-\ldots x_{n+1})\right)\longrightarrow {\rm{Ext}}^1\left(\Omega_{C'}^1,\mathcal{O}_{C'}(-x_1-\ldots x_{n-1}-f(x_n))\right)$$
is surjective.
\end{lemma}

\begin{proof}
Let $p:\mathcal{C}_{g,n}\to\overline{\mathcal{M}}_{g,n}$ be the universal curve. Recall that \cite{Kn} the isomorphism $\overline{\mathcal{M}}_{g,n+1}\overset{\sim}{\to}\mathcal{C}_{g,n}$ is induced by stable reduction after forgetting the last marking. Moreover, the map $\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ factors as $\overline{\mathcal{M}}_{g,n+1}\overset{\sim}{\to}\mathcal{C}_{g,n}\to\overline{\mathcal{M}}_{g,n}$:
$$
\left[C,x_1\ldots,x_{n},x_{n+1}\right]\mapsto\left[C',f(x_1),\ldots,f(x_{n}),f(x_{n+1})\right]\mapsto\left[C',f(x_1),\ldots,f(x_{n})\right],
$$
where $f:C\to C'$ is the stable reduction after forgetting the last marking. The morphism $p:\mathcal{C}_{g,n}\to\overline{\mathcal{M}}_{g,n}$ is smooth away from the nodal locus
$$\left\{[C,x_1,\ldots,x_n,p]\in\mathcal{C}_{g,n}\left|\;\;p\in C_{sing}\right. \right\}.$$
In particular, since we are under the assumption that no rational bridge gets contracted, $f(x_{n+1})\not\in C'_{sing}$ and the differential of $p$ at $\left[C',f(x_1),\ldots,f(x_{n}),f(x_{n+1})\right]$ has maximal rank. Since $\pi$ is given by $p$ composed with an isomorphism, the differential of $\pi$ at $\left[C,x_1\ldots,x_{n},x_{n+1}\right]$ has also maximal rank.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:smHgn}]
Let $\left(C,x_1,\ldots,x_{n}\right)$ be a stable $n$-pointed hyperelliptic curve and $C'$ the stable model of the unpointed curve. By definition, $C'$ is hyperelliptic. Let $\sigma$ be the hyperelliptic involution. Recall that $\overline{\mathcal{H}}_{g,n}$ is defined as the stack-theoretic preimage of $\overline{\mathcal{H}}_g$ under the forgetful map $\pi:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_g$. In particular
$$T_{\left[C,x_1,\ldots,x_{n}\right]}\overline{\mathcal{H}}_{g,n}=d\pi^{-1}\left({\rm{Ext}}^1\left(\Omega_{C'}^1, \mathcal{O}_{C'}\right)^\sigma\right),$$
where
$$d\pi:{\rm{Ext}}^1\left(\Omega_{C}^1, \mathcal{O}_{C}\left(x_1+\ldots+x_n\right)\right)\longrightarrow {\rm{Ext}}^1\left(\Omega_{C'}^1, \mathcal{O}_{C'}\right)$$
is the differential of the forgetful map $\pi$ given by the compositon \eqref{sec2:eq10}. Smoothness of $\overline{\mathcal{M}}_{g,n}$ implies that
$${\rm{ker}}(d\pi)-{\rm{coker}}(d\pi)=n$$
and we only have to check the following transversality:
\begin{equation}
\label{sec2:eq11}
{\rm{codim}}\left({\rm{im}}(d\pi),{\rm{Ext}}^1\left(\Omega_{C'}^1, \mathcal{O}_{C'}\right)\right)={\rm{codim}}\left({\rm{im}}(d\pi)^\sigma,{\rm{Ext}}^1\left(\Omega_{C'}^1, \mathcal{O}_{C'}\right)^\sigma\right).
\end{equation}
Here ${\rm{im}}(d\pi)^\sigma$ stands for the intersection ${\rm{im}}(d\pi)\cap {\rm{Ext}}^1\left(\Omega_{C'}^1, \mathcal{O}_{C'}\right)^\sigma$. Let
\begin{equation}
\label{sec2:eq12}
d\pi^\vee:H^0\left(C', \Omega_{C'}^1\otimes\omega_{C'}\right)\longrightarrow H^0\left(C, \Omega_{C}^1\otimes\omega_{C}\left(\overline{x}\right)\right)
\end{equation}
be the dual of the map $d\pi$ and we denote by $(d\pi^\vee)^\sigma$ the restriction to the $\sigma$-invariant subspace. Then transversality \eqref{sec2:eq11} reads
$$\ker\left(d\pi^\vee\right)=\ker\left((d\pi^\vee)^\sigma\right).$$
The map \eqref{sec2:eq12} factors as follows
\begin{equation}
\label{sec2:eq13}
H^0\left(C', \Omega_{C'}^1\otimes\omega_{C'}\right)\overset{\alpha}\longrightarrow H^0\left(C', f_*\Omega_{C}^1\otimes\omega_{C}\right)\overset{\sim}{\to}H^0\left(C, \Omega_{C}^1\otimes\omega_{C}\right)\overset{\beta}{\to}H^0\left(C, \Omega_{C}^1\otimes\omega_{C}(\overline{x})\right).
\end{equation}
Since the markings are away from the nodes, ${\rm{Tor}}^1\left(\Omega_C^1,\mathcal{O}_{\overline{x}}\right)=0$ and $\beta$ is injective. By Lemma \ref{sec2:lemma1} we can assume $f$ only contracts rational bridges. In particular, one observes that the kernel of the map $\Omega_{C'}^1\to f_*\Omega_{C}^1$ is a direct sum of sky-scraper sheaves supported at the nodes of $C'$. Recall that the nodal locus is $\sigma$-invariant and therefore, sections in $\ker(\alpha)$ are also $\sigma$-invariant. This gives us the missing inclusion $\ker(d\pi^\vee)\subset \ker((d\pi^\vee)^\sigma)$.
\end{proof}

\section{Intermediate type}
\label{sec:int_type}

In this section we provide a geometric constructions of a nef or moving curve $\frak{M}$ in $\overline{\mathcal{H}}_{g,n}$ and show that when $n=4g+6$, the curve $\frak{M}$ intersects the canonical divisor $K_{\overline{\mathcal{H}}_{g,n}}$ trivially, proving that $\overline{\mathcal{H}}_{g,4g+6}$ is of intermediate type. In Section \ref{section:M} we realise $\frak{M}$ as a curve coming from the Hurwitz correspondence, cf. Definition \ref{def:M}. We will use the many splittings of these curves to produce non trivial conditions on elements in the canonical linear system. We produce just enough conditions to show that any effective divisor in $\left|mK_{\overline{\bf{H}}_{g,n}}\right|$ must be supported on a fixed set of divisors giving us rigidity. \\

Recall that a general hyperelliptic curve $C\in \overline{\mathcal{H}}_g$ can be embedded in ${\mathbb P}^1\times {\mathbb P}^1$ with degree $(2,g+1)$. Moreover,
$$h^0\left(\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}(2,g+1)\right)=4g+6$$
and for general points $p_1,\ldots,p_{4g+4}\in C$, the linear system
$$\Gamma=\left|\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}(C)\otimes \mathcal{I}_{p_1+\ldots+p_{4g+4}}\right|$$
induces a map
$$\Gamma={\mathbb P}^1\longrightarrow\overline{\mathcal{H}}_{g,4g+4}.$$
The universal surface
\begin{equation}
\label{eq:univS}
\begin{tikzcd}
\mathcal{S}\arrow[r]\arrow[d]&\overline{\mathcal{H}}_{g,4g+5}\arrow[d]\\
\Gamma\arrow[r]&\overline{\mathcal{H}}_{g,4g+5}
\end{tikzcd}
\end{equation}
is the blow-up of ${\mathbb P}^1\times{\mathbb P}^1$ at the base locus of $\Gamma$ consisting on exactly $\{p_1,\ldots,p_{4g+4}\}$.

\begin{lemma}
\label{int_type:lemma1}
Every curve in $\Gamma$ is stable and irreducible.
\end{lemma}
\begin{proof}
Since $\Gamma$ is a general pencil in $\left|\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}(2,g+1)\right|$, singular curves on $\Gamma$ are at worst $1$-nodal, cf. \cite[Corollary 2.10]{Vo}. Very ampleness of $\mathcal{O}_{{\mathbb P}^1\times{\mathbb P}^1}(C)$ together with \cite[Thm. A]{BL} gives us that all curves in $\Gamma$ are $2$-connected, so they must be irreducible.
\end{proof}

As pointed out in \cite[Prop. 2.1]{AB} diagram \eqref{eq:univS} already shows that $\overline{\mathcal{H}}_{g,4g+5}$ is dominated by a family of rational surfaces birational to blow-ups of a fixed quadric at $4g+4$ points. Let $L$ be a line in ${\mathbb P}^1\times {\mathbb P}^1$ with degree $(0,1)$ and $L'$ its proper transform on $\mathcal{S}$. Observe that $L'$ intersects the fiber of $\mathcal{S}\to \Gamma$ at $L\cdot C=2$ points. Consider the following diagram

\begin{equation}
\label{eq:univS2}
\begin{tikzcd}
\mathcal{X}\arrow[r]\arrow[d]&\overline{\mathcal{H}}_{g,4g+6}\arrow[d]\\
L'\arrow[r]&\overline{\mathcal{H}}_{g,4g+5}.
\end{tikzcd}
\end{equation}

The map $\mathcal{X}\to L'$ is given by the base change of $\mathcal{S}\to\Gamma$ along the $2$-to-$1$ map $L'\to\Gamma$, and the map $\tau:\mathcal{X}\to\mathcal{S}$ is branched over two fibers of $\mathcal{S}\to \Gamma$. Since $L'$ is a covering curve for $\mathcal{S}$, the pull-back $\frak{M}=\tau^*L'$ is a covering curve for $\mathcal{X}$ and therefore for $\mathcal{H}_{g,4g+6}$. Now \cite[Prop. 2.4]{AB} and Lemma \ref{int_type:lemma1} reads:

\begin{proposition}
\label{prop:M}
The curve $\frak{M}$ intersects the generators of ${\rm{Pic}}_{{\mathbb Q}}\left(\overline{\mathcal{H}}_{g,4g+6}\right)$ as follows: for all $i=1,\ldots,4g+6$
$$\frak{M}\cdot \psi_i=4,\;\;\; \frak{M}\cdot \eta_{irr}=4(8g+4),$$
and all other intersections are zero.
\end{proposition}

