text:
\section{Introduction}

Donaldson-Thomas (DT) theory was introduced in \cite{Tho00} as an enumerative theory which gives a virtual count of stable coherent sheaves with fixed topological invariants on certain 3-folds, including Calabi-Yau threefolds.  In the Calabi-Yau case, this moduli problem supports a perfect obstruction theory of virtual dimension zero in good situations, and so defines a degree zero class in the Chow ring of the moduli space --- the virtual fundamental class of \cite{BF97}.  \ccblue{When the moduli space is proper,} the DT invariant is defined as the degree of this virtual fundamental class.  This theory was applied to Hilbert schemes of points in \cite{Tho00} by identifying the Hilbert scheme with the moduli space of their corresponding ideal sheaves.

\begin{comment}
Donaldson-Thomas (DT) theory was defined in \cite{Tho00}. It is an enumerative theory which gives a virtual count of ideal sheaves associated to one dimensional schemes on a Calabi-Yau 3-fold. More precisely, the DT moduli space parametrizes Gieseker stable coherent sheaves on a Calabi-Yau 3-fold with fixed topological invariants. Using the technique of \cite{BF97}, this moduli problem defines a degree zero class in the Chow ring of the moduli space - the virtual fundamental class. Intuitively speaking, this class corresponds to the locus in the moduli space where the obstruction vanishes. Then the degree of this virtual fundamental class is defined to be the DT invariant \cite{Tho00}.
\end{comment}

It turns out that there is a rich structure underlying the DT invariant. The DT moduli space supports a symmetric obstruction theory, which implies that \ccblue{when the moduli space is proper} the DT invariant is a weighted Euler characteristic of the moduli space, the weighting being given by a constructible function on the moduli space called the Behrend function \cite{Beh09}.  These ideas are applied to show that for a projective threefold $X$, the generating function of the DT invariants of $Hilb^n(X)$ can be expressed in terms of the MacMahon function and the Euler characteristic of $X$ \cite{BF08, LP09, MNOP06}.

If the moduli space can be globally realized as a degeneracy locus, the value of its Behrend function at a point can be expressed in terms of the Euler characteristic of the Milnor fiber at that point \cite{Beh09,PP98}.  Replacing the Milnor fibers with their motivic incarnations, one is led to the notion of Motivic DT invariants.

Motivic DT theory was introduced and developed in \cite{KS,BJM19}, and had been studied by several authors, see e.g.\ \cite{BBS13,Dav,MMNS12,Nag}. One of the first computations of motivic DT invariants was given for $Hilb^n(X)$, where $X$ is a Calabi-Yau threefold  \cite{BBS13}. This computation relied on the fact that $Hilb^n(\mathbb{C}^3)$ can be realized as a degeneracy locus with an equivariant torus action. The authors introduced a notion of motivic DT invariants of $Hilb^n(X)$ based on their computations for $Hilb^n(\mathbb{C}^3)$.

Later, a general formalism for motivic DT invariants was given in \cite{BJM19}, using the notions of a d-critical locus and orientation  introduced in \cite{Joy15}. The notion of orientation is extracted from the notion of orientation data introduced in \cite{KS}.

Now let $X$ be a local toric surface $\omega_S$, the total space of the canonical bundle of a smooth, complete toric surface $S$.
In this paper, we define a d-critical locus structure on $Hilb^n(X)$ if $X=\omega_{\mathbb{P}^2}$ or $X=\omega_{\mathbb{F}_n}$, and endow it with an orientation following \ccblue{\cite{Dav}}.  We then show that the motivic DT invariants of $Hilb^n(X)$ as defined using the d-critical locus and orientation agree with the motivic DT invariants as defined in \cite{BBS13}. We therefore show that the results of \cite{BBS13} are valid in the context of the more general theory for these Calabi-Yau threefolds.

A natural question arises which we do not address further in this paper:

\smallskip\noindent
The notion of a d-critical locus was introduced as a classical truncation of the notion of a $-1$-shifted symplectic structure in derived algebraic geometry.  It was shown in \cite{PTVV13} that the derived moduli stack of sheaves on a \ccblue{projective} Calabi-Yau threefold has a canonical $-1$-shifted symplectic structure. \ccblue{For \ccred{a} similar result in the noncompact case, see \cite{BD19}.} Does the classical truncation of the canonical $-1$-shifted symplectic structure on the derived Hilbert scheme agree with our d-critical locus structure? This would imply that our motivic result is true for the total space of the canonical bundle of any smooth toric surface.

\subsection{Outline of the paper}
In Section 2 we review background material including the definition of motivic DT invariants following \cite{BJM19}.  Section 3 begins with the needed toric geometry.  We then prove that there is a canonical  d-critical locus structure on an open subset of $Hilb^n(\omega_S)$ for any smooth toric surface $S$ (Proposition~\ref{prop:opendcrit}).  We use this result and its proof to show that there is a natural d-critical locus structure on all of $Hilb^n(\omega_S)$ induced by copies of $Hilb^n(\mathbb{C}^3)$'s, for $S=\mathbb{P}^2$ and $S=\mathbb{F}_n$ (Theorem~\ref{thm:dcrit}). In Section 4, we exhibit an orientation on $Hilb^n(\omega_S)$ associated to this d-critical structure. Finally in Section 5, using the d-critical locus structure constructed in Section 3 and the orientation in Section 4, we recover the computation of $Hilb^n(\omega_S)$ in \cite{BBS13}.

\subsection{Notations}
All schemes in this paper are assumed to be separated and of finite type over $\mathbb{C}$. We use $X$ to denote a smooth quasi-projective Calabi-Yau 3-fold. For a smooth surface $S$, we denote the total space of its canonical bundle by $\omega_S$, and the projection from $\omega_S$ to $S$ by $\pi: \omega_S\rightarrow S$.

\subsection{Acknowledgements}
We thank Ben Davison for helpful discussions on this subject and motivic DT theory in general and Tony Pantev for helpful conversations about $-1$-shifted symplectic structures. We are also grateful for Bal\'azs Szendr\H{o}i for helpful comments on a previous version of this paper. The research of the first author is supported in part by NSF grant DMS--1802242, as well as by NSF grant DMS-1440140 while in residence at MSRI in Spring, 2018. Both authors would like to thank MSRI for the excellent working environment.  The second author is grateful for the mini course given by Kai Behrend during the program Enumerative Geometry beyond Numbers at MSRI, where she first learned this problem.   She would also like to thank Guanying Peng for helpful conversations on the special case of local $\mathbb{P}^2$. Part of the work was done while the second author was a postdoc at CMSA, Harvard.  She would like to thank CMSA for the excellent working environment.

\section{Background}

\subsection{Introduction to motivic DT invariants}

In this section, we review the main ingredients of motivic Donaldson-Thomas (DT) invariants as defined by Bussi, Joyce and Meinhardt \cite{BJM19}.

First we recall the definition of the monodromic Grothendieck ring, where the motivic invariant associated to a moduli space lives. Let $S$ be a separated scheme of finite type over $\mathbb{C}$. The Grothendieck group of $S$-schemes is the abelian group generated by isomorphism classes of $S$-schemes $[X\rightarrow S]$, modulo the scissor relations:
\begin{equation*}
[T\to S]=[T'\to S]+[T\textbackslash T'\to S]
\end{equation*}
for $T'$ a closed $S$-subscheme of $T$.

Now we consider an equivariant version of the Grothendieck ring. Let $\mu_n$ be the group of $n$th roots of unity. A $\mu_n$ action on a scheme $T$ is called \emph{good} if every $\mu_n$ orbit is contained in an affine subscheme of $T$. Note that there are maps
$\mu_m\rightarrow \mu_n$ if $n|m$,
defined by $x\rightarrow x^{m/n}$ forming an inverse system.
Denote $\varprojlim \mu_n$ by $\hat{\mu}$.
\begin{definition} (\ccblue{\cite{DF00, BJM19}})
The monodromic Grothendieck group $K_0^{\hat{\mu}}(S)$ is the abelian group generated by isomorphism classes $[T\to S, \hat{\sigma}]$, where $T$ is a separated $S$-scheme of finite type and
\begin{equation*}
\hat{\sigma}:\hat{\mu}\times T\rightarrow T
\end{equation*} is a $\hat{\mu}$ action which factors through a good $\mu_n$ action \ccblue{for some $n$}. The relations of $K_0^{\hat{\mu}}(S)$ are given by

(i) $[T_1, \hat{\sigma_1}]= [T_2, \hat{\sigma_2}]$ for $T_1$ and $T_2$ equivariantly isomorphic as $S$-schemes,

(ii) $[T, \hat{\sigma}]= [T', \hat{\sigma}|_{T'}]+[T\setminus T', \hat{\sigma}|_{T\setminus T',}]$ for any $T'\subset T$ closed $\hat{\sigma}$ invariant $S$-subscheme,

(iii) $[T\times \mathbb{A}^n, \hat{\sigma}\times\hat{\tau}_1]= [T\times \mathbb{A}^n, \hat{\sigma}\times\hat{\tau}_2]$ for any linear $\hat{\mu}$ actions $\hat{\tau}_1$, $\hat{\tau}_2$ on $\mathbb{A}^n$.
\end{definition}

\begin{comment}
equivariant isomorphisms as $S$-schemes, equivariant scissor relationships and the equivalence of any linear $\hat{\mu}$ action on $\mathbb{A}^n$.

\begin{definition}
The monodromic Grothendieck group $K_0^{\hat{\mu}}(X)$ is an abelian group generated by elements $[S, \hat{\sigma}]$, where $S$ is a separated $X$-scheme of finite type, and $\hat{\sigma}:\hat{\mu}\times S\rightarrow S$ a $\hat{\mu}$ action with the relations:

(i) $[S, \hat{\sigma}]\simeq [T, \hat{\tau}]$, if $S\simeq T$ as $X$ schemes, and the isomorphism is $\hat{\mu}$ equivariant,

(ii) $[S, \hat{\sigma}]\simeq [T, \hat{\sigma}|_T]+[S\setminus T, \hat{\sigma}|_{S\setminus T,}]$, if $T\subset S$ is a closed $\hat{\sigma}$ invariant $X$ subscheme,

(iii) $[S\times \mathbb{A}^n, \hat{\sigma}\times\hat{\tau}_1]\simeq [S\times \mathbb{A}^n, \hat{\sigma}\times\hat{\tau}_2]$ for any linear $\hat{\mu}$ actions $\hat{\tau}_1$, $\hat{\tau}_2$ on $\mathbb{A}^n$.
\end{definition}
\end{comment}
Note that there is an obvious commutative ring structure on $K_0^{\hat{\mu}}(S)$ defined by
\begin{equation*}
[T_1, \hat{\sigma}]\cdot[T_2, \hat{\tau}]=[T_1\times_S T_2, \hat{\sigma}\times \hat{\tau}].
\end{equation*}
We denote by $\mathbb{L}$ the element $[\mathbb{A}^1\times S\rightarrow S,\hat{\iota}]$, where $\hat{\iota}$ is the trivial action. Then we obtain a ring
\begin{equation*}
\mathcal{M}^{\hat{\mu}}_S=K_0^{\hat{\mu}}(S)[\mathbb{L}^{-1}]
\end{equation*}
by formally joining the inverse of $\mathbb{L}$ under this multiplication.
\ccblue{ Note that there is a map $\pi_{\hat{\mu}}: \mathcal{M}^{\hat{\mu}}_S\rightarrow \mathcal{M}_S$ of $\mathcal{M}_S$ modules, defined by taking generators to the corresponding orbit space:
\begin{equation*}
\pi_{\hat{\mu}}: [T\rightarrow S, \hat{\sigma}]\rightarrow [T/{\hat{\sigma}}].
\end{equation*}
Let $f:S_1\rightarrow S_2$ be a morphism between $\mathbb{C}$-schemes. The \ccred{notions} of pushforward and pullback on $K_0(S)$ are defined by $f_*([T\rightarrow S_1])=[T\rightarrow S_2]$ by composing the map $T\rightarrow S_1$ with the map $f$, and $f^*[T'\rightarrow S_2]=[T'\times_{S_2}S_1\rightarrow S_1]$. They \ccred{induce a} pushforward and \ccred{a} pullback on $\mathcal{M}^{\hat{\mu}}_S$ in the obvious way.}

There is a less obvious product on $K^{\hat{\mu}}_0(S)$ or $\mathcal{M}^{\hat{\mu}}_S$, called the convolution product. It is denoted by '$\odot$' in \cite{BJM19}. Since we will not explicitly use it in this paper, we refer to \cite{Loo02}, \cite{DF99} and \cite{BJM19} for its definition.

The motivic DT invariants we consider here take values in a quotient of $(\mathcal{M}^{\hat{\mu}}_S, \odot)$ denoted by $(\overline{\mathcal{M}}^{\hat{\mu}}_S, \odot)$. To define it, we first write
\begin{equation*}
\mathbb{L}^{1/2}=[S,\hat{\iota}]-[S\times\mu_2,\mu_2]
\end{equation*}
with the natural $\mu_2$ action.  The notation is justified since the square of the right hand side (using $\odot$) can be checked to be $\mathbb{L}$.  It follows immediately that $\mathbb{L}^{-1/2}\in \mathcal{M}^{\hat{\mu}}_S$ as well.  Then to a
principal $\mathbb{Z}_2$ bundle $P$ over a scheme $T$ we associate a motive:
\begin{equation*}
\Upsilon(P)=\mathbb{L}^{-1/2}\odot([T, \hat{\iota}]-[P, \hat{\rho}])\in \mathcal{M}^{\hat{\mu}}_T,
\end{equation*}
where $\hat{\rho}$ is induced by the $\mathbb{Z}_2$ action on $P$. Then $(\overline{\mathcal{M}}^{\hat{\mu}}_S, \odot)$ is obtained from $(\mathcal{M}^{\hat{\mu}}_S, \odot)$ by taking the quotient by the ideal generated by elements $\phi_*(\Upsilon(P\otimes_{\mathbb{Z}_2}Q)-\Upsilon(P)\odot\Upsilon(Q))$, for all $\mathbb{C}$ scheme morphisms $\phi: Z\rightarrow S$ and $P$, $Q$ principal $\mathbb{Z}_2$ bundles on $Z$.  Note that if $P$ is a trivial $\mathbb{Z}_2$ bundle, then $\Upsilon(P)=1$ is trivial as well.

\bigskip

Now we review the definition of motivic DT invariants. First consider the special case when the moduli space $X$ can be globally realized as a degeneracy locus, i.e. $X:=\{df=0\}\subset U$ where $f:U\to \mathbb{C}$ is a regular function on a smooth scheme $U$. Then the motivic DT invariant can be defined as the motivic vanishing cycle \ccblue{up to a \ccred{factor of}  a power of $\mathbb{L}^{1/2}$}. We will not work with motivic vanishing cycles directly in this paper, and we simply refer to \cite{Loo02}, \cite{DF99}, \cite{BJM19} for their precise definition.

More generally, the motivic invariants can still be defined if the moduli space can be covered by degeneracy loci which satisfy a compatibility condition. The precise \ccblue{terminology} for this structure is called a \emph{d-critical locus} structure \cite{Joy15}.
Let $Y$ be a $\mathbb{C}$ scheme locally of finite type. The following theorem is from \cite{Joy15}:

\begin{theorem} (\cite{Joy15}, Theorem 2.1)
	\label{thm1}
There exists a sheaf $\mathcal{S}_Y$ of $\mathbb{C}$ vector spaces, uniquely characterized by two properties.

(i) Suppose $R\subset Y$ is a Zariski open subset of $Y$, and $i:R\hookrightarrow U$ a closed embedding in some smooth scheme $U$. Define the sheaf of ideals $I_{R, U}$ by the following exact sequence of vector spaces on $R$.
\begin{equation*}
0\rightarrow I_{R, U}\rightarrow i^{-1}(O_U)\rightarrow O_Y|_R\rightarrow 0.
\end{equation*}

Then there is an exact sequence of sheaves of vector spaces on $R$:
\begin{equation*}
0\rightarrow \mathcal{S}_Y|_R\xrightarrow{\iota_{R, U}} \frac{i^{-1}(O_U)}{I_{R, U}^2}\xrightarrow{d}\frac{i^{-1}(T^*U)}{I_{R, U}\cdot i^{-1}(T^*U)},
\end{equation*}
where $\iota_{R, U}$ is a morphism of sheaves of vector spaces, and $d$ is the differential map.

(ii) Let $R\subset S\subset Y$ be Zariski open inclusions, and $i: R\hookrightarrow U$, $j: S\hookrightarrow V$ closed embeddings in smooth schemes $U$ and $V$. Let $\Phi: U\to V$ be a morphism satisfies $\Phi\circ i=j|_R$. Then the following diagram commutes:
\[\begin{tikzcd}
0\arrow{r} & \mathcal{S}_Y|_R \arrow{r}{\iota_{S, V}|_R} \arrow[swap]{d}{id} & \frac{j^{-1}(O_V)}{I^2_{S, V}}|_R\arrow{r}{d} \arrow{d}{i^{-1}(\Phi^\sharp)} & \frac{j^{-1}(T^*V)}{I_{S, V}\cdot j^{-1}(T^*V)}|_R \arrow{d}{i^{-1}(d\Phi)}\\
0\arrow{r} & \mathcal{S}_Y|_R \arrow{r}{\iota_{R, U}} & \frac{i^{-1}(O_U)}{I^2_{R, U}}\arrow{r}{d} & \frac{i^{-1}(T^*U)}{I_{R, U}\cdot i^{-1}(T^*U)}.
\end{tikzcd}
\]
\end{theorem}
Let $\mathcal{S}_Y^0$ be the kernel of the composition
\begin{equation*}
\mathcal{S}_Y\rightarrow O_Y\rightarrow O_{Y_{red}}\ccred{,}
\end{equation*}
\ccblue{where the map $\mathcal{S}_Y\rightarrow O_Y$ is locally defined by composing $\iota_{R, U}$ with $i^{-1}(O_U)\rightarrow O_Y|_R$. }

Then the sheaf $\mathcal{S}_Y$ has a canonical decomposition
\begin{equation*}
\mathcal{S}_Y\simeq \mathbb{C}_Y\oplus \mathcal{S}_Y^0.
\end{equation*}

\begin{definition} (\cite{Joy15} Definition 2.5)
\label{dcrit}
An algebraic d-critical locus over $\mathbb{C}$ is a pair $(Y, s)$, where $Y$ is a $\mathbb{C}$ scheme and $s\in H^0(\mathcal{S}_Y^0)$ such that the following is satisfied: for every point $y\in Y$, there is a Zariski open neighborhood $R$ of $y$ with a closed embedding $i:R\hookrightarrow U$ into a smooth scheme $U$, such that $i(R)=\{df=0\}\subset U$ for $f: U\rightarrow \mathbb{C}$ a regular function on $U$. Furthermore, $\iota_{R, U}(s|_R)=i^{-1}(f)+I_{R, U}^2$.
\end{definition}
Using the notation in the definition, the charts $(R, U, f, i)$'s are called critical charts of $(Y, s)$.