From \eqref{eq:can_class} we obtain
\begin{equation}
\label{sec3:eq3}
\frak{M}\cdot K_{\overline{\mathcal{H}}_{g,4g+6}}=c_*\frak{M}\cdot K_{\overline{\bf{H}}_{g,n}}=0.
\end{equation}
The curve $\frak{M}$ forms a covering curve for $\overline{\mathcal{H}}_{g,4g+6}$, in particular $\frak{M}$ is nef. This together with \cite[Thm. 1.1]{Sch} give:
\begin{corollary}
For all $g\geq 2$, the moduli space $\overline{\mathcal{H}}_{g,4g+6}$ is of intermediate Kodaira dimension.
\end{corollary}
\begin{proof}
The divisor $mK_{\overline{\bf{H}}_{g,4g+6}}$ is effective for some $m>0$, cf. \cite{Sch}. On the other hand, for any $m>0$, if $mK_{\overline{\bf{H}}_{g,4g+6}}$ is big, then there exists and effective divisor $E$ and an ample divisor $A$ such that
$mK_{\overline{\bf{H}}_{g,4g+6}}=E+A.$
But $\frak{M}$ is nef and $A$ is ample, thus $\frak{M}\cdot (E+A)>0$ which contradicts equation \eqref{sec3:eq3}.
\end{proof}

\section{Hurwitz spaces and the conjugate and Weierstrass divisors}
\label{Section:Hur}

Let ${\rm{Hur}}_{g,2}^n$ be the Hurwitz space parameterising marked degree two covers
$$\left[C\overset{2:1}{\longrightarrow}{\mathbb P}^1, p_1,\ldots,p_n\right]\in{\rm{Hur}}_{g,2}^n.$$
We denote by $\overline{{\rm{Hur}}}_{g,2}^n$ the compactification by admissible covers defined in \cite{HM}. We impose the conditions that the markings $p_i$ cannot be ramification points of the cover and any two markings $p_i, p_j$ on $C$ are not allowed to be mapped to the same point on the target rational curve. More concretely, for a scheme $T$, objects in $\overline{{\rm{Hur}}}_{g,2}^n(T)$ are isomorphism classes of flat $T$-families;
\begin{equation}
\label{eqn:univ_ad.cover}
\begin{tikzcd}
\mathcal{C}\arrow[rr, "f_T"]\arrow[dr]&&\mathcal{P}\arrow[dl]\\
&T,\arrow[bend left=40, ul, "\sigma_j"]&
\end{tikzcd}
\end{equation}
where
\begin{itemize}
\item[(i)] The map $\mathcal{C}\longrightarrow T$ is a family of $n$-marked semi-stable curves of genus $g$ with $\sigma_1,\ldots,\sigma_n:T\longrightarrow\mathcal{C}$ the $n$ disjoint sections not intersecting the nodes and Weierstrass points of the fibers. With the extra condition that no two points in $\sigma_1(t),\ldots,\sigma_n(t)$ are conjugate.
\item[(ii)] The family $\mathcal{P}\longrightarrow T$ is fiberwise a tree of rational curves and $f_T:\mathcal{C}\to\mathcal{P}$ is fiberwise an admissible cover of degree $2$.
\item[(iii)] The map $\mathcal{P}\longrightarrow T$ together with the induced sections $\tau_1,\ldots,\tau_n:T\longrightarrow\mathcal{P}$, and the relative branch divisor away from the nodes is an element in the $S_{2g+2}$-quotient of $\overline{\mathcal{M}}_{0,2g+2+n}(T)$, where the symmetric group acts by permuting the first markings.

\end{itemize}

We will denote by $\overline{\mathcal{M}}_{g,\left[n\right]+m}$ the $S_n$-quotient of $\overline{\mathcal{M}}_{g,n+m}$, where $S_n$ acts by permuting the first $n$ marked points. For every subset $S\subset\{n+1,\ldots,n+m\}$, the divisor
$$\sum_{\substack{T\subset \{1,\ldots,n\}\\|T|=t}}\delta_{i:T\cup S}\in {\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{M}}_{g,n+m}\right)$$
is $S_n$-invariant and descends to a divisor
$$\delta_{i:\left[t\right]+S}\in{\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{M}}_{g,\left[n\right]+m}\right).$$
When $m=0$ we simply write $\delta_{i:\left[t\right]}$. We denote by $\psi_j$ the usual $\psi$-classes on the ordered markings $j=n+1,\ldots,n+m$ and $\psi_\star$ the class in $\overline{\mathcal{M}}_{g,\left[n\right]+m}$ induced by the $S_n$-invariant class $\psi_1+\ldots+\psi_n$ on $\overline{\mathcal{M}}_{g,n+m}$. Recall that ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{M}}_{g,\left[n\right]+m}\right)$ is generated by $\lambda, \psi_\star, \psi_1,\ldots,\psi_m$ and boundary divisors $\delta_{i:\left[t\right]+S}$.\\

The space $\overline{{\rm{Hur}}}_{g,2}^n$ comes with two natural forgetful morphisms that we call $\phi$ and $\pi$. The map $\phi:\overline{{\rm{Hur}}}_{g,2}^n\to\overline{\mathcal{H}}_{g,n}$ forgets the cover and remembers the stable model of the pointed source curve. The second map $\pi:\overline{{\rm{Hur}}}_{g,2}^n\to\overline{\mathcal{M}}_{0,[2g+2]+n}$ is a forgetful morphism that remembers only the target rational curve together with the image of the markings and ramification points. They sit the the following commutative diagram:
\begin{equation}
\label{eqn:Hurwitz}
\begin{tikzcd}
&\overline{{\rm{Hur}}}_{g,2}^n\arrow[dl, "\phi"']\arrow[dr, "\pi"]&\\
\overline{\mathcal{H}}_{g,n}\arrow[d]&&\overline{\mathcal{M}}_{0,\left[2g+2\right]+n}\arrow[d]\\
\overline{\mathcal{H}}_g\arrow[rr, "\alpha"]&&\overline{\mathcal{M}}_{0,\left[2g+2\right]}.
\end{tikzcd}
\end{equation}

The map at the bottom is induced by the isomorphism between $\overline{\mathcal{H}}_g$ and $\overline{{\rm{Hur}}}_{g,2}$, cf. \cite[Prop. 2.2]{EH}.\\

The condition that marked hyperelliptic conjugate points are not allowed to be mapped to the same point makes the map $\pi$ finite of degree $2^{n-1}$ for $n>1$ and of degree $1$ for $n=0$. With this extra condition, the target marked curve is always stable. On the other hand, $\phi$ is only generically finite.

\begin{figure}[h]
\label{figure:Pic1}
\caption{Marked admissible double cover in the boundary $E_{(S_1,S_2)}$ (filled points are unordered).}
\includegraphics[scale=0.15]{Pic1}
\end{figure}

As usual, we will denote by $\psi_j$ the $\psi$-class in
$\overline{\mathcal{H}}_{g,n}$, $\overline{{\rm{Hur}}}_{g,2}^n$, and $\overline{\mathcal{M}}_{0,\left[2g+2\right]+n}$ corresponding to the $j$-th ordered marking. More precisely, let $p:\mathcal{C}\to B$ be a family parameterised by any of the moduli spaces in question, and $\sigma_j$ is a section. Then $\psi_j$ on $B$ is the first Chern class of $\sigma_j^*\Omega_p^1$, where $\Omega_p^1$ is the sheaf of relative $1$-forms of the family. Finally for any $3$-partition of the set of ordered markings
$$\mu=\left(S_1,S_2,S_3\right)\quad\hbox{with}\quad S_1\coprod S_2\coprod S_3=\{1,\ldots,n\},$$
we denote by $E_{\mu}$ or $E_{\left(S_1,S_2\right)}$ the divisor on $\overline{{\rm{Hur}}}_{g,2}^n$ whose general element corresponds to a cover having two rational tails exchanged by the hyperelliptic involution, where the markings indexed by $S_1$ sit on one rational tail, the ones indexed by $S_2$ on the conjugate rational tail, and the markings indexed by $S_3$ sit on the genus $g$ component. Observe that $E_{(S_1,S_2)}=E_{(S_2,S_1)}$. See Figure 2.\\

\begin{remark}
\label{rmk:exc_loc}
The composition ${\rm{Hur}}_{g,2}^n\to\overline{\mathcal{H}}_{g,n}\hookrightarrow \overline{\mathcal{M}}_{g,n}$ factors through $\overline{\mathcal{M}}_{g,[2g+2]+n}$ by remembering the ramification points. By looking at fibers of the aforementioned map one concludes that the exceptional locus of $\phi$ is exactly the union
$${\rm{exc}}(\phi)=\bigcup_{\substack{\mu=(S_1,S_2,S_3)\\\left|S_1\cup S_2\right|>1}} E_{\mu}\cup \bigcup_{\left|S\right|\geq 1}\pi^{-1}\left(\delta_{0:[2]+S}\right).$$
\end{remark}

Assume $n\geq 2$ and let $D_{\{j,k\}}=\pi_{\{j,k\}}^*D$, where
$$\pi_{\{j,k\}}:\overline{\mathcal{H}}_{g,n}\longrightarrow\overline{\mathcal{H}}_{g,2}$$
is the forgetful morphism that forgets all but the $j$th and $k$th marked points and $D$ is the class of the conjugate divisor in ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,2}\right)$. That is, the class of the closure of the locus of $2$-pointed hyperelliptic curves $\left[C,p_j,p_k\right]\in\overline{\mathcal{H}}_{g,2}$ such that $p_j$ and $p_k$ are conjugate points. The identities \eqref{eqn:pull-back} generalise as follows.