Consider a moduli space which has a d-critical locus structure. In particular it is locally covered by degeneracy loci, and one can associate motivic vanishing cycles to the local charts. However, these locally defined invariants do not naturally agree on intersections, see e.g. \cite{BJM19} Example 3.5.
\ccblue{It turns out} \ccred{that} these locally defined invariants can be modified using an orientation, and they then agree and can be glued to a global motivic invariant.

To give the definition of an orientation, we first recall the notion of the virtual canonical bundle of a d-critical locus.
\begin{theorem}
\label{defK}
Let $(Y, s)$ be a d-critical locus, and let $Y^{red}$ be $Y$ with its reduced structure. Then there exists a line bundle $K_{Y, s}$ on $Y^{red}$ which is uniquely defined by two properties.

(i) If $(R, U, f, i)$ is a critical chart on $(Y, s)$, there is a natural isomorphism
\begin{equation}
\label{equiota}
\iota_{R, U, f, i}: K_{Y, s}|_{R^{red}}\rightarrow i^*(K_U^{\otimes 2})|_{R^{red}},
\end{equation}
where $K_U$ is the usual canonical bundle of $U$.

(ii) Let $\Phi: (R, U, f, i)\rightarrow (S, V, g, j)$ be an embedding of critical charts on $(Y, s)$.  Then there is an isomorphism of line bundles on $Crit(f)^{red}$:
\begin{equation*}
J_\Phi:K^{\otimes^2}_U|_{Crit(f)^{red}}\rightarrow \Phi|^*_{Crit(f)^{red}}(K_V^{\otimes^2}).
\end{equation*}
such that
\begin{equation*}
i|^*_{R^{red}}(J_\Phi):i^*(K_U^{\otimes^2})|_{R^{red}}\rightarrow j^*(K_V^{\otimes^2})|_{R^{red}},
\end{equation*}
is an isomorphism and
\begin{equation*}
\iota_{S, V, g, j}|_{R^{red}}=i|^*_{R^{red}}(J_\Phi)\circ \iota_{R, U, f, i}: K_{X, s}|_{R^{red}}\rightarrow j^*(K_V^{\otimes^2})|_{R^{red}}.
\end{equation*}
\end{theorem}
Here $J_\Phi$ is induced by the isomorphism $K_U\otimes \Lambda^nN^*_{UV}\rightarrow \Phi^*(K_V)$.  For details of the definition of $J_\Phi$ see \cite{Joy15}, Def. 2.26 or \cite{BJM19}, section 4.

\begin{definition} (\cite{Joy15} Definition 2.31)
\label{defOD}
Let $(Y, s)$ be a d-critical locus. An \emph{orientation on $(Y,s)$} is given by a line bundle $L$ on $Y^{red}$ with $L^{\otimes2}\simeq K_{Y, s}$ .
\end{definition}

With these two additional structures, a global motivic DT invariant can always be defined:

\begin{theorem} (\cite{BJM19} Theorem 5.10)
\label{thm1.1}
Let $(Y,s)$ be a finite type algebraic d-critical locus with a choice of orientation $K^{1/2}_{Y,s}$. There exists a unique motive $MF_{Y, s}\in \overline{\mathcal{M}}_Y^{\hat{\mu}}$ with the property that if $(R, U, f, i)$ is a critical chart on $(Y, s)$, then
\begin{equation*}
MF_{Y, s}|_R=i^*(\mathbb{L}^{-dimU/2}\odot MF^\phi_{U,f})\odot\Upsilon(Q_{R, U, f, i}) \in \overline{\mathcal{M}}_R^{\hat{\mu}}.
\end{equation*}
\end{theorem}

Here, the term $MF^\phi_{U,f}$ is the motivic vanishing cycle associated to the critical chart $(R, U, f, i)$. The term $Q_{R, U, f, i}$ is the principal $\mathbb{Z}_2$ bundle of local isomorphisms $\alpha_R:K_{Y, s}^{1/2}|_{R^{red}}\to i^*(K_U)|_{R^{red}}$ such that $\alpha_R^{\otimes 2}=\iota_{R, U, f, i}$ in equation \ref{equiota}.
We would like to point out that the motivic vanishing cycle used in this paper is \cgreen{the class of $f^{-1}(0)$ with the trivial action minus the motivic nearby cycle, which is the negative of the definition used in \cite{BBS13}}.
\begin{comment}
The line bundle $L$ is the choice of orientation $K^{1/2}_{Y,s}$.
Following the terminology of \cite{BBS13}, we call the motivic invariant in the above Theorem the relative motivic class. We call the pushforward of the relative motivic class to a point the absolute motivic class, and denote it by $[Y, s]^L$.
\end{comment}
\subsection{Construction of $Hilb^n(\mathbb{C}^3)$}\label{subsec:hilb}

We next briefly recall the construction of $Hilb^n(\mathbb{C}^3)$ used in \cite{BBS13}, in particular its presentation as a degeneracy locus.

To a subscheme $Z\subset \mathbb{C}^3$ with Hilbert polynomial $P_Z=n$ we can associate
an $n$-dimensional vector space $V_n=H^0(\mathcal{O}_Z)$, three pairwise commuting linear maps
\begin{equation*}
X, Y, Z: V_n\to V_n
\end{equation*}
defined as multiplication by $x$, $y$, $z\in \mathbb{C}[x, y, z]$, and a vector $v\in V_n$ corresponding to $1\in H^0(\mathcal{O}_Z)$.  The vector is $v$ cyclic for the action of $\mathbb{C}[X, Y, Z]$ on $V_n$: $\mathbb{C}[X,Y,Z]\cdot v=V_n$.

Now consider the space of triples of $n\times n$ matrices and a vector in $V_n$:
\begin{equation*}
Hom(V_n, V_n)^3\times V_n.
\end{equation*}
This space is a quasiprojective variety and admits a $GL(V_n)$ action induced from the action of $GL(V_n)$ on $V_n$. The character $\chi: GL(V_n)\rightarrow \mathbb{C}^*$ defined by $\chi(g)=det(g)$ defines a linearization of the trivial bundle. Let $U$ be the stable locus of the linearization. It turns out that $U$ consists of the points $(X, Y, Z, v)$ where $v$ is cyclic for the action of $X, Y, Z$. Consider the GIT quotient
\begin{equation*}
NHilb^n(\mathbb{C}^3):=Hom(V_n, V_n)^3\times V_n\sslash GL_n=U/GL_n
\end{equation*}
with respect to this linearization.
Let $W:NHilb^n(\mathbb{C}^3)\rightarrow \mathbb{C}$ be the function on $NHilb^n(\mathbb{C}^3)$ defined by $W(X, Y, Z, v)=tr([X, Y]Z)$. The condition $\{dW=0\}$ is equivalent to the condition that $X$, $Y$ and $Z$ pairwise commute. Then
the locus $\{dW=0\}$ is isomorphic to $Hilb^n(\mathbb{C}^3)$. As the construction suggests, we call $NHilb^n(\mathbb{C}^3)$ the noncommutative Hilbert scheme.

\medskip\noindent
{\bf Remark.} While any nonzero scalar multiple of $W$ would work just as well as $W$ in defining $Hilb^n(\mathbb{C}^3)$, the construction of \cite{BBS13} makes clear that the choice of Calabi-Yau form $dx\wedge dy\wedge dz$ on $\mathbb{C}^3$ determines the normalization of $W$ described above.  More precisely, any determinant 1 linear change of coordinates in $T=\mathrm{span}\{x,y,z\}$ will leave $W$ unchanged.

\section{D-critical Locus Structure on $Hilb^n(\omega_S)$}\label{sec:dcrit}

We begin this section with a lightning review of some standard notions and notations of toric geometry.  We knowingly omit some important details and simply refer the reader to \cite{CLS11} if more detail is needed.

We let $N$ be an $n$-dimensional lattice and $\Sigma$ a fan in $N_\mathbb{R}=N\otimes\mathbb{R}$, giving rise to a normal toric variety $X_\Sigma$ with torus $T=N\otimes_{\mathbb{Z}}\mathbb{C}^*$. We can describe $X_\Sigma$ by a quotient construction.  Let $v_1,\ldots, v_r$ be the primitive integral generators of the 1-dimensional cones in $\Sigma(1)$.  We have an exceptional closed subset $Z(\Sigma)\subset\mathbb{C}^r$ and the group $G=Hom(A_{n-1}(X_\Sigma),\mathbb{C}^*)$ acts on $\mathbb{C}^r-Z(\Sigma)$ by an action which will be described below.  Then $X_\Sigma$ is identified with the quotient
\begin{equation*}
X_\Sigma=\left(\mathbb{C}^r-Z(\Sigma)
\right)/G.
\end{equation*}
If $\sigma$ is a cone in $\Sigma$, then we have an affine toric variety $X_\sigma$, which is identified with an open subset of $X_\Sigma$ by
\begin{equation*}
X_\sigma=\left\{[(x_1,\ldots,x_r)]\in X_\Sigma\mid x_i\ne 0{\rm \ if\ }v_i\not\in\sigma
\right\},
\end{equation*}
where $[(x_1,\ldots,x_r)]$ denotes equivalence class of $(x_1,\ldots,x_r)\in \mathbb{C}^r-Z(\Sigma)$ mod $G$.

For $i=1\ldots r$ we have $T$-invariant divisors $D_i\subset X_\Sigma$ defined by $x_i=0$, where $(x_1,\ldots,x_r)$ are coordinates on $\mathbb{C}^r$.  The action of $G$ on $\mathbb{C}^r$ is given by
\begin{equation*}
g\cdot(x_1,\ldots,x_r)=(g([D_1])x_1,\ldots,g([D_r])x_r)
\end{equation*}
where $[D_i]$ is the divisor class of $D_i$.

Now suppose that $X_\Sigma$ is smooth.  Then $A_{n-1}(X_\Sigma)$ is torsion-free and consequently $G\simeq(\mathbb{C}^*)^{r-n}$.  Furthermore, if $\sigma\in\Sigma(n)$ is a maximal cone generated by $v_{i_1},\ldots,v_{i_n}$, the affine toric variety $X_\sigma$ is isomorphic to $\mathbb{C}^n$ via the isomorphism
\begin{equation}\label{eq:affinecoordinates}
\mathbb{C}^n\to X_\sigma,\qquad (x_1,\ldots,x_n)\mapsto (1,\ldots,1,x_1,1,\ldots,1,x_2,1\ldots,1,x_n,1\ldots,1),
\end{equation}
where $x_j$ is in the $i_j$th position and all other entries are 1.

We are concerned with the case where $X_\Sigma$ is a smooth Calabi-Yau threefold.  For any smooth toric variety we have
\begin{equation*}
\mathcal{O}(K_{X_\Sigma})\simeq\mathcal{O}\left(-\sum_{i=1}^rD_i\right),
\end{equation*}
so the Calabi-Yau condition says that $\sum_{i=1}^rD_i\sim 0$.  It is convenient to express this condition in terms of the homogeneous coordinate ring
\begin{equation*}
S=\mathbb{C}[x_1,\ldots,x_r].
\end{equation*}
The ring $S$ is graded by $A_{n-1}(X_\Sigma)=A_2(X_\Sigma)$, with $x_i$ having degree $[D_i]\in A_2(X_\Sigma)$.  The Calabi-Yau condition translates into the statement that $x_1\cdots x_r$ has degree $0$.

We now construct a nowhere-vanishing $T$-invariant holomorphic form $\Omega$ on $X_\Sigma$, well-defined up to sign.   We have holomorphic $r$-form on $\mathbb{C}^r$
\begin{equation*}
\widehat{\Omega}=dx_1\wedge\cdots\wedge dx_r,
\end{equation*}
well-defined up to a sign depending on the chosen ordering of $\Sigma(1)$.  The action of $G\simeq(\mathbb{C}^*)^{r-3}$ on $\mathbb{C}^r$ determines a rank $r-3$ lattice of holomorphic vector fields on $\mathbb{C}^r$.   More precisely, let $\Lambda\subset\mathbb{Z}^r$ be the lattice of relations
\begin{equation}\label{eqn:Lambda}
\Lambda=\left\{(a_1,\ldots,a_r)\in\mathbb{Z}^r\mid \sum a_i v_i=0
\right\}.
\end{equation}
Then $\lambda=(a_1,\ldots,a_r)$ determines the vector field
\begin{equation*}
\chi_\lambda=\sum a_i x_i\frac{\partial}{\partial x_i}
\end{equation*}
on $\mathbb{C}^r$.  The relation $\lambda$ determines a 1-parameter subgroup of $G$
\begin{equation*}
t\mapsto ([D_i]\mapsto t^{a_i}).
\end{equation*}
The fact that $\lambda=(a_1,\ldots,a_r)$ satisfies (\ref{eqn:Lambda}) implies that $[D_i]\mapsto t^{a_i}$ is compatible with linear equivalence and so describes an element of $G=Hom(A_{n-1}(X_\Sigma),\mathbb{C}^*)$.
The vector field $\chi_\lambda$ is then seen to be the infinitesimal action of this 1-parameter subgroup.

Choosing generators
$\chi_1,\ldots,\chi_{r-3}$ for the lattice of vector fields, we define
\begin{equation}\label{eqn:omega}
\Omega = i_{\chi_1}\cdots i_{\chi_{r-3}}(\widehat{\Omega}).
\end{equation}
For each $j=1,\ldots r-3$ we have $i_{\chi_j}\Omega=0$, so that $\Omega$ is $G$-invariant and therefore pulls back from a holomorphic 3-form on $X_\Sigma$.  We abuse notation and denote this 3-form on $X_\Sigma$ by $\Omega$ as well.  In other words, $\Omega$ is the unique holomorphic 3-form $\Omega$ on $X_\Sigma$ whose pullback to $\mathbb{C}^r-Z(\Sigma)$ is the right hand side of (\ref{eqn:omega}).  The Calabi-Yau condition guarantees that $\Omega$ is a 3-form rather than merely a 3-form valued in a line bundle.

We now specialize further to local toric surfaces, the total space $\omega_S$ of the canonical bundle of a smooth complete toric surface $S$.  Since $S$ is a toric variety, we have a rank 2 lattice $N'$ and a complete fan $\Sigma'$ in $N'_\mathbb{R}$ with edges spanned by primitive integral vectors $v_1',\ldots,v_{r-1}'$.  We form the rank 3 lattice $N=N'\oplus \mathbb{Z}$, identify the fan $\Sigma'$ with the collection of cones $\{\sigma'\times\{1\}\mid \sigma'\in\Sigma'\}$ in the affine hyperplane of $N_\mathbb{R}$ defined by setting the last coordinate to 1.  We then form the fan $\Sigma$ as the collection of cones in $N_\mathbb{R}$ over the cones ${\sigma'}\times\{1\}$.  In particular, the primitive integral generators of the 1-dimensional cones of $\Sigma$ are identified with
\begin{equation}\label{eq:edges}
v_1=(v_1',1),\ldots,v_{r-1}=(v_{r-1}',1), v_r=(\mathbf{0},1),
\end{equation}
where $\mathbf{0}$ is the zero of $N$.
We introduce the convention of replacing $x_r$ by the variable $p$, to emphasize the distinguished role of $v_r$.

\medskip\noindent
{\bf Example.}
Local $\mathbb{P}^2$ can be described by a fan $\Sigma$ with edges generated by the vectors
\begin{equation*}
v_1=(1,0,1),\ v_2=(0,1,1),\ v_3=(-1,-1,1),\ v_4=(0,0,1).
\end{equation*}
We have maximal cones $\sigma_j$ for $j=1,2,3$.  Each $\sigma_j$ is spanned by the set of vectors $\{v_1,\ldots,v_4\}-\{v_j\}$.

In this case, the lattice $\Lambda$ is spanned by $(1,1,1-3)$, corresponding to the relation $v_1+v_2+v_3-3v_4=0$.  We get the generating vector field $\chi=x_1(\partial/\partial x_1)+x_2(\partial/\partial x_2)+x_3(\partial/\partial x_3)-3p(\partial/\partial p)$.  Putting $\Omega=i_\chi\left(dx_1\wedge dx_2\wedge dx_3\wedge dp
\right)$, we get
\begin{equation*}
\Omega=x_1 dx_2\wedge dx_3\wedge dp - x_2 dx_1\wedge dx_3\wedge dp+x_3dx_1\wedge dx_2\wedge dp+3pdx_1\wedge dx_2\wedge dx_3.
\end{equation*}

By respectively setting $x_i=1$ for $i=1,2,3$, we obtain
\begin{equation*}
\Omega|_{\sigma_1} = dx_2\wedge dx_3\wedge dp,\ \Omega|_{\sigma_2} = dx_3\wedge dx_1\wedge dp,\ \Omega|_{\sigma_3} = dx_1\wedge dx_2\wedge dp.
\end{equation*}
Accordingly, we can transfer the construction of Section~\ref{subsec:hilb} from $Hilb^n(\mathbb{C}^3)$ to each $Hilb^n(X_{\sigma_i})$, using the respective coordinate systems $(x_2,x_3,p), (x_3,x_1,p),(x_1,x_2,p)$.\footnote{We have abused notation here.  The relationship between the three local coordinates and homogeneous coordinates is given by three separate applications of (\ref{eq:affinecoordinates}).  It follows that the coordinate changes will be nontrivial, even for the variables in different coordinate systems denoted by the same symbol.} For example we can describe $NHilb^n(X_{\sigma_1})$ as a GIT quotient of a space of $n\times n$ matrices $X_2,X_3,P$ together with a vector $v$, and define $Hilb^n(X_{\sigma_1})$ as the degeneracy locus of $W_1=Tr([X_2,X_3]P)$.  Similarly, we can define $Hilb^n(X_{\sigma_2})$  as the degeneracy locus of $W_2=Tr([X_3,X_1]P)$ in $NHilb^n(X_{\sigma_2})$ and $Hilb^n(X_{\sigma_3})$  as the degeneracy locus of $W_3=Tr([X_1,X_2]P)$ in $NHilb^n(X_{\sigma_3})$.  This description gives the data $(Hilb^n(X_{\sigma_j}), NHilb^n(X_{\sigma_j}), W_j, i_j)$, where $i_j:Hilb^n(X_{\sigma_j})\hookrightarrow NHilb^n(X_{\sigma_j})$ is the inclusion.

We will show that the $(Hilb^n(X_{\sigma_j}), NHilb^n(X_{\sigma_j}), W_j, i_j)$ are among the d-critical charts of a d-critical locus on $Hilb^n(\omega_{\mathbb{P}^2})$. The construction of the d-critical locus structure proceeds in two steps.

First, we show that the sections $s_j$ of $S^0_{Hilb^n(X_{\sigma_j})}$ induced by the $W_j$ agree on pairwise intersections $Hilb^n(X_{\sigma_j})\cap Hilb^n(X_{\sigma_k})\subset Hilb^n(\omega_{\mathbb{P}^2})$.  We will show this more generally for any local toric Calabi-Yau threefold in Proposition~\ref{prop:opendcrit} below.  But this is still not enough to construct a d-critical locus, since the $Hilb^n(X_{\sigma_j})$ do not cover $Hilb^n(\omega_{\mathbb{P}^2})$.