\begin{proposition}\label{prop:pullback}
The following identities hold in ${\rm{Pic}}_{\mathbb Q}\left(\overline{{\rm{Hur}}}_{g,2}^n\right)$:
\begin{equation*}
\begin{array}{rcl}
2\phi^*\delta_{i:S}&=&\pi^*\delta_{0:[2i+1]+S} \;\;\;\hbox{for $1\leq i\leq g-1$},\\
\phi^*\eta_{i:S}&=&\pi^*\delta_{0:[2i+2]+S} \;\;\;\hbox{for $1\leq i\leq g-2$},\\
\phi^*\left(\frac{1}{2}\eta_{\text{irr}}\right)&=&\pi^*\left(\sum_{S\subset[n]}\delta_{0:[2]+S}\right)\\
\phi^*D_{j,k}&=&E_{\left(\{j\}, \{k\}\right)}.
\end{array}
\end{equation*}
For $\psi$-classes\footnote{We use the same notation; $\psi_i$ for the $\psi$ class on $\overline{\mathcal{H}}_{g,n}$, ${\rm{Hur}}_{g,2}^n$, and $\overline{\mathcal{M}}_{0, [n]+m}$ corresponding to the $i$-th marking.} we have $\pi^*\psi_i=\psi_i$ and
$$\phi^*\psi_i=\psi_i-\sum_{i\neq S}E_{(\{i\},S)}-\delta_{0:[1]+\{i\}}.$$
\end{proposition}

\begin{proof}
The proof of first three identities is analogous to the one of the identities \eqref{eqn:pull-back}, and the forth identity is immediate. Let $\mathcal{P}\to B$ be a family of rational curves over some base $B$, and $\tau_1,\ldots,\tau_n$ the sections corresponding to the $n$ ordered markings. There is a unique family over $B$ of admissible double covers $f_B:\mathcal{C}\to\mathcal{P}$ ramified over the $2g+2$ unordered sections. The preimage $f_B^{-1}(\tau_i(B))\subset \mathcal{C}$ of $\tau_i(B)\subset \mathcal{P}$ via $f_B$ is a two-to-one cover of $\tau_i(B)$ and therefore of $B$. In order to lift the sections $\tau_1,\ldots,\tau_n$ to a section of $\mathcal{C}\to B$ we base change $f_B$ via the $2^n$-to-one cover
$$\pi_B:\widetilde{B}:=f_B^{-1}(\tau_1(B))\times_B\ldots\times_Bf_B^{-1}(\tau_n(B))\longrightarrow B.$$
Moreover, the sections don't intersect and $\tau_i(B)$ is not allowed to intersect the branch points making $\pi_B$ \'{e}tale. Consider the diagram

\begin{equation}
\begin{tikzcd}
\widetilde{C}\arrow[r, "\tilde{\pi}_B"]\arrow[d, "p"]&\mathcal{C}\arrow[r, "f_B"]\arrow[d, "q"]&\mathcal{P}\arrow[dl, "\rho"]\\
\widetilde{B}\arrow[bend left=40, u, "\sigma_i"]\arrow[r, "\pi_B"]&B.\arrow[bend right=40, ur, "\tau_i"']&
\end{tikzcd}
\end{equation}
The square is cartesian, the map $\pi_B$ \'{e}tale, and $f_B^*\Omega_{\rho}^1\left({\rm{log}}R\right)=\Omega_q^1({\rm{log}} Br)$, where $Br$ is the relative branch divisor and $R$ the relative ramification. Since the sections are disjoint from branch and ramification one obtains
$$\pi_B^*\tau_i^*\Omega_\rho^1=\sigma_i^*\Omega_{p}^1.$$
This gives us $\pi^*\psi_i=\psi_i$. For the last identity we observe that the map
$$\overline{{\rm{Hur}}}_{g,2}^n\to\overline{\mathcal{H}}_{g,n}\hookrightarrow\overline{\mathcal{M}}_{g,n}$$
factors through the forgetful morphism $p:\overline{\mathcal{M}}_{g,[2g+2]+n}\to\overline{\mathcal{M}}_{g,n}$. Let us call $r:\overline{{\rm{Hur}}}_{g,2}^n\to\overline{\mathcal{M}}_{g,[2g+2]+n}$ the map that remembers the ramification. By standard pull-back formulas for forgetful morphisms \cite[Lemma 4.28, Ch. XVII]{ACG} one has
$$p^*\psi_i=\psi_i-\sum_{t=1}^{2g+2}\delta_{0:[t]+\{i\}}.$$
At this point we observe that $r^*\delta_{0:[t]+\{i\}}=0$ when $t\neq 1$. This is because if non of the ordered markings collide, the geometry of the admissible cover forces the rational components to have exactly two ramification points. This can happen if either two ramification points come together in which case we obtain a rational bridge, generically that is a curve with two non-separating nodes so disjoint from $\delta_{0:[2]+\{i\}}$. This leaves as the only possibility one rational tail ramified at the node and at the unordered marking. We have
$$\phi^*\psi_i=r^*\psi_i-r^*\delta_{0:[1]+\{i\}}.$$
Finally, let $\mathcal{C}\to B$ be a family of admissible covers with $2g+2$ unordered sections corresponding to the ramification points and $n$ ordered sections. If we forget the map to $\mathcal{P}$, the only semistable and non-stable fibers are those with a rational tail having exactly one ordered marking on them. If one assumes $B$ is not contained in any $E$-divisor, since sections intersect any fiber transversally, the the stable reduction map ${\rm{st}}:\mathcal{C}\to \widehat{\mathcal{C}}$ is the blow up at the points ${\rm{st}}(\sigma_j(b))$, where $\sigma_j(b)$ lies on a rational tail with no other marking. This gives us on $B$ the relation
$$r^*\psi_i=\psi_i-\sum_{i\neq S}E_{(\{i\}, S)}.$$
\end{proof}

We observe at this point that the situation for $n=1$ is significantly simplified by the following diagram (see also \cite[Section 9.2]{Ru})
\begin{equation}
\label{eqn:nis1}
\begin{tikzcd}
\overline{\mathcal{M}}_{0,\left[2g+2\right]+1}\arrow[dr, "p"]\arrow[rr,"\beta"]&&\Hbarc{g}{1}\arrow[dl, "\gamma"']\\
&\overline{\mathcal{M}}_{0,\left[2g+2\right]}.&
\end{tikzcd}
\end{equation}
The map $\beta$ sends a point in $\overline{\mathcal{M}}_{0,\left[2g+2\right]+1}$ to its unique degree two admissible cover ramified over the $2g+2$ unordered points and the last ordered point is sent to any of the two preimages. The map is well-defined on ${\mathbb C}$-points but not on families that is why we have to declare the target of to be the coarse space.

\begin{proposition}\label{prop:pullback2}
The following hold
\begin{equation*}
\begin{array}{lll}
\gamma^*\delta_{0:[2]}=\frac{1}{2}\eta_{irr}\\
\gamma^*\delta_{0:[2i+2]}=\eta_{i:1}+\eta_{i:0}\\
\gamma^*\delta_{0:[2i+1]}=2\left(\delta_{i:1}+\delta_{i:0}\right)\\
p^*\delta_{0:[j]}=\begin{cases}\delta_{0:[j]+1}&\text{if $j=g+1$,}\\
\delta_{0:[j]+1}+\delta_{0:[j]}&\text{otherwise,}  \end{cases}
\end{array}
\quad
\begin{array}{lll}
\beta^*\psi=\psi_1-\frac{1}{2}\delta_{0:[1]+\{1\}} \\
\beta^*\delta_{i:\varnothing}=\frac{1}{2}\delta_{0:[2i+1]}\\
\beta^*\eta_{irr}=2\left(\delta_{0:[2]}+ \delta_{0:[2]+\{1\}}\right)\\
\beta^*\eta_{i:\varnothing}=\delta_{0:[2i+2]}.
\end{array}
\end{equation*}
\end{proposition}
\begin{proof}
Consider the following commutative diagram:
\begin{equation}
\label{prop:pullback.eq1}
\begin{tikzcd}
\overline{{\rm{Hur}}}_{g,2}^1\arrow[r, "\phi"]\arrow[d, "\pi"]&\overline{\mathcal{H}}_{g,1}\arrow[d, "c"]\\
\overline{\mathcal{M}}_{0,[2g+2]+1}\arrow[r, "p"]&\Hbarc{g}{1}.
\end{tikzcd}
\end{equation}
Both $\pi$ and $\phi$ are generically finite and surjective, so $\pi^*$ and $\phi^*$ are injective on Picard groups with ${\mathbb Q}$ coefficients and is enough to check equalities on $\overline{{\rm{Hur}}}_{g,2}^1$. Recall that the $\psi$-classes on $\Hbarc{g}{n}$ are defined as the push forward via $c$ of $\psi$-classes on $\overline{\mathcal{H}}_{g,n}$. For the $\psi$-class, by Proposition \ref{prop:pullback} we have
$$\pi^*p^*\psi=\phi^*c^*\psi=\phi^*\psi=\pi^*\left(\psi_1-\frac{1}{2}\delta_{0:[1]+\{1\}}\right).$$
The rest follow immediately from Proposition \ref{prop:pullback}.
\end{proof}
As consequence we also obtain the class of the Weierstrass divisor.
\begin{proposition}
\label{class:Weierstrass}
The class of the Weierstrass divisor $W$ in ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,1}\right)$ is given by
$$W=  \frac{g+1}{g-1}\psi-\frac{\eta_0}{2(2g+1)(g-1)}-\sum_{i=1}^{g-2}\frac{(i+1)(2i+1)}{(2g+1)(g-1)}\eta_{i:\varnothing}-\sum_{i=1}^{g-1}\frac{2i(2i+1)}{(2g+1)(g-1)}\delta_{i:\varnothing}.$$
\end{proposition}

\begin{proof}
From Remark \ref{rmk:exc_loc} it follows that the exceptional locus of $\beta$ is $\delta_{0:[2]+\{1\}}$. Hence $\beta^*W=\frac{1}{2}\delta_{0:[1]+\{1\}}+f\delta_{0:[2]+\{1\}}$ for some $f$. Consider the curve $B_{2g,\varnothing}\in N_1\left(\overline{\mathcal{M}}_{0,[2g+2]+1}\right)$ obtained by gluing two fixed rational curves at fixed points, where one component has $2g$ fixed unordered points and the other component has one ordered point and the remaining two unordered points where one of them moves freely on the component. See Figure 3. The intersection of $B_{2g,\varnothing}$ with standard divisors of $\overline{\mathcal{M}}_{0,[2g+2]+1}$ is a particular case of Proposition~\ref{prop:intersections_BiS}. Observe that the curve $B_{2g,\varnothing}$ is contracted by $\beta$ and hence $B_{2g,\varnothing}\cdot\beta^*W=0$ giving $f=1$. Let
$$W=c_\psi\psi+c_0\eta_{irr}+\sum e_{i}\eta_{i:\varnothing}+\sum d_i\delta_{i:\varnothing}$$
in ${\rm{Pic}}_{\mathbb Q}\left(\Hbarc{g}{1}\right)$. By Proposition~\ref{prop:pullback2} we obtain
$$\frac{1}{2}\delta_{0:[1]+\{1\}}+\delta_{0:[2]+\{1\}}=c_\psi(\psi_1-\frac{1}{2}\delta_{0:[1]+\{1\}}) +2c_0(\delta_{0:[2]}+\delta_{0:[2]+\{1\}})+\sum e_i\delta_{0:[2i+2]}+\sum \frac{d_i}{2}\delta_{0:[2i+1]}$$
and hence
$$c_\psi\psi_1=\left(\frac{1}{2}+\frac{c_\psi}{2}\right)\delta_{0:[1]+\{1\}}+(1-2c_0)\delta_{0:[2]+\{1\}}-2c_0\delta_{0:[2]}-\sum e_i\delta_{0:[2i+2]}-\sum \frac{d_i}{2}\delta_{0:[2i+1]}.$$
However,
$$\psi_1
=\sum_{j=2}^{2g+1}\frac{j(j-1)}{(2g+2)(2g+1)}\delta_{0:[j]} $$
(see~\cite{FarkasGibney} for example) and further, the boundary divisors form a basis for ${\rm{Pic}}_{\mathbb Q}\left(\Mbar{0}{[2g+2]+1}\right)$. Hence we obtain the class of $W$ in ${\rm{Pic}}_{\mathbb Q}\left(\Hbarc{g}{1}\right)$ by equating the coefficients on each side. This pulls back to the stack to give the result.
\end{proof}