Second, we construct a cover of $\omega_{\mathbb{P}^2}$ by open subsets $U_\alpha\simeq\mathbb{C}^3$ such that the $Hilb^n(U_\alpha)$ cover $Hilb^n(\omega_{\mathbb{P}^2})$.  Furthermore, automorphisms of $\mathbb{P}^2$ induce isomorphisms $U_\alpha\simeq X_{\sigma_j}$.  These isomorphisms are used to construct the data $(Hilb^n(U_\alpha), NHilb^n(U_\alpha), W_\alpha, i_\alpha)$.  Finally, a comparison to the first step shows that the sections $s_\alpha$ of $S^0_{Hilb^n(U_\alpha)}$ are pairwise compatible.  We will elaborate on this step more generally in the proof of Theorem~\ref{thm:dcrit} below.

\medskip
Returning to the general case, we have the following easy lemma.  The coordinates $(x,y,z)$ on $X_\sigma$ in the statement of the lemma are given by (\ref{eq:affinecoordinates}).

\begin{lemma}\label{lem:plus1}
Let $\Omega$ be the holomorphic 3-form (\ref{eqn:omega}) on the local Calabi-Yau threefold $\omega_S$ and let $\sigma$ be a maximal cone in a fan for $\omega_S$.  Then we can order the coordinates$(x,y,z)$ on $X_\sigma\simeq\mathbb{C}^3$ so that $\Omega|_{X_\sigma}=dx\wedge dy\wedge dz$.
\end{lemma}

We had already seen this for $\omega_{\mathbb{P}^2}$ by direct calculation.

\begin{proof}
 Reordering $\Sigma(1)$ if necessary, we can assume that the edges of $\sigma$ are generated by $v_{r-2},v_{r-1}$, and $v_r$.  We have coordinates $(x_{r-2},x_{r-1},x_r)$ on $X_\sigma$.  Choosing generators $\lambda_1,\ldots,\lambda_{r-3}$ of the lattice (\ref{eqn:Lambda}) and writing $\lambda_i=(a_{i1},\ldots,a_{ir})$, we get $\Omega|_{X_\sigma}=\pm\det(A)\, dx_{r-2}\wedge dx_{r-1}\wedge dx_r$, where $A$ is the $(r-3)\times (r-3)$ matrix with entries $a_{ij}, 1\le i,j\le r-3$.  Choosing a different set of generators in the definition of $\Omega$ can only change $\Omega|_{X_\sigma}$ by a sign as remarked earlier.  However, since $\{v_{r-2},v_{r-1},v_r\}$ is a basis for $N$, it follows that we can find generators $\lambda_i$ with $A$ the identity matrix, in which case $\det(A)=1$.
\end{proof}

Now fixing a choice of $\Omega$ on $\omega_S$, for each maximal cone $\sigma$ we choose coordinates $(x,y,z)$ on $X_\sigma$ so that Lemma~\ref{lem:plus1} holds, noting that the coordinates $(x,y,z)$ can undergo a cyclic permutation and still satisfy Lemma~\ref{lem:plus1}.  We repeat the description of $Hilb^n(\mathbb{C}^3)$ in this context, introducing endomorphisms $X_\sigma,Y_\sigma,Z_\sigma\in Hom(V_n,V_n)$ and putting $NHilb(X_\sigma)=Hom(V_n,V_n)^3\times V_n\sslash GL_n$ as before. This determines the data $(Hilb^n(X_{\sigma}), NHilb^n(X_{\sigma}), W_\sigma, i_\sigma)$, where $W_\sigma$ is the function on $NHilb(X_\sigma)$  given by $Tr([X_\sigma,Y_\sigma]Z_\sigma)$.   Since $W_\sigma$ is unchanged by a cyclic permutation of $(X_\sigma,Y_\sigma,Z_\sigma)$, we conclude that $(Hilb^n(X_{\sigma}), NHilb^n(X_{\sigma}), W_\sigma, i_\sigma)$ only depends on $\sigma$.

Let $s_\sigma\in S^0_{Hilb^n(X_\sigma)}$ be the section induced by $W_\sigma$.

\begin{proposition}\label{prop:opendcrit}
Given maximal cones $\sigma$ and $\sigma'$, the sections $s_\sigma$ and $s_{\sigma'}$ agree on $Hilb^n(X_\sigma)\cap Hilb^n(X_{\sigma'})$.
\end{proposition}

We therefore have a canonical d-critical locus structure on $\cup_{\sigma}Hilb^n(X_\sigma)\subset Hilb^n(\omega_S)$ with critical charts $(Hilb^n(X_{\sigma}), NHilb^n(X_{\sigma}), W_\sigma, i_\sigma)$.

\begin{proof}
We begin by observing that the proposition can be reduced to the case when $\sigma$ and $\sigma'$ share a codimension~1 face.
If $\sigma$ and $\sigma'$ do not share a codimension~1 face, then
we can choose maximal cones $\sigma_1,\sigma_2,\ldots,\sigma_m$ with $m\ge 3$ and $\sigma_1=\sigma$ and $\sigma_m=\sigma'$ such that each pair $\sigma_i,\sigma_{i+1}$ has a common codimension~1 face.  Furthermore, $X_\sigma\cap X_{\sigma'}=X_{\rho}\simeq \mathbb{C}\times(\mathbb{C}^*)^2$, where $\rho$ is the 1-dimensional cone generated by $v_r=(0,0,1)$ in the notation (\ref{eq:edges}).  Since $\rho$ is a face of every $\sigma_i$, we have $X_\rho\subset X_{\sigma_i}$ for each $i$.  Correspondingly, $Hilb^n(X_\sigma)\cap Hilb^n(X_{\sigma'})=Hilb^n(X_\rho)\subset Hilb^n(X_{\sigma_i})$ for all $i$.  Once we have proven the proposition for cones sharing a codimension~1 face, it will follow immediately that on $Hilb^n(X_\rho)$ we have $s_\sigma=s_{\sigma_2}=s_{\sigma_3}=\ldots=s_{\sigma'}$ as required.

The common codimension~1 face of $\sigma$ and $\sigma'$ is spanned by $v_r$ and some $v_i=(v_i',1)$ for $i<r$.  The vector $v_i'\in N'$ corresponds to a torus-invariant curve $D'_i$ in the toric surface $S$.  We let $f$ be the self-intersection of $D'_i$ in $S$, so that $v'_{i-1}+fv'_i+v'_{i+1}=0$.\footnote{If $i=1$ or $i=r-1$, we cyclically reorder the $v'_j$ so that $v'_{i+1}$ makes sense.}
It follows that $v_{i-1}+fv_i+v_{i+1}=(f+2)v_r$.  Using the basis $\{v_{i-1},v_i,v_r\}$ for $N$ to identify $N$ with $\mathbb{Z}^3$, we have, reordering $\sigma$ and $\sigma'$ if necessary
\begin{equation*}
\sigma=\mathrm{span}\{(1,0,0),(0,1,0),(0,0,1)\},\ \sigma'=\mathrm{span}\{(0,1,0),(-1,-f,f+2),(0,0,1)\}.
\end{equation*}

The coordinate change between $X_\sigma$ and $X_{\sigma'}$ is given by
\begin{equation}\label{eq:change}
(x',y',z')=(x^{-f}y,x^{-1},x^{f+2}z).
\end{equation}
We have ordered the coordinates so that $dx'\wedge dy'\wedge dz'=dx\wedge dy \wedge dz$.  If $\Omega|_{X_\sigma}=-dx\wedge dy\wedge dz$, we simply reorder the coordinates appropriately without impacting the rest of the argument.

We next transfer the construction of Section~\ref{subsec:hilb} to $X_{\sigma}$ and to $X_{\sigma'}$ while providing some more detail.
We let $X,Y$ and $Z$ be three $n\times n$ matrices with indeterminate entries $X(k, l)$, $Y(k, l)$ and $Z(k, l)$, representing multiplication by $x$, $y$ and $z$, respectively. Let $v_1,..., v_n$ be $n$ additional indeterminates and put
\begin{equation*}
A=\mathbb{C}[X(k, l), Y(k, l), Z(k, l), v_1,...,v_n].
\end{equation*}
Denote by $U$ the open subset $(Spec(A))^{ss}$ of $Spec(A)$ with respect to the natural $GL_n$ action and put $NH=U/GL(n)$.  Put $d=\det(X)$ and let $A_d=A[d^{-1}]$.
We denote $U\cap SpecA_{d}$ by $U_{d}$. We similarly define
\begin{equation*}
A'=\mathbb{C}[X'(k,l),Y'(k,l),Z'(k,l),v'_1,\ldots,v'_n],
\end{equation*}
$U'=(Spec(A'))^{ss}$, $NH'=U'/GL_n$, $A'_{d'}=A'[det({X'})^{-1}]$, and $U'_{d'}=U'\cap SpecA_{d'}$.

There is an isomorphism
\begin{equation*}
\label{equ3.4}
\phi: A'_{d'}\simeq A_{d}
\end{equation*}
which identifies
\begin{equation*}
X'=X^{-f}Y,
Y'=X^{-1},
Z'=X^{f+2}Z,
v'_i=v_i,\ i=1,\ldots n
\end{equation*}
induced by the change of coordinates (\ref{eq:change}) after making a choice in the order of matrix multiplication. This induces the isomorphism of open subsets:
\begin{equation}
\label{equ3.5}
U_{d}\cap \phi^{-1}(U_{d'})\simeq \phi(U_{d})\cap U'_{d'},
\end{equation}
where we have abused notation slightly by using $\phi$ to also denote the induced map $Spec(A_{d})\to Spec(A'_{d'})$.

To simplify notation, we put $R=Hilb^n(X_\sigma)$ and $R'=Hilb^n(X_{\sigma'})$.  Let $i:R\to NH$ and $i':R'\to NH'$ be the inclusions.  Let $R\cap R'$ be the intersection of $R$ and $R'$ taken inside $Hilb^n(\omega_S)$. Note that $U_{d}$ and $U'_{d'}$ are invariant under the $GL(V_n)$ actions. Hence $(U_{d}\cap \phi^{-1}(U'_{d'}))/GL(V_n)$ is an open neighborhood of $R\cap R'$ in $U_{d}/GL(V_n)\subset NH$, and $(\phi(U_{d})\cap U')/GL(V_n)$ is an open neighborhood of $R\cap R'$ in $U'_{d"}/GL(V_n)\subset NH'$. Then (\ref{equ3.5}) induces an isomorphism
\begin{equation}
\label{equ3.6}
i^{-1}(O_{N})/I_{R, N}^2|_{R\cap R'}\simeq {i'}^{-1}(O_{N'})/I_{R', N'}^2|_{R\cap R'}.
\end{equation}

We have the exact sequence of sheaves on $N$
\begin{equation*}
0\to K\to i^{-1}(O_{N})/I_{R, N}^2\xrightarrow{d} i^{-1}(T^*N)/I_{R, N}\cdot i^{-1}(T^*N)
\end{equation*}
where $K$ is defined as the kernel, and is naturally identified with the sheaf $\mathcal{S}_{R}$.  We have a similar exact sequence defining a sheaf $K'$ on $R'$.

We deduce the following commutative diagram, where the vertical arrows are isomorphisms
\begin{equation}
\label{eq:isomk}
\begin{array}{ccccccc}
0&\to& \left(K\right)|_{R\cap R'}&\to& \left(i^{-1}(O_{N})/I_{R, N}^2\right)|_{R\cap R'}&\xrightarrow{d}& \left(i^{-1}(T^*N)/I_{R, N}\cdot i^{-1}(T^*N)\right)|_{R\cap R}\\
&&\downarrow&&\downarrow&&\downarrow\\
0&\to& \left(K'\right)|_{R\cap R'}&\to& \left({i'}^{-1}(O_{N'})/I_{R', N'}^2\right)|_{R\cap R'}&\xrightarrow{d}& \left({i'}^{-1}(T^*N')/I_{R', N'}\cdot {i'}^{-1}(T^*N')\right)|_{R\cap R'}
\end{array}.
\end{equation}
We show that the sections $s_\sigma\in H^0(K)$ and $s_{\sigma'}\in H^0(K')$ agree on $R\cap R'$ according to the leftmost isomorphism in (\ref{eq:isomk}).  For this, it suffices to show that their images in $i^{-1}(O_{N})/I_{R, N}^2$ and ${i'}^{-1}(O_{N'})/I_{R', N'}^2$ agree on $R\cap R'$ via the isomorphism (\ref{equ3.6}).  We proceed to verify this equality.

We have the potentials $W_\sigma=tr([X,Y]Z])$ on $U$ and $W_{\sigma'}=tr([X',Y']Z])$ on $U'$.
Denote the image of $i^{-1}(W_\sigma)$ in the sheaf $i^{-1}(O_{N})/I_{R, N}^2$ by $i^{-1}(W_\sigma)+I_{R, N}^2$, with analagous notation for ${i'}^{-1}(W_{\sigma'})$. Then we claim that:

\begin{equation}\label{eq:compatible}
i^{-1}(W_\sigma)+I_{R, N}^2|_{R\cap R'}={i'}^{-1}(W_{\sigma'})+I_{R', N'}^2|_{R\cap R'}\rm{\  under\ the\ identification\ (\ref{equ3.6})}.
\end{equation}
This is precisely the assertion of the proposition that the sections $s_\sigma$ and $s_{\sigma'}$ agree on $R\cap R'$.

We now demonstrate (\ref{eq:compatible}).
Under the isomorphism (\ref{equ3.4}), we have
\begin{equation*}
W_\sigma|_{Spec(A_{d})}=tr(XYZ-YXZ)
\end{equation*}
\begin{equation}\label{eq:w'}
W_{\sigma'}|_{Spec(A'_{d'})}=tr(X^{-f}YX^{f+1}Z-X^{-(f+1)}YX^{f+2}Z).
\end{equation}
Making use of the identity $tr(MN)=tr(NM)$ for $n\times n$ matrices $M$ and $N$, we can rewrite (\ref{eq:w'}) as
\begin{equation}\label{eq:w'2}
W_{\sigma'}|_{Spec(A'_{d'})}=tr(X^{f+1}ZX^{-f}Y-X^{f+2}ZX^{-(f+1)}Y).
\end{equation}
Since (\ref{eq:w'2}) can be obtained from (\ref{eq:w'}) by exchanging \cgreen{$Y$ and $Z$}, replacing $f$ by $-f-2$, and multiplying by $-1$, we can and will assume that $f\ge -1$.  If $f=-1$, then $W_\sigma|_{Spec(A_{d})}=W_{\sigma'}|_{Spec(A'_{d'})}$.  If $f=0$ we compute
\begin{equation}\label{eq:f0}
W_\sigma-W_{\sigma'}=tr\left(XYZ-2YXZ+X^{-1}YX^2Z\right)=tr\left(X^{-1}[X,Y][Z,X]\right),
\end{equation}
which is visibly in $I_{R, N}^2|_{R\cap R'}$.

We proceed by induction on $f$, assuming the inductive hypothesis
\begin{equation}\label{eq:f}
tr\left(\left(X^{-f}YX^{f+1}Z-X^{-(f+1)}YX^{f+2}Z\right)-\left(XYZ-YXZ\right)\right)\in I_{R, N}^2|_{R\cap R'}.
\end{equation}
The substitution $(X,Y,Z)\mapsto (X,YX,Z)$ preserves $I_{R, N}$ and hence $I_{R,N}^2$, since
\begin{equation*}
[X,YX]=-X[X,Y],\ [Z,YX]=[Z,Y]X+Y[Z,X].
\end{equation*}
Making this substitution in (\ref{eq:f}) we get
\begin{equation}\label{eq:induct}
tr\left(\left(X^{-f}YX^{f+2}Z-X^{-(f+1)}YX^{f+3}Z\right)-\left(XYXZ-YX^2Z\right)\right)\in I_{R, N}^2|_{R\cap R'}.
\end{equation}
We write
\begin{equation*}
X^{-(f+1)}YX^{f+2}Z-X^{-(f+2)}YX^{f+3}Z=X^{-1}\left(X^{-f}YX^{f+2}Z-X^{-(f+1)}YX^{f+3}Z
\right),
\end{equation*}
so that we have by (\ref{eq:induct}) modulo $I_{R, N}^2|_{R\cap R'}$
\begin{equation*}
tr(X^{-(f+1)}YX^{f+2}Z-X^{-(f+2)}YX^{f+3}Z)\equiv tr(YXZ-X^{-1}YX^2Z)
\end{equation*}
and therefore
\begin{equation}\label{eq:almostplus1}
tr\left(\left(X^{(-f+1)}YX^{f+2}Z-X^{-(f+2)}YX^{f+3}Z\right)-\left(YXZ-X^{-1}YX^2Z\right)\right)\in I_{R, N}^2|_{R\cap R'}.
\end{equation}
We also have
\begin{equation*}
Tr\left(XYZ-2YXZ+X^{-1}YX^2Z\right)\in I_{R, N}^2|_{R\cap R'}.
\end{equation*}
by (\ref{eq:f0}).   Subtracting this last equation from (\ref{eq:almostplus1}), we obtain (\ref{eq:f}) with $f$ replaced by $f+1$, proving the inductive step, and we are done.
\end{proof}

We now come to the main result of this section.

\begin{theorem}\label{thm:dcrit}
Suppose that $S=\mathbb{P}^2$ or $S=\mathbb{F}_n$.  Then $Hilb^n(\omega_S)$ has a natural d-critical locus structure.
The critical charts are all isomorphic to $(Hilb^n(\mathbb{C}^3),NHilb^n(\mathbb{C}^3),W,i)$.
\end{theorem}

\begin{proof}
Let $\pi:\omega_S\to S$ be the projection.  Our strategy is to exhibit an open cover $\{U_\alpha\}$ of $S$ satisfying the two properties:
\begin{enumerate}
\item There exists an automorphism of $\omega_S$ which preserves $\Omega$ and takes $V_\alpha:=\pi^{-1}(U_\alpha)$ to $X_{\sigma_\alpha}$ for some maximal cone $\sigma_\alpha$.
\item Any finite subset of $S$ is contained in some $U_\alpha$.
\end{enumerate}
The first condition allows us to define charts
\begin{equation*}
(Hilb^n(V_\alpha),NHilb^n(V_\alpha),W_\alpha,i_\alpha)
\end{equation*}
by using the automorphism to transfer the chart $(Hilb^n(X_{\sigma_\alpha})),NHilb^n(g(X_{\sigma_\alpha})),W_{\sigma_\alpha},i_{\sigma_\alpha})$ from $Hilb^n(X_{\sigma_\alpha})$ to $Hilb^n(V_\alpha)$.   The second condition implies that $\{Hilb^n(V_\alpha)\}$ is an open cover of $Hilb^n(\omega_S)$. To see this, let $Z\in Hilb^n(\omega_S)$.  Thinking of $Z$ as a length $n$ subscheme of $\omega_S$, we choose a $U_\alpha$ containing the finite set $\pi(Z)$.  Then $Z\subset V_\alpha$, so that $Z\in Hilb^n(V_\alpha)$.

The rest of the proof consist of choosing the $U_\alpha$ and automorphisms so that the sections $s_\alpha\in S^0_{Hilb^n(V_\alpha)}$ agree on pairwise intersections.
We work out the details for $S=\mathbb{P}^2$ and $S=\mathbb{F}_n$ separately.