This allows us to compute the class of the conjugate divisor in ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,2}\right)$

\begin{proposition}\label{class:conj}
The class of the divisor of conjugate points $D$ in ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,2}\right)$ is given by
$$D=  \frac{1}{g-1}(\psi_1+\psi_2)-\frac{\eta_0}{2(2g+1)(g-1)}-\frac{g+1}{g-1}\delta_{0:\{1,2\}}+\sum_{i=1}^{g-2}(e_{i:0}\eta_{i:\varnothing}+e_{i:1}\eta_{i:\{1\}})+\sum_{i=1}^{g-1}(d_{i:0}\delta_{i:\varnothing}+d_{i:1}\delta_{i:\{1\}})$$
where
$$e_{i:0}=\frac{-(i+1)(2i+1)}{(2g+1)(g-1)},\hspace{0.2cm}e_{i:1}=\frac{2gi-2i^2-2i-1}{(2g+1)(g-1)},\hspace{0.2cm}d_{i:0}=\frac{-2i(2i+1)}{(2g+1)(g-1)},\hspace{0.2cm}d_{i:1}=\frac{4gi-4i^2-2g-1}{(2g+1)(g-1)}.
$$
\end{proposition}
\begin{proof}
Consider the map
\begin{eqnarray*}
\begin{array}{cccc}
g:&\Hbars{g}{1}&\rightarrow& \Hbars{g}{2}\\
&[C,p]&\mapsto&[C\bigcup_{p=q}\mathbb{P}^1,p_1,p_2]
\end{array}
\end{eqnarray*}
that glues in the rational tail $[\mathbb{P}^1,q,p_1,p_2]$ by gluing points $p$ and $q$ to form a node. The pullback of the generators of the Picard group~\cite{AC} are
\begin{equation*}
g^*\eta_0=\eta_0,\hspace{0.3cm}g^*\delta_{0:\{1,2\}}=-\psi,\hspace{0.3cm}g^*\delta_{i:\varnothing}=\delta_{i:\varnothing},\hspace{0.3cm}g^*\psi_j=0,\hspace{0.3cm}g^*\eta_{i:\varnothing}=\eta_{i:\varnothing}
\end{equation*}
and the pullback of all other boundary divisors is zero. Further, $g^*D=W$ and hence we obtain from the Proposition~\ref{class:Weierstrass} the coefficients in $D$ of the generators $\eta_0,\eta_{i:\varnothing},\delta_{0:\{1,2\}}, $ and $\delta_{i:\varnothing}$.

Consider the forgetful morphism $\pi:\Hbars{g}{2}\longrightarrow\Hbars{g}{1}$ that forgets the second marked point. Over a point in the interior $\mathcal{H}_{g,1}$ the fibre of $\pi$ is irreducible. However, over a general point of $\delta_{i:\{1\}}$ the fibre is reducible. We denote by $F_{\delta_{i:\{1\}}}$ the class of the irreducible component of this curve that is contained in $\delta_{i:\{1\}}$ in $\Hbars{g}{2}$. It is a well-troaden exercise in intersection theory to compute the intersections of this curve class with the generators as (see for example~\cite{HM})
$$ F_{\delta_{i:\{1\}}}\cdot\psi_2=2(g-i)-1,\hspace{0.3cm}F_{\delta_{i:\{1\}}}\cdot\delta_{g-i:\varnothing}=1,\hspace{0.3cm}F_{\delta_{i:\{1\}}}\cdot\delta_{i:\{1\}}=-1    $$
and the intersection with all other generators is zero. Similarly, we denote by $F_{\eta_{i:\{1\}}}$ class of the curve that is the irreducible component contained in $\eta_{i:\{1\}}$ in $\Hbars{g}{2}$ of the fibre of $\pi$ over a general point in $\eta_{i:\{1\}}$. The intersections with the generators are given by
$$ F_{\eta_{i:\{1\}}}\cdot\psi_2=2(g-i-1),\hspace{0.3cm}F_{\eta_{i:\{1\}}}\cdot\eta_{g-i-1:\varnothing}=1,\hspace{0.3cm}F_{\eta_{i:\{1\}}}\cdot\eta_{i:\{1\}}=-1        $$
and the intersection with all other generator is zero.

The remaining coefficients and hence the full class of $D$ then follows from the set-theoretic observation that
$$F_{\delta_{i:\{1\}}}\cdot D=F_{\eta_{i:\{1\}}}\cdot D=0.$$
\end{proof}

We stress again that the classes of $W$ and $D$ were computed independently in~\cite{EH} by the method of test curves and indeed agree with our computation.

\section{Magic curves}
\label{section:M}

Let $S$ be a subset of the last $n$ markings $S\subset \{2g+2+1,\ldots,2g+2+n\}$, $i$ a non-negative integer with $0\leq i\leq 2g$, and $I\subset\{1,\ldots,2g+2\}$ any subset of cardinality $i$ on the first $2g+2$ markings. Let
\begin{equation}
\label{eqn:bound.map}
\xi_{I,S}:\overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\times\overline{\mathcal{M}}_{0,I\cup S\cup\{q\}}\longrightarrow\overline{\mathcal{M}}_{0,2g+2+n}
\end{equation}
be the boundary map that glues two marked rational curves
$$\left[R_1, p_{I^c}, q_{S^c}, p\right]\in \overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\quad\hbox{and}\quad\left[R_2,p_I,q_S, q\right]\in\overline{\mathcal{M}}_{0,I\cup S\cup\{q\}}$$
by identifying $p$ and $q$. Here $p_I$ is short notation for the set of points $p_i$ with $i\in I$, and similarly for the rest.\\

Let $j\in I^c$ and
\begin{equation}
\label{eqn:definitionBis}
\overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\longrightarrow \overline{\mathcal{M}}_{0,I^c\cup S^c\cup \{p\}-\{j\}}
\end{equation}
the corresponding forgetful morphism. We define the curve $B_{i, S}$ in $N_1\left(\overline{\mathcal{M}}_{0,\left[2g+2\right]+n}\right)$ as the push-forward of a fiber of \eqref{eqn:definitionBis};
$$F\times\{pt\}\subset \overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\times\overline{\mathcal{M}}_{0,I\cup S\cup\{q\}}$$
by the composition
$$\overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\times\overline{\mathcal{M}}_{0,I\cup S\cup\{q\}}\overset{\xi_{I,S}}{\longrightarrow}\overline{\mathcal{M}}_{0,2g+2+n}\longrightarrow\overline{\mathcal{M}}_{0,\left[2g+2\right]+n}.$$
Observe that since we are taking the $S_{2g+2}$-quotient, the definition of $B_{i,S}$ is independent of the set $I$ and $j\in I^c$, and only depends on the choice of $S$ and the cardinality $\left|I\right|=i$. Informally, the curve $B_{i,S}$ is obtained by gluing two fixed rational curves at fixed points, where one component has fixed points indexed by $S$ together with $i$-many fixed unordered points and the other component has fixed ordered points indexed by $S^c$ and $2g+2-i$ unordered points where one of them moves freely on the component and the rest are fixed. When $i+|S|\leq1$, the curve $B_{i,S}$ consist on a smooth rational curve with one of the unordered marked points moving freely and the rest fixed.\\

\begin{figure}[h]
\label{figure:BiS}
\caption{Curve $B_{i,S}$ (filled points are unordered).}
\includegraphics[scale=0.15]{BiS}
\end{figure}

\begin{proposition}
\label{prop:intersections_BiS}
Let $n$ be a non-negative integer, $0\leq i\leq 2g$, and $g\geq 2$. The curve $B_{i,S}\in N_1\left(\overline{\mathcal{M}}_{0,\left[2g+2\right]+n}\right)$ intersects the generators of ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{M}}_{0,\left[2g+2\right]+n}\right)$ as follows:

\begin{equation*}
\begin{array}{rcl}
B_{i,S}\cdot \delta_{0:\left[2\right]+T}&=&2\cdot\left(2g+1-i\right)\;\;\;\hbox{if}\;\;\;i\neq 2,\\
B_{i,S}\cdot \delta_{0:\left[i\right]+S}&=&-1\;\;\;\hbox{if}\;\;\;i+\left|S\right|\geq2,\\
B_{i,S}\cdot\delta_{0:\left[i+1\right]+S}&=&1\;\;\;\hbox{if}\;\;\;i+\left|S\right|\geq1,\\
B_{i,S}\cdot \psi_k&=&1\;\;\;\hbox{if}\;\;\;k\in S^c,\\
B_{i,S}\cdot \psi_\star&=&\left\{\begin{array}{cc}2(2g-i)+1+\left|S^c\right|&\hbox{if }i+\left|S\right|\geq 2,\\2(2g-i)+\left|S^c\right|&\hbox{otherwise,}\end{array}\right.\\
B_{i,S}\cdot\delta_{0:\left[1\right]+\{j\}}&=&1\;\;\;\hbox{if}\;\;\;S^c\neq \varnothing\hbox{ and }j\in S^c,
\end{array}
\end{equation*}
and all other intersections are trivial.
\end{proposition}

\begin{proof}
Recall that the pull-back of $\psi_\star$ by the $S_{2g+2}$-quotient morphism is $\psi_1+\ldots+\psi_{2g+2}$. By construction and the projection formula,
\begin{equation}
\label{eqn:int1_BiS}
B_{i,S}\cdot \psi_\star=F\cdot\theta_{I,S}^*\left(\psi_1+\ldots+\psi_{2g+2}\right),
\end{equation}
where $I$ is any subset of $\{1,\ldots,2g+2\}$ of cardinality at most $2g$, $\theta_{I,S}$ is the restriction of the boundary map \eqref{eqn:bound.map} to
$$\overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\times \{pt\}\subset\overline{\mathcal{M}}_{0,I^c\cup S^c\cup\{p\}}\times\overline{\mathcal{M}}_{0,I\cup S\cup\{q\}},$$
and $F$ is the class of a fiber of the forgetful map \eqref{eqn:definitionBis} obtained by forgetting some marking $j\in I^c$. Recall that $F\cdot \psi_k=1$ if $k\neq j$. Similarly, by \cite[Lemma 4.38, Section 17]{ACG},
$$
\theta_{I,S}^*\psi_k=\left\{\begin{array}{ccc}\psi_k&\hbox{if}&k\in I^c,\\
0&\hbox{otherwise.}&\end{array}\right.
$$
Thus, for $k\in I^c$,
$$F\cdot \psi_k=1\;\;\hbox{for }k\neq j\quad\hbox{and}\quad F\cdot \psi_j=\left\{\begin{array}{ccc}-2+\left|I^c\right|+\left|S^c\right|&\hbox{if}&i+\left|S\right|\geq 2,\\
-3+\left|I^c\right|+\left|S^c\right|&\hbox{otherwise.}&\end{array}\right.$$
This together with \eqref{eqn:int1_BiS} imply
$$B_{i,S}\cdot \psi_\star=\left\{\begin{array}{cc}2(2g-i)+1+\left|S^c\right|&\hbox{if }i+\left|S\right|\geq 2,\\2(2g-i)+\left|S^c\right|&\hbox{otherwise,}\end{array}\right.\quad\hbox{and}\quad B_{i,S}\cdot \psi_k=1\;\;\hbox{for }k\in S^c.$$
Similarly, by projection formula
\begin{equation}
\label{eqn:int2_BiS}
B_{i,S}\cdot \delta_{0:\left[i\right]+S}=F\cdot\theta_{I,S}^*\left(\sum_{\substack{J\subset\{1+\ldots+2g+2\}\\ \left|J\right|=i}}\delta_{0:J\cup S}\right)
\end{equation}
and \cite[Lemma 4.38, Section 17]{ACG} reads
$$\theta_{I,S}^*\delta_{0,J\cup S}=\left\{\begin{array}{ccc}-\psi_p&\hbox{if}&J=I\hbox{ or }J=I^c\hbox{ and }S=\varnothing\\0&\hbox{otherwise,}&\end{array}\right.$$
and $F\cdot\psi_p=1$. Recall that if $S=\varnothing$, then $\delta_{0,I}=\delta_{0,I^c}$ and we count it only once in the sum on the right hand side of \eqref{eqn:int2_BiS}. Thus,
$$B_{i,S}\cdot \delta_{0:\left[i\right]+S}=-1.$$
The remaining intersections are clear from the construction, having into consideration that the $S_n$-quotient $\overline{\mathcal{M}}_{0,n+m}\to\overline{\mathcal{M}}_{0,[n]+m}$ is simply ramified over $\delta_{0:[2]+T}$, thus the factor of $2$ in the first equality.
\end{proof}

Proposition \ref{prop:intersections_BiS} implies
$$B_{0,\varnothing}=B_{i,S}+B_{2g+1-i,S^c}$$
for any $i\leq 2g$ and $S$. Geometrically, for each $i$ and $S$, a smooth $\left(\left[2g+2\right]+n\right)$-pointed rational curve can be deformed to a curve on the boundary $\delta_{0:\left[i\right]+S}$, where the the free unordered point on the smooth curve, after degenerating will move freely on both components.\\

We now define the corresponding curve classes in $\overline{\mathcal{H}}_{g,n}$ whose numerics will give us our main result.

\begin{definition}
\label{def:M}
Recall diagram \eqref{eqn:Hurwitz}. For subsets $S, \{a,b\}\subset \{1,\ldots,n\}$, we define the curve classes
$$
M:=\frac{1}{2^{n-1}}\phi_*\pi^* B_{0,\varnothing},\quad
B_{i,S}^\eta:=\frac{1}{2^{n-1}}\phi_*\pi^* B_{2i+2,S},
$$
$$
B_{i,S}^\delta:=\frac{1}{2^{n-1}}\phi_*\pi^* B_{2i+1,S},\quad\hbox{and}\quad B_{\{a,b\}}:=\frac{1}{2^{n-1}}\phi_*\pi^* B_{0,\{a,b\}}-B_{0,\{a,b\}}^\delta.
$$
\end{definition}
\begin{remark}
Note that the curve $M$ has also the geometric incarnation given by the curve constructed in Proposition \ref{prop:M}, more precisely, $\frak{M}=4M$.
\end{remark}

One observes that  $B_{i,S}^\delta$ and $B_{g-i,S^c}^\delta$ are in general different curves and both cover the divisor $\delta_{i:S}$ in $\overline{\mathcal{H}}_{g,n}$. Similarly $B_{i,S}^\eta$ and $B_{g-i,S^c}^\eta$ are covering curves for $\eta_{i:S}$, while

\begin{proposition}
\label{porp:magic_curves}
The curve classes defined above intersect the generators of ${\rm{Pic}}_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,n}\right)$ as follows:
\begin{equation*}
\begin{array}{ll}
M\cdot\psi_j=1& M\cdot\eta_{\text{irr}}=8g+4,
\end{array}
\end{equation*}

\begin{equation*}
\begin{array}{ll}
B_{i,S}^\delta\cdot\psi_k=\left\{\begin{array}{ccc}0&\hbox{ if }&k\in S,\\ 1&\hbox{otherwise},&\end{array}\right.& B_{i,S}^\delta\cdot\delta_{i:S}=-1,\\
B_{i,S}^\delta\cdot\eta_{\rm{irr}}=8(g-i),& B_{i,S}^\delta\cdot\eta_{i: S}=2,
\end{array}
\end{equation*}

\begin{equation*}
\begin{array}{ll}
B_{i,S}^\eta\cdot\psi_k=\left\{\begin{array}{ccc}0&\hbox{ if }&k\in S,\\ 1&\hbox{otherwise},&\end{array}\right.& B_{i,S}^\eta\cdot\delta_{i+1:S}=1,\\
B_{i,S}^\eta\cdot\eta_{\rm{irr}}=8(g-i)-4,& B_{i,S}^\eta\cdot\eta_{i: S}=-2,
\end{array}
\end{equation*}

\begin{equation*}
\begin{array}{ll}
B_{0,S}^\delta\cdot\psi_k=\left\{\begin{array}{ccc}0&\hbox{ if }&k\in S,\\ 1&\hbox{otherwise},&\end{array}\right.& B_{0,S}^\delta\cdot\delta_{0:S}=-1,\\
B_{0,S}^\delta\cdot\eta_{\rm{irr}}=8g+4,&
\end{array}
\end{equation*}

\begin{equation*}
\begin{array}{ll}
B_{\{a,b\}}\cdot\psi_k=\left\{\begin{array}{ccc}2&\hbox{ if }&k\in \{a,b\},\\ 1&\hbox{otherwise},&\end{array}\right.& B_{\{a,b\}}\cdot\delta_{0:\{a,b\}}=1,\\
B_{\{a,b\}}\cdot\eta_{\rm{irr}}=8g+4,&
\end{array}
\end{equation*}
and other intersections are trivial.
\end{proposition}

\begin{proof}
Observe via Proposition~\ref{prop:pullback} and the projection formula
\begin{equation}
\begin{array}{rl}
M\cdot\eta_{\text{irr}}=\frac{1}{2^{n-1}}\phi_*\pi^* B_{0,\varnothing}\cdot \eta_{\text{irr}}&=\frac{1}{2^{n-1}}\pi^* B_{0,\varnothing}\cdot  \phi^*\eta_{\text{irr}}\\
&=\frac{1}{2^{n-1}}\pi^* B_{0,\varnothing}\cdot 2(\pi^*\delta_{[2]}+\sum_{S\subset[n]}e_S\pi^*\delta_{[2]+S})\\
&=\frac{1}{2^{n-2}}\pi_*\pi^* B_{0,\varnothing}\cdot\delta_{[2]}\\
&=\frac{1}{2^{n-2}}\deg(\pi)\left(B_{0,\varnothing}\cdot\delta_{[2]}\right)\\
&=2 \left(B_{0,\varnothing}\cdot\delta_{[2]}\right)\\
&=8g+4
\end{array}
\end{equation}
by Proposition~\ref{prop:intersections_BiS}. Set theoretically it is clear that $M$ has zero intersection with all other boundary divisors. The intersections with the $\psi_j$ are obtained from the observation that $M\cdot D_{a,b}=0$ for all $\{a,b\}\subset [n]$.

Similarly, via Proposition~\ref{prop:pullback} and the projection formula we obtain
\begin{equation}
\begin{array}{rl}
B_{i,S}^\delta\cdot\eta_{\text{irr}}=\frac{1}{2^{n-1}}\phi_*\pi^* B_{2i+1,S}\cdot \eta_{\text{irr}}&=\pi^* B_{2i+1,S}\cdot  \phi^*\eta_{\text{irr}}\\
&=\frac{1}{2^{n-1}}\pi^* B_{2i+1,S}\cdot 2(\pi^*\delta_{[2]}+\sum_{S\subset[n]}e_S\pi^*\delta_{[2]+S})\\
&=\frac{1}{2^{n-2}}\pi_*\pi^* B_{2i+1,S}\cdot\delta_{[2]}\\
&=\frac{1}{2^{n-2}}\deg(\pi)\left(B_{2i+1,S}\cdot\delta_{[2]}\right)\\
&=2\left(B_{2i+1,S}\cdot\delta_{[2]}\right)\\
&=8(g-i)
\end{array}
\end{equation}
by Proposition~\ref{prop:intersections_BiS}. Set theoretically it is clear that $B_{i,S}^\delta$ has zero intersection with all other boundary divisors outside of $\eta_{i:S}$ and $\delta_{i:S}$. Further Proposition~\ref{prop:pullback} and the projection formula yield
\begin{equation}
\begin{array}{rl}
B_{i,S}^\delta\cdot\eta_{i:S}=\frac{1}{2^{n-1}}\phi_*\pi^* B_{2i+1,S}\cdot \eta_{i:S}&=\frac{1}{2^{n-1}} \pi^* B_{2i+1,S}\cdot  \phi^*\eta_{i:S}\\
&=\frac{1}{2^{n-1}}\pi^* B_{2i+1,S}\cdot \pi^*\delta_{[2i+2]+S}\\
&=\frac{1}{2^{n-1}}\pi_*\pi^* B_{2i+1,S}\cdot\delta_{[2i+2]+S}\\
&=\frac{1}{2^{n-1}}\deg(\pi)\left( B_{2i+1,S}\cdot\delta_{[2i+2]+S}  \right)\\
&=B_{2i+1,S}\cdot\delta_{[2i+2]+S} \\
&=2
\end{array}
\end{equation}
by Proposition~\ref{prop:intersections_BiS}. While
\begin{equation}
\begin{array}{rl}
B_{i,S}^\delta\cdot\delta_{i:S}=\frac{1}{2^{n-1}}\phi_*\pi^* B_{2i+1,S}\cdot \delta_{i:S}&=\frac{1}{2^{n-1}}\pi^* B_{2i+1,S}\cdot  \phi^*\delta_{i:S}\\
&=\frac{1}{2^{n-1}}\pi^* B_{2i+1,S}\cdot\frac{1}{2}\pi^*\delta_{[2i+1]+S}\\
&=\frac{1}{2^{n-1}}\pi_*\pi^* B_{2i+1,S}\cdot\frac{1}{2}\delta_{[2i+1]+S}\\
&=\frac{1}{2^n}\deg(\pi)\left(B_{2i+1,S}\cdot\delta_{[2i+1]+S} \right)\\
&=\frac{1}{2}\left(B_{2i+1,S}\cdot\delta_{[2i+1]+S} \right)\\
&=-1
\end{array}
\end{equation}
by Proposition~\ref{prop:intersections_BiS}.