\smallskip\noindent
$S=\mathbb{P}^2$:  For a line $\ell\subset \mathbb{P}^2$, we let $U_\ell=\mathbb{P}^2-\ell$ be its complement.  Then $\{U_\ell\}$ is an open cover of $\mathbb{P}^2$ which clearly satisfies the second property.   To check the first property, fix an equation for $\ell$ and let $A$ be a determinant~1 linear transformation of $\mathrm{span}(x_1,x_2,x_3)$ which induces an automorphism of $\mathbb{P}^2$  taking $\ell$ to one of the torus invariant lines.
In terms of the homogeneous coordinates $(x_1,x_2,x_3,p)$ for $\omega_{\mathbb{P}^2}$, the automorphisms we consider are  $\phi_A(\mathbf{x},p)=(A\mathbf{x},p)$ where $\mathbf{x}=(x_1,x_2,x_3)$.
 Since both $\widehat{\Omega}=dx_1\wedge dx_2\wedge dx_3\wedge dp$ and $v$ are invariant under $A$, it follows that $\Omega$ is invariant under the automorphisms $\phi_A$.

Given any pair of lines $\ell,\ell'$, we can find equations of $\ell$ and $\ell'$ which are taken to $x_1$ and $x_2$ respectively by $A$. Then $\phi_A$ identifies
$Hilb^n(V_\ell)\cap Hilb^n(V_{\ell'})$ with $Hilb^n(X_{\sigma_1})\cap Hilb^n(X_{\sigma_2})$ using the labeling of the cones $\sigma_i$ introduced in the example earlier in this section.  The compatibility of the critical charts of $Hilb^n(V_\ell)$ and $Hilb^n(V_{\ell'})$ follows immediately from the compatibility of the critical charts of $Hilb^n(X_{\sigma_1})$ and $Hilb^n(X_{\sigma_2})$ proven in Proposition~\ref{prop:opendcrit}.

\smallskip\noindent
$S=\mathbb{F}_n$:  We describe $\mathbb{F}_n$ as the toric surface associated to the complete fan with 1-dimensional cones spanned by the vectors
\begin{equation}
v'_1=(1,0),\ v'_2=(0,1),\ v'_3=(-1,n),\ v'_4=(0,-1).
\end{equation}
The torus-invariant curves $D'_1$ and $D'_3$ are fibers of $\mathbb{F}_n$, the curve $D'_2$ is the section with $(D'_2)^2=-n$, and $D'_4$ is a section with $(D'_4)^2=n$.  The lattice of vectors fields associated to the quotient construction of $\omega_{\mathbb{F}_n}$ is generated by
\begin{equation*}
\chi_1=x_2(\partial/\partial x_2)\! +\! x_4(\partial/\partial x_4)\! -\! 2p(\partial/\partial p), \chi_2=x_1(\partial/\partial x_1)\! +\! x_3(\partial/\partial x_3)\! +\! nx_4(\partial/\partial x_4)\! -\! (n+2)p(\partial/\partial p).
\end{equation*}

The automorphisms of $\omega_{\mathbb{F}_n}$ which we use are of two types, described in terms of homogeneous coordinates by
\begin{enumerate}
\item $(x_1,x_2,x_3,x_4,p)\mapsto (ax_1+bx_3,x_2,cx_1+dx_3,x_4,p)$, where $ad-bc=1$
\item $(x_1,x_2,x_3,x_4,p)\mapsto (x_1,x_2,x_3,x_4+ex_2x_3^n,p)$.
\end{enumerate}

These automorphisms preserve $\widehat{\Omega}$ and the $\chi_i$, hence they preserve $\Omega$ as well.

Let $\phi:\mathbb{F}_n\to \mathbb{P}^1$ be the map exhibiting $\mathbb{F}_n$ as a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$.   In homogeneous coordinates, we have $\phi(x_1,x_2,x_3,x_4)=(x_1,x_3)$.

Given a point $p=(a,b)\in \mathbb{P}^1$,
we have a 1-parameter family of sections of $S_{a,b,e}$ of $\mathbb{F}_n$ described parametrically  as $\{(x_1,1,x_3,e(bx_1-ax_3)^n)\mid (x_1,x_3)\in \mathbb{P}^1\}$.
Intrinsically, these sections are the irreducible members of the linear system $|D'_4|$ which intersect $D'_4$ only in the point $(a,1,b,0)$ of $D'_4$, with multiplicity $n$.  Since $S_{\lambda a,\lambda b,\lambda^{-n}e}=S_{a,b,e}$, we see that this 1-parameter family of sections only depends on $p$ and not on a choice of homogenous coordinates $(a,b)$ of $p$.     We use the cover $\{U_{a,b,e}\}=\{\phi^{-1}(\mathbb{P}^1-(a,b))-S_{a,b,e}\}$ of $\mathbb{F}_n$.  Let $V_{a,b,e}=\pi^{-1}(U_{a,b,e})$.

To verify the first property required of $\{U_\alpha\}$, we can find an automorphism of the first type mapping $V_{a,b,e}$ to $V_{1,0,e'}$.  Then an automorphism of the second type maps $V_{1,0,e'}$ to $V_{1,0,0}$.  But $V_{1,0,0}$ is equal to $X_\sigma$, where $\sigma$ is the maximal cone generated by $\{v_1=(1,0,1),v_2=(0,1,1),v_5=(0,1,1)\}$.   We use these automorphisms to identify the function $W_\sigma$ on $NHilb^n(X_\sigma)$ with a function $W_{a,b,e}$ on $NHilb^n(V_{a,b,e})$.  Let $s_{a,b,e}$ be the section of $S^0_{Hilb^n(V_{a,b,e})}$ induced by $W_{a,b,e}$.

Given a finite subset $T$ of $\mathbb{F}_n$, we first choose $p=(a,b)\in \mathbb{P}^1-\phi(T)$.  Then we can find an $e$ such that $S_{a,b,e}\cap T$ is empty, so that $T\subset U_{a,b,e}$.  This verifies the second property required of $\{U_\alpha\}$.

It remains to show pairwise compatibility of the sections $s_{a,b,e}$ of $S^0_{Hilb^n(V_{a,b,e})}$.  Consider distinct open subsets $V_{a,b,e}$ and $V_{a',b',e'}$.  There are two cases to consider:
\begin{enumerate}
\item $(a,b)= (a',b')$ as points of $\mathbb{P}^1$, or
\item $(a,b)\ne(a',b')$ as points of $\mathbb{P}^1$.
\end{enumerate}
In the first case, we can find a determinant 1 linear transformation of $\mathrm{span}(x_1,x_3)$ taking $(a,b)$ to $(1,0)$.  Then we use an automorphism of the second type to take $V_{a,b,e}$ to $V_{1,0,0}$.  This automorphism takes $V_{a',b',e'}$ to $V_{1,0,e''}$ for some $e''\ne0$.  We are reduced to comparing $V_{1,0,0}$ and $V_{1,0,e''}$.  On $V_{1,0,0}$ we have affine coordinates $(x_1,x_2,p)$ following (\ref{eq:affinecoordinates}).  Our construction gives us affine coordinates $(x_1',x_2',p')$ on $U_{1,0,e''}$.  We relate $(x_1',x_2',p')$ to $(x_1,x_2,p)$ using full homogenous coordinates.  We have
\begin{equation*}
(x_1',x_2',1,1,p)=(x_1,x_2,1,1+e''x_2,p)=(x_1,x_2(1+e''x_2)^{-1},1,1,(1+e''x_2)^2p),
\end{equation*}
where we have used the $(\mathbb{C}^*)^2$ of the quotient construction in the last step.  This gives the coordinate change as
$(x_1',x_2',p')=(x_1,x_2(1+e''x_2)^{-1},(1+e''x_2)^2p)$.  Promoting $x_1,x_2,p,x_1',x_2',p'$ to elements of $End(V_n)$, we compute

\begin{equation*}
[X_1',X_2']P'-[X_1,X_2]P=[X_1,X_2(1+e''X_2)^{-1}](1+e''X_2)^2P-[X_1,X_2]P,
\end{equation*}
which simplifies to
\begin{equation*}
e''X_1X_2^2P+X_2X_1P-X_2(1+e''X_2)^{-1}X_1(1+e''X_2)^2P.
\end{equation*}
\cgreen{Using the properties of the trace, we see that this last expression has the same trace as that of
\begin{equation*}
[X_1(1+e''X_2),X_2(1+e''X_2)^{-1}]\ [X_1,P].
\end{equation*}
We only have to show that this trace is in the square of the ideal $I$  generated by the entries of the commutators of $X_1,X_2$, and $P$, appropriately localized so that $1+e''X_2$ is invertible.  We only have to show that the entries of $[X_1(1+e''X_2),X_2(1+e''X_2)^{-1}]$ are in $I$, or equivalently that $X_1(1+e''X_2)X_2(1+e''X_2)^{-1}$ is congruent to $X_2(1+e''X_2)^{-1}X_1(1+e''X_2)$ modulo $I$.  We can do this by commuting adjacent terms successively.  For example, to show that  $(1+e''X_2)^{-1}X_1$ is congruent to $X_1(1+e''X_2)^{-1}$ moduli $I$, we only have to observe that $X_1(1+e''X_2)$ is congruent to $(1+e''X_2)X_1$ modulo $I$, and then multiply by $(1+e''X_2)^{-1}$ on the left and on the right.
}

We abbreviate our notation by putting $H_{a,b,e}=Hilb^n(V_{a,b,e})$ and $NH_{a,b,e}=NHilb^n(V_{a,b,e})$.
Thus $W_{1,0,0}=W_{1,0,e''}$ on $H_{1,0,0}\cap H_{1,0,e''}$, hence $s_{1,0,0}=s_{1,0,e''}$ on $H_{1,0,0}\cap H_{1,0,e''}$ as well.