The intersection numbers for $B_{i,S}^\eta$ can be computed in the same way. A further check on these computations arises from the observation that in $\Mbar{0}{[2g+2]+n}$,
$$B_{2i+2,S} + B_{2g-2i-1,S^c}=B_{0,\varnothing}$$
and hence by $\phi_*\pi^*$ we obtain
$$B^\eta_{i,S}+B^\delta_{g-i-1,S^c}=M. $$

Now consider the gluing morphism
$$\theta:\Hbars{g}{n+1-|S|}\to\Hbars{g}{n}$$
be the boundary map that associates to any $\left(\{r\}\cup\{1\ldots,n\}\setminus S\right)$-pointed stable hyperelliptic curve curve of genus $g$, the $n$-pointed curve obtained by gluing to it a fixed general $S\cup\{s\}$-pointed rational tail by identifying $r$ with $s$. We have
$$\theta^*\psi_j=\begin{cases} 0&\text{for $j\in S$}\\ \psi_j&\text{for $j\notin S$}\end{cases}\quad \theta^*\delta_{0:S}=-\psi_r\quad \theta^*\delta_{i:T\sqcup S}=\delta_{i:T}\quad \theta^*\eta_{i:T\sqcup S}=\eta_{i:T}\quad\theta^*\eta_{\text{irr}}=\eta_{\text{irr}}.$$
The intersection numbers for the curve $B_{0:S}^\delta$ follow from the observation that $B_{0:S}^\delta=\theta_*M$ and an application of the projection formula.

The intersection numbers for $B_{\{a,b\}}$ follow from the formula
$$B_{\{a,b\}}:=\phi_*\pi^* B_{0,\{a,b\}}-B_{0,\{a,b\}}^\delta$$
and an application of the projection formula to the first term on the right.
\end{proof}

An immediate consequence is the already mentioned splitting of the curve $M$ that will be key in the proof of Theorem \ref{thm1}.
\begin{corollary}
\label{corollary:splittingM}
For any $1\leq i\leq g$ and $S\subset \{1,\ldots,n\}$,
$$M=B^\eta_{i,S}+B^\delta_{g-i-1,S^c}.$$
\end{corollary}

\begin{remark}
As our results rely on these intersection numbers, we provide a number of cross-checks. Observe in $\Mbar{0}{[2g+2]+n}$ we have
$$B_{0,\varnothing}=B_{1,\{1\}}+B_{2g,\{2,\dots,n\}}$$
and further $\phi_*\pi^*B_{2g,\{2,\dots,n\}}=0$ hence
$$\phi_*\pi^*B_{0,\varnothing}=\phi_*\pi^*B_{1,\{1\}}$$

With this observation we can exploit the gluing morphisms of $\Mbar{g}{n}$ that do not exist for $\overline{\mathcal{H}}_{g,n}$ as the curve $\phi_*\pi^*B_{1,\{1\}}$ is contained in the loci in $\overline{\mathcal{H}}_{g,n}$ where the first marked point is a Weierstrass point. Consider the following diagram.
\begin{equation}
\label{eqn:Gluing}
\begin{tikzcd}
\overline{\mathcal{H}}_{g-i,(n-s)+1}\arrow[d,"i"]&&\overline{\mathcal{H}}_{g,n}\arrow[d,"i"]\\
\Mbar{g-i}{(n-s)+1}\arrow[rr,"\pi_S"]&&\Mbar{g}{n}
\end{tikzcd}
\end{equation}
Then under the obvious appropriate labelling
$$i_*B^\delta_{i,S}=\pi_S{}_*i_*\phi_*\pi^*B_{1,\{1\}}=\pi_S{}_*i_*[M].$$
Then intersection numbers of $M$ and the projection formula provide
\begin{equation*}
\begin{array}{llll}
B^\delta_{i,S}\cdot i^*\psi_j=\begin{cases}0&\text{ for $j\in S$}\\
1& \text{ for $j\notin S$,} \end{cases}&B^\delta_{i,S}\cdot i^*\delta_{\rm{irr}}=8(g-i)+4,&  B^\delta_{i,S}\cdot\delta_{i:S}=-1.&
\end{array}
\end{equation*}
It is clear $B^\delta_{i,S}\cdot\eta_0=8(g-i)$ and hence by Equation~\ref{sec2:eq2} we obtain $B^\delta_{i,S}\cdot\eta_{i:S}=2$.

More cross-checks arise from the observation that
$$B_{i,S}^\delta\cdot W_j=B_{i,S}^\eta\cdot W_j=0  $$
for $i\geq1$ and $j\in S$. Where $W_j=\pi_j^*W$ for $\pi_j:\Hbars{g}{n}\longrightarrow \Hbars{g}{1}$ the morphism that forgets all but the $j$th marked point.
\end{remark}

\section{The Kodaira dimension of $\Hbars{g}{n}$}
\label{section:KodDim}

Finally we are in a position to prove Theorem \ref{thm1}. As mentioned in the introduction, the curve $M\in N_1\left(\overline{\mathcal{H}}_{g,n}\right)$ has the remarkable property that if intersects trivially an effective divisor, then the divisor must be rigid. The reason for this is the abundance of splittings stated in Corollary \ref{corollary:splittingM} and the following Lemma.

\begin{lemma}
\label{lem:splitting}
Let $X$ be a normal projective variety and $M\in N_1\left(X\right)$ a nef curve class such that
$$M=\beta_1+\beta_2,$$
where $\beta_i$ is the class of a curve that sweeps out a divisor $D_i$ with $\beta_i\cdot D_i<0$. Then, for any divisor $D$ such that $M\cdot D=0$ and $D$ is either extremal in the pseudo-effective cone, or effective and not supported on $D_1\cup D_2$, one must have
$$\beta_1\cdot D=0\quad\hbox{and}\quad\beta_2\cdot D=0.$$
\end{lemma}
\begin{proof}
By contradiction and without loss of generality assume $\beta_1\cdot D>0$, then $\beta_2\cdot D<0$. Since $\beta_2$ is a covering curve for $D_2$, the divisor  $D_2$ lies in the base locus of all effective divisors with negative intersection with $\beta_2$. This provides a contradiction if $D$ is effective and not supported on $D_1\cup D_2$. Further, this implies
$$D-\left(\frac{\beta_2\cdot D}{\beta_2\cdot D_2}\right)D_2$$
is pseudo-effective, showing $D$ cannot be extremal in the pseudo-effective cone completing the contradiction.
\end{proof}

This lemma when combined with the decompositions of $M$ into covering curves given in Corollary~\ref{corollary:splittingM}, and the covering curves we have constructed for the rational tail divisors gives the following proposition.

\begin{proposition}\label{prop:rigid}
If $D$ is a pseudo-effective divisor in $\overline{\mathcal{H}}_{g,n}$ and $M\cdot D=0$ then $D$ is effective, rigid and has a unique rigid pseudo-effective decomposition
$$D=\sum_{{\{a,b\}\subset[n]}}c_{\{a,b\}}D_{\{a,b\}}+\sum_{i:S} c_{i:S}^\delta\delta_{i:S}+\sum_{i:S}c_{i:S}^\eta\eta_{i:S}$$
with all constants non-negative.
\end{proposition}

\begin{proof}
Assume $D$ pseudo-effective, $M\cdot D=0$ and $D$ can be expresses as a pseudo-effective decomposition not of this form. Then there exists a pseudo-effective decomposition of $D$ containing either an extremal pseudo-effective divisor, or an effective divisor not supported on the collection of divisors given. Let such a divisor be denoted $G$. We show that the class of $G$ is trivial and hence provide a contradiction to the existence of such a decomposition.

Necessarily $M\cdot G=0$ as $M$ is nef and hence must have zero intersection with every term in any pseudo-effective decomposition of $D$. Then Corollary~\ref{corollary:splittingM} and Lemma~\ref{lem:splitting}
imply
$$B^\eta_{i,S}\cdot G=B^\delta_{i,S}\cdot G=0$$
for all $i\geq 1$ and $S$. Similarly,
$$B_{\{a,b\}}\cdot G=B^\delta_{0,\{a,b\}}\cdot G=0$$
for any $\{a,b\}\subset\{1,\ldots,n\}$.

But these conditions provide a full rank set of conditions on the coefficients of $G$ for the generators $\psi_i,\eta_{\text{irr}}, \eta_{i:S}$ and $\delta_{i:S}$ for $i\geq1$, and $\delta_{0:S}$ for $|S|=2$. Hence the coefficients of each in the class of $G$ must be zero and we have
$$G=\sum_{|S|\geq 3} c_S\delta_{0:S}$$
for $c_S\in \mathbb{R}$. However, then we have
$$B^\delta_{0:S}\cdot G=-c_S$$
for each $S$. But the curves $B^\delta_{0:S}$ are covering curves for the divisors $\delta_{0:S}$. Hence any $c_S>0$ would imply that
$$G-\left(\frac{B^\delta_{0:S}\cdot G}{B^\delta_{0:S}\cdot \delta_{0:S}}\right)\delta_{0:S}$$
is pseudo-effective. Further, if $G$ is effective, $G$ must be supported on $\delta_{0:S}$ contradicting the assumptions on $G$. But if all $c_S\leq 0$ then also $-G$ is effective, hence $G$ is trivial providing the required contradiction.
\end{proof}

The moduli space $\overline{\mathcal{H}}_{g,n}$  is uniruled for $n<4g+6$ by~\cite[Prop 4.1]{Be} and \cite[Prop. 2.1]{AB}. Here we observe this as for nef curve $M$ we have $M\cdot K_{\overline{\mathcal{H}}_{g,n}}<0$ for $n<4g+6$, hence the canonical divisor is not pseudo-effective. Finally, we prove the following.

\begin{theorem}
For all $g\geq2$, the moduli space $\overline{\mathcal{H}}_{g,n}$ has Kodaira dimension zero for $n=4g+6$.
\end{theorem}

\begin{proof}
When $n=4g+6$, the canonical divisor is known to be effective. Hence as $M\cdot K_{\overline{\mathcal{H}}_{g,n}}=0$ the result follows from Proposition~\ref{prop:rigid}.
\end{proof}

\section{The effective cone of $\Hbars{g}{n}$}
\label{section:eff}

In this section we completely classify the structure of the pseudo-effective cone of $\Hbars{g}{n}$ and further give a complete answer the the Mori dream space question for these spaces. We closely follow~\cite{Mu} and identify a nef curve that forms an extremal ray of the dual cone of nef curves, at which this cone is non-polyhedral.

\begin{lemma}
For $g\geq 2$, the cone $\overline{\text{Eff}}^1(\overline{\mathcal{H}}_g)$ is polyhedral and generated by the irreducible components of the boundary.
\end{lemma}

\begin{proof}
$\overline{\text{Eff}}(\Mbar{0}{[2g+2]})$ is polyhedral and generated by the boundary divisors, cf. \cite[Thm. 1.3]{KM}. Hence the isomorphism $\alpha$ implies the result.
\end{proof}

\begin{lemma}
For $g\geq 2$, the cone $\overline{\text{Eff}}^1(\Hbars{g}{1})$ is polyhedral and generated by the irreducible components of the boundary and the Weierstrass divisor.
\end{lemma}

\begin{proof}
Consider the map
$$p: \Mbar{0}{[2g+2]+1}\longrightarrow \Hbarc{g}{1}.$$
As $\overline{\text{Eff}}(\Mbar{0}{[2g+2]+1})$ is polyhedral and generated by the boundary divisors~\cite[Prop. 5.3]{Ru} and
$$p_*\delta_{[1]+1}=W$$
we have
$$\overline{\text{Eff}}(\Hbarc{g}{1})=p_*(\overline{\text{Eff}}(\Mbar{0}{[2g+2]+1}))$$
is generated by the irreducible components of the boundary and the Weierstrass divisor.
\end{proof}

The case of $n\geq2$ is more interesting. We follow closely~\cite{Mu} which proves the result in the case of $\Mbar{g}{n}$ for $g\geq2$ and $n\geq 2$. Denote by $[F]$ the curve class of the general fibre of the forgetful morphism
$$\Hbars{g}{n}\longrightarrow\Hbars{g}{n-1}.$$
Irreducible curves with this class sweep out an open Zariski dense subset of $\Hbars{g}{n}$ and hence $[F]\cdot[D]\geq0$ for any pseudo-effective $[D]$, that is, $[F]$ is a moving or nef curve.\\

Following the strategy of~\cite{Mu} we show that $[F]$ is in fact an extremal ray of $\overline{\text{Nef}}_1(\Hbars{g}{n})$ the cone of nef or moving curve classes which is dual to $\overline{\text{Eff}}^1(\Hbars{g}{n})$. However, we show the corank of pseudo-effective divisors $[D]$ with $[F]\cdot[D]=0$ is at least two in ${\rm{Pic}}(\Hbars{g}{n})$ and hence this curve forms an extremal but non-polyhedral edge of $\overline{\text{Nef}}_1(\Hbars{g}{n})$. \\

For any nef curve class $[B]$ we define the pseudoeffective dual space
$$[B]^\vee:=\{[D]\in \overline{\text{Eff}}^1(\Mbar{g}{n})\hspace{0.3cm}\big|\hspace{0.3cm} [B]\cdot[D]=0\}.$$
Our first Proposition bounds the corank of this dual space for the curve $[F]$.

\begin{proposition}\label{prop:corank}
For $g\geq 2$ and $n\geq2$,
$$\rho(\Hbars{g}{n})-n\leq\text{rank}([F]^{\vee}\otimes\mathbb{R})\leq \rho(\Hbars{g}{n})-2,$$
where $\rho(\Hbars{g}{n})$ denotes the Picard number of $\Hbars{g}{n}$.
\end{proposition}

\begin{proof}
Consider an irreducible effective divisor $D$ in $\Hbars{g}{n}$ that intersects the interior $\mathcal{H}_{g,n}$ with $[F]\cdot [D]=0$. For any point $[C,p_1,\dots,p_{n}]\in \mathcal{H}_{g,n}$ contained in $D$, denote by $B$ the curve that is the fibre of $[C,p_1,\dots,p_{n-1}]$ under the forgetful morphism $\pi:\Hbars{g}{n}\longrightarrow \Hbars{g}{n-1}$ that forgets the $n$th point. As $[B]=[F]$ we have $[B]\cdot[D]=0$ so necessarily as $B$ is irreducible and intersects $D$ set theoretically, $B$ is contained in the support of $D$. Hence the divisor $D$ is supported on the pullback of an irreducible effective divisor under $\pi$. Further, if $D$ is an irreducible divisor in $\Hbars{g}{n}$ that does not intersect the interior $\mathcal{H}_{g,n}$ with $[F]\cdot [D]=0$ then $D$ is an irreducible component of the boundary outside of $\delta_{0:\{i,n\}}$ for $i=1,\dots,n-1$. This shows
$$\text{rank}([F]^{\vee}\otimes\mathbb{R})\geq \rho(\Hbars{g}{n})-n.$$

A key observation from~\cite{Mu} is that the intersection in $\Hbars{g}{2}$ of the surface equal to the general fibre of the forgetful morphism to $\mathcal{H}_g$ with any effective divisor offers a potential way to obtain a covering curve for the divisor. More formally, let $C$ be a general smooth hyperelliptic genus $g$ curve and
$$i:C\times C\longrightarrow \Hbars{g}{2}$$
the natural morphism. For a fixed divisor $[D]$ in $\Hbars{g}{2}$, define
$$[B_D]:=i_*i^*[D].$$

Further, Proposition 1.6 of~\cite{Mu} holds verbatim when replacing $\Mbar{g}{2}$ by $\Hbars{g}{2}$ and we have that if
$$[B_D]\cdot [D]=(2g-2)\left((4g-4)c_{\psi_1}c_{\psi_2}+(c_{\psi_1}+c_{\psi_2})^2-c_{0:\{1,2\}}^2  \right)<0$$
then $[D]$ contains a rigid strictly effective component.\\

Our argument for the lower bound of the pseudoeffective corank of $[F]$ in the case of $n=2$ implies that $\text{rank}([F]^{\vee}\otimes\mathbb{R})= \rho(\Hbars{g}{2})-1$ if and only if there exists a strictly pseudoeffective class in $\overline{\text{Eff}}(\Hbars{g}{2})$ with zero intersection with $[F]$. That is, this would imply the existence of a $[D]\in\overline{\text{Eff}}(\Hbars{g}{2})\setminus \text{Eff}(\Hbars{g}{2})$ with class
$$[D]=((1-2g)a-b)\psi_1+a\psi_2+b\delta_{0:\{1,2\}}+\text{other boundary components}$$
for $a,b\in\mathbb{R}$ with $a\ne 0$.

However, for such a class we obtain
$$[B_D]\cdot [D]=-8a^2(g-1)^2g<0$$
implying the existence of a strictly effective divisor class $[E]$ with non-zero $\psi_2$ coefficient and $[F]\cdot[E]=0$, hence providing a contradiction.\\

The last remaining task is to extent this upperbound to the case $n\geq3$. Though the fibre of the forgetful morphism $\Hbars{g}{n}\longrightarrow\Mbar{g}{n-1}$ over a point on the interior $\Hbars{g}{n-1}$ is irreducible, for $n\geq3$ the fibre over a general point in $\delta_{0:\{1,\dots,n-1\}}$ is reducible with two irreducible components. As a result, $[F]$ can be expressed as an effective sum of these two curves which form covering curves for $\delta_{0:\{1,\dots,n-1\}}$ and $\delta_{0:\{1,\dots,n\}}$. Any extremal $[D]\in \overline{\text{Eff}}^1(\Hbars{g}{n})$ outside of $\delta_{0:\{1,\dots,n-1\}}$ or $\delta_{0:\{1,\dots,n\}}$ must have nonnegative intersection with each of these covering curves and if further $[F]\cdot[D]=0$ then each of these intersections must in fact be zero. \\

Let $\varphi:\Hbars{g}{2}\longrightarrow \Hbars{g}{n}$ be the morphism gluing in a general $n$-pointed rational curve at the first marked point. For such a $[D]$ we obtain $[F]\cdot\pi^*[D]=0$. The proof of Lemma 3.3 from~\cite{Mu} for $\Mbar{g}{n}$ holds verbatim for $\Hbars{g}{n}$ and shows in this situation $\pi^*[D]$ is pseudoeffective and hence by pulling back the class of $[D]$ we have $c_{\psi_n}=0$. This provides a second linearly independent condition on $[F]^\vee\otimes\mathbb{R}$ for $n\geq 3$ completing the proof of the upperbound.
\end{proof}

The remaining task is to show that $[F]$ is indeed extremal in $\overline{\text{Nef}}_1(\Hbars{g}{n})$. We do this through the use of well-chosen families of effective divisor classes such that for any potential nef decomposition of the class $[F]$ we can provide an effective divisor class with negative intersection with a summand.\\

Define $D_k$ and $E_k$ as the following loci in $\mathcal{H}_{g,2}$ defined via conditions in the Jacobian of hyperelliptic curves
\small
\begin{equation*}
D_k:={\left\{ [C,p_1,p_2]\in\mathcal{H}_{g,2} \big| \exists [C,p_1,p_2,q_1,\dots,q_{g-1}]\in\mathcal{H}_{g,g+1}, \mathcal{O}_C((k(g-1)+1)p_1-p_2-k\sum_{i=1}^{g-1}q_i)\sim\mathcal{O}_C   \right\}}
\end{equation*}
\begin{equation*}
E_k:={\left\{ [C,p_1,p_2]\in\mathcal{H}_{g,2} \big| \exists [C,p_1,p_2,q_1,\dots,q_{g-1}]\in\mathcal{H}_{g,g+1}, \mathcal{O}_C((k(g-1)-1)p_1+p_2-k\sum_{i=1}^{g-1}q_i)\sim\mathcal{O}_C   \right\}}.
\end{equation*}
\normalsize
Taking the closure we obtain two families of divisors $\overline{D}_k$ and $\overline{E}_k$ for $k\geq 2$ in $\Hbars{g}{2}$. The following lemma proves these are indeed effective conditions.

\begin{lemma}\label{lem:eff}
The divisors $\overline{E}_k$ and $\overline{D}_k$ are effective in $N^1_{\mathbb Q}\left(\overline{\mathcal{H}}_{g,2}\right)$.
\end{lemma}

\begin{proof}
By \cite{Mu} both are effective in $\overline{\mathcal{M}}_{g,2}$ so it is enough to show that there is a single 2-pointed hyperelliptic curve $[C,p_1,p_2]\in \overline{\mathcal{H}}_{g,2}$ not contained in $\overline{D}_k$, resp. $\overline{E}_k$. Let $[C,q]$ be a general pointed hyperelliptic curve of genus $g\geq 1$ where $q$ is a Weierstrass point if $g\geq2$. We will prove the following more general statement by induction on $g$. For any integers $n_1,n_2$, not both zero, there exist points $p_1,p_2$ in $C$ such that for any set of points $q_1,\ldots,q_{g-1}$ with $r\leq g-1$ and $q_i\neq p_j$ when $n_j\neq0$
$$\mathcal{O}_{C}\left(n_1p_1+n_2p_2-k\sum_{i=1}^r q_i-(n_1+n_2-kr)q\right)\neq \mathcal{O}_C.$$
The lemma then will follow choosing $n_1=k(g-1)+1$ and $n_2=-1$ for $\overline{D}_k$ and $n_1=k(g-1)-1$ and $n_2=1$ for $\overline{E}_k$.\\

When $g=1$ the condition becomes a non-trivial torsion condition on three points on an elliptic curve and hence the statement clearly holds. Now let $C$ be the hyperelliptic curve of genus $g$ obtained by gluing $C_1$ and $C_2$ along a Weierstrass point $q$ and $n_1,n_2$ to integers not both zero. We assume $C_1$ has genus $1$ and therefore $C_2$ has genus $g-1$. Take $p_1, p_1'$ in $C_1$ satisfying the induction hypothesis for integers $(n_1,0)$ and $p_2,p_2'\in C_2$ satisfying the induction hypothesis for integers $(n_2,0)$. Further, observe that as there is no condition on $p_1'$ and $p_2'$ this will hold for these points chosen freely in the curves $C_1$ and $C_2$ respectively. Then for any set of $r\leq g-1$ points, in $C$, say $q_1,\ldots,q_{r_1}\in C_1$ and $q_1',\ldots,q_{r_2}'\in C_2$ with $r=r_1+r_2$.
The equation in the Jacobian of $C$ then becomes
\begin{equation}\label{sec7:eqEff}\mathcal{O}_{C_1}\left(n_1p_1-k\sum_{i=1}^{r_1}q_i-(n_1-kr_1)q\right)= \mathcal{O}_{C_1}\quad\hbox{and}\quad\mathcal{O}_{C_2}\left(n_2p_2-k\sum_{i=1}^{r_2} q_i'-(n_2-k{r_2})q\right)= \mathcal{O}_{C_2}.\end{equation}
But either $r_1<1$ or $r_2<g-1$. Thus, by induction hypothesis there are no points
$$q_1,\ldots,q_{r_1},q_1',\ldots,q_{r_2}'$$
with $q_i\neq p_1$ if $n_1\neq 0$ and $q_j'\neq p_2$ if $n_2\neq 0$ satisfying \eqref{sec7:eqEff}.
\end{proof}

The classes of the corresponding divisors in $\Mbar{g}{2}$ we computed in~\cite[Corollary 4.2]{Mu}.The statement in the proof of Lemma~\ref{lem:eff} further shows that $\delta_{0:\{1,2\}}$ is not an effective component of the pullback of these divisors from $\Mbar{g}{2}$. That is, under the inclusion
$$i:\Hbars{g}{2}\longrightarrow \Mbar{g}{2}$$
 we obtain the classes of these divisors in ${\rm{Pic}}(\Hbars{g}{2})$ as \small
$$  [ \overline{D}_k]=\frac{1}{2}(gk+1)(gk-k+1)k^{2g-2}\psi_1+\frac{1}{2}(1-k)k^{2g-2}\psi_2 -\frac{1}{2}(gk^{2g}-k^{2g}+2)g\delta_{0:\{1,2\}}        +\dots$$ \normalsize
and \small
$$   [\overline{E}_k]=\frac{1}{2}(gk-1)(gk-k-1)k^{2g-2}\psi_1+\frac{1}{2}(k+1)k^{2g-2}\psi_2-\frac{1}{2}(gk^{2g}-k^{2g}+2)g\delta_{0:\{1,2\}}+\dots    $$ \normalsize

\begin{proposition}\label{prop:extremal}
The curve class $[F]$ is extremal in $\overline{\text{Nef}}_1(\Hbars{g}{n})$ for $g\geq2$ and $n\geq1$.
\end{proposition}

\begin{proof}
Consider the case of $n=2$. If $[F]$ is not extremal it is the sum of nef curve classes with zero intersection with $[F]^\vee$. Hence a non-trivial decomposition of $[F]$ into a sum of nef curves implies the existence of a nef curve class $[F^t]$ for a fixed $t\in\mathbb{R}\setminus\{0\}$ with intersections
$$[F^t]\cdot\psi_1=1,\hspace{1cm}[F^t]\cdot \psi_2=(2g-1)+t,\hspace{1cm}[F^t]\cdot\delta_{0:\{1,2\}}=1$$
and zero intersection with the other generators of ${\rm{Pic}}(\Hbars{g}{2})$.  But for any fixed $t>0$
$$ [F^t]\cdot [\overline{D}_k]= (k^{2g - 2}-1) g +t\frac{1}{2}(1-k)k^{2g-2} =\frac{-t}{2}k^{2g-1}+\mathcal{O}(k^{2g-2})<0 \text{  for  }k\gg0,$$
and for any fixed $t<0$
$$ [F^t]\cdot [\overline{E}_k]=(k^{2g - 2}-1) g   +t \frac{1}{2}k^{2g-2}(k+1)=\frac{t}{2}k^{2g-1}+\mathcal{O}(k^{2g-2}) <0 \text{  for  }k\gg0.$$
Hence $[F^t]\in \overline{\text{Nef}}_1(\Hbars{g}{2})$ if and only if $t=0$. For $n\geq 3$ we consider the the forgetful morphisms
$$\pi_j:\Hbars{g}{n}\longrightarrow\Hbars{g}{2}$$
for $j=1,\dots,n-1$ that forget all but the $j$th and $n$th points. As ${\pi_j}_*[F]=[F]$ is an extremal nef curve and the pushforward of a nef curve is nef, if $[F']$ is a nef curve class appearing in a nef decomposition of $[F]$ then ${\pi_j}_*[F']=k_j[F]$ for some $0\leq k_j\leq 1$.

Hence if $[F]$ is not an extremal nef curve class, there must exist a nef curve with class $[F^{\underline{t}}]$ for $\underline{t}=(t_1,\dots,t_{n-1})\in \mathbb{R}^{n-1}$ with $\underline{t}\ne \underline{0}$ where
 $$[F^{\underline{t}}]\cdot\psi_i=[F^{\underline{t}}]\cdot\delta_{0:\{i,n\}}=1+t_i,\text{ for $i=1,\dots,n-1,$} \hspace{1cm}[F^{\underline{t}}]\cdot \psi_n=2g-(n-3),  $$

By the observation that~\cite{AC}
 $$\pi_j^*\psi_1=\psi_j-\sum_{j\in S,n\notin S}\delta_{0:S},\hspace{0.5cm}\pi_j^*\psi_2=\psi_n-\sum_{n\in S,j\notin S}\delta_{0:S},\hspace{0.5cm}\pi_j^*\delta_{0:\{1,2\}}=\sum_{j,n\in S}\delta_{0:S},$$
 and an application of the projection formula for each $j$ we obtain
 $$ 2g-1-\sum_{i\ne j}t_i=k_j(2g-1)\hspace{0.5cm}\text{ and }\hspace{0.5cm}1+t_j=k_j.$$
 Giving $n-1$ linearly independent relations in the $t_j$, hence the requiring all $t_i=0$ and providing the contradiction.
\end{proof}

Proposition~\ref{prop:corank} and Proposition~\ref{prop:extremal} provide the main result of this section.

\begin{proposition}
For $g\geq 2$ and $n\geq2$, the cone $\overline{\text{Eff}}^1(\Hbars{g}{n})$ is non-polyhedral.
\end{proposition}

\begin{proof}
The curve class $[F]$ forms an extremal ray of $\overline{\text{Nef}}_1(\Hbars{g}{n})$ where the corank of the pseudoeffective dual space is at least two. Hence $\overline{\text{Nef}}_1(\Hbars{g}{n})$ and the dual $\overline{\text{Eff}}(\Hbars{g}{n})$ are non-polyhedral.
\end{proof}

\begin{remark}
A long standing question is whether symmetric quotients of $\overline{\mathcal{M}}_{0,n}$ are Mori dream spaces. This is only known for the full symmetric quotient when $n\leq 7$, cf. \cite{Mo}. If both $\overline{\mathcal{M}}_{0,[2g+2]}$ and $\overline{\mathcal{M}}_{0,[2g+2]+1}$ are Mori dream spaces, by~\cite[Thm 1.1]{O}, the isomorphism $\overline{\mathcal{M}}_{0,[2g+2]}\overset{\sim}{\to} \overline{{\bf{H}}}_{g}$ and the morphism $ \overline{\mathcal{M}}_{0,[2g+2]+1}{\to}\overline{{\bf{H}}}_{g,1}$ would show $\overline{{\bf{H}}}_{g}$ and $\Hbarc{g}{1}$ are also Mori dream spaces completing the classification for the moduli of pointed hyperelliptic curves.
 \end{remark}

redpajama_set_name: RedPajamaArXiv