In the second case, we can find a determinant 1 linear transformation of $\mathrm{span}(x_1,x_3)$ taking $(a,b)$ to $(1,0)$
and $(a',b')$ to $(0,1)$.  Then we use an automorphism of the second type to take $V_{a,b,e}$ to $V_{1,0,0}$.  This automorphism takes $V_{a',b',e'}$ to $V_{0,1,e''}$ for some $e''$.  We are reduced to comparing $s_{1,0,0}$ and $s_{0,1,e''}$ on $H_{1,0,0}\cap H_{0,1,e''}$.

If $e''=0$, we are already done by Proposition~\ref{prop:opendcrit}. For $e''\ne0$, we first note that $H_{1,0,0}\cap H_{0,1,e''}= H_{1,0,0}\cap H_{0,1,0}\cap H_{0,1,e''}$.  It follows that the three sections $s_{1,0,0}$ on $H_{1,0,0}$, $s_{0,1,0}$ on $H_{0,1,0}$, and $s_{0,1,e''}$ on $H_{0,1,e''}$ are all defined on $H_{1,0,0}\cap H_{0,1,e'}$  and so can be compared there.  The first two agree on $H_{1,0,0}\cap H_{0,1,0}$ by Proposition~\ref{prop:opendcrit} as just noted, and for the last two, they agree on $H_{0,1,0}\cap H_{0,1,e''}$ by a straightforward calculation analogous to what was just done above.  It follows that $s_{1,0,0}$ and $s_{0,1,e''}$ agree on $H_{1,0,0}\cap H_{0,1,e''}$ and we are done.
\end{proof}

\smallskip\noindent
{\bf Remark.}  Our methods are easily adapted to construct a d-critical locus structure on the Hilbert schemes of local $\mathbb{P}^1$.

\begin{comment}
We let $d_n$ be the sum of the terms in the expansion of $W_1-W_0$ which are of degree $n$ in the entries of $E$.
We easily check that $d_n=0$ for $n=0, 1$. For $n\geq 2$, $d_n$ is given by:
\begin{equation*}
\begin{split}
d_n=tr((E_0)^nP_0Z_0-2(E_0)^{n-1}P_0E_0Z_0+(E_0)^{n-2}P_0(E_0)^2Z_0).
\end{split}
\end{equation*}
By the cyclic property of the trace,
\begin{equation*}
d_n=tr(-(E_0)^{n-2}(E_0P_0-P_0E_0)(E_0Z_0-Z_0E_0)).
\end{equation*}

Hence in $\widehat{A_0}$, we have
$$W_1=W_0-tr((\sum_{n=0}^\infty E_0^n)(E_0P_0-P_0E_0)(E_0Z_0-Z_0E_0)).$$

Writing $\widehat{A_0}\simeq k[{Z_0}(k, l), {P_0}(k, l), v_1,...,v_n][[\delta_{ij}-{Y_0}(k, l)]]$, it clear that $\widehat{A_0}$ is an integral domain and the natural map $A_0\to \widehat{A_0}$ is injective.  Furthermore $Y_0$ is invertible when considered as matrix with entries in $\widehat{A_0}$: its inverse is given by
$$Y_0^{-1}=\sum_{n=0}^\infty E_0^n.$$
Thus $\det(Y_0)$ is invertible in $\widehat{A_0}$ (uniquely invertible since $\widehat{A_0}$ is a domain), we deduce a ring homomorphism
\begin{equation*}
A_{01}=A_0[\det(Y_0)^{-1}]\to \widehat{A_0}
\end{equation*}
which is immediately seen to be injective.

Hence in $A_{01}$, we have
\begin{equation*}
W_1=W_0-tr(Y_0^{-1}(E_0P_0-P_0E_0)(E_0Z_0-Z_0E_0)=W_0-tr(Y_0^{-1}(P_0Y_0-Y_0P_0)(Z_0Y_0-Y_0Z_0),
\end{equation*}
which implies that
\begin{equation*}
i_0^{-1}(W_0)+I_{R_0, N_0}^2=i_1^{-1}(W_1)+I_{R_1, N_1}^2.
\end{equation*}

\begin{comment}
We prove the equality of local sections for the other charts in a similar way.
Consider the case $i=1, 2$, and $W_1=tr(x_1[z_1, t_1])$ and $W_2=tr(x_2[t_2, y_2])$. Then we have
\begin{lemma}
\label{lemma11}
$i_1^{-1}(W_1)+I_{R_1, N_1}^2|_{R_1\cap R_2}=i_2^{-1}(W_2)+I_{R_2, N_2}^2|_{R_1\cap R_2}$.
\end{lemma}

\begin{proof}
The argument is exactly the same as Lemma \ref{lemma10}, except for the computation of the difference between $W_1$ and $W_2$. Let $\widehat{A_1}$ be the completion of $A_1$ at the ideal $I=(Id-z_1)$. Denoting the matrix of indeterminants $Id-z_1$ by $e_1$. Then $det(z_1)$ is invertible in $\widehat{A_1}$ whose inverse is given by
$$det(\sum_{n=0}^\infty e_1^n).$$  We have

\begin{equation*}
W_1=tr(x_1(1-e_1)t_1-x_1t_1(1-e_1)),
\end{equation*}

\begin{equation*}
W_2=tr((1-e_1)x_1t_1-(1-e_1)^2x_1(1+e_1+e_1^2+...)t_1).
\end{equation*}

We denote the image of $W_2-W_1$ in $A_1/{(e_1)^{n+1}}$ by $d_n$. Then $d_n=0$ for $n=0, 1$. For $n\geq 2$, $d_n$ is given by:

\begin{equation*}
\begin{split}
d_n=tr(-e_1^{n-2}(e_1^2t_1x_1-2e_1t_1e_1x_1+t_1e_1^2x_1))\\
=tr(-e_1^{n-2}(e_1t_1-t_1e_1)(e_1x_1-x_1e_1)).
\end{split}
\end{equation*}

\end{proof}

Then consider the case when $i=0, 2$, and $W_0=tr(y_0[z_0, t_0])$ and $W_2=tr(x_2[t_2, y_2])$. We have

\begin{lemma}
\label{lemma12}
$i_2^{-1}(W_2)+I_{R_2, N_2}^2|_{R_2\cap R_0}=i_0^{-1}(W_0)+I_{R_0, N_0}^2|_{R_2\cap R_0}$.
\end{lemma}
\begin{proof}
Again, the argument is exactly the same as in Lemma \ref{lemma10}, except for the computation of the difference between $W_0$ and $W_2$. We abuse notation to denote the completion of $A_0$ at the ideal $I=(Id-z_0)$ also by $\widehat{A_0}$. Denoting the matrix of indeterminants $Id-z_0$ by $f_0$. Then $det(z_0)$ is invertible in $\widehat{A_0}$ whose inverse is given by
$$det(\sum_{n=0}^\infty f_0^n).$$
We have

\begin{equation*}
W_0=tr(y_0(1-f_0)t_0-y_0t_0(1-f_0)),
\end{equation*}

\begin{equation*}
W_2=tr(-(1+f_0+f_0^2+...)t_0(1-f_0)^2y_0+(1-f_0)y_0t_0),
\end{equation*}

We denote the image of $W_2-W_1$ in $A_0/{(f_0)^{n+1}}$ by $d_n$. Then $d_n=0$ for $n=0, 1$. For $n\geq 2$, $d_n$ is given by:

\begin{equation*}
\begin{split}
d_n=tr(-f_0^{n-2}(f_0^2t_0y_0-2f_0t_0f_0y_0+t_0f_0^2y_0))\\
=tr(-f_0^{n-2}(f_0t_0-t_0f_0)(f_0y_0-y_0f_0)).
\end{split}
\end{equation*}

\end{proof}

Next, we consider a general $\mathbb{C}_\ell$'s. Choosing any homogenous equation $ax+by+cz=0$ of $\ell$, we can find a determinant~1 linear transformation of the span of $\{x,y,z\}$ taking $x$ to $ax+by+cz$. In this way, we transfer the d-critical locus structure from $Hilb^n(\mathbb{C}_{\ell_x})$ to $Hilb^n(\mathbb{C}_{\ell})$.
Furthermore, our previous discussion shows that this d-critical locus structure is independent of the choice of equation for $\ell$ and the choice of a determinant~1 linear transformation.

Now, given another $\mathbb{C}_{\ell'}$, we get a d-critical locus structure on $Hilb^n(\mathbb{C}_{\ell'})$ and we want to show that this agrees with the $Hilb^n(\mathbb{C}_{\ell})$ on the intersection $Hilb^n(\mathbb{C}_{\ell})\cap Hilb^n(\mathbb{C}_{\ell'})\subset Hilb^n(\omega_S$).  But this is easy.  We choose an equation $dx+ey+fz$ for $\ell'$ and find a determinant~1 linear transformation of the span of $\{x,y,z\}$ taking $x$ to $ax+by+cz$ and $y$ to $dx+ey+fz$.  The desired compatibility then follows from Lemma~\ref{lemma10}.

The relations with the coordinates in $C_0$ are given by
\begin{equation*}
y_\ell=\frac{y_0}{a+by_0+cz_0},
\end{equation*}
\begin{equation*}
z_\ell=\frac{z_0}{a+by_0+cz_0},
\end{equation*}
\begin{equation*}
t_\ell=\frac{1}{a}t_0(a+by_0+cz_0)^3.
\end{equation*}
The relations with the coordinates in $C_1$ are given by
\begin{equation*}
y_\ell=\frac{1}{-ax_1+b+cz_1},
\end{equation*}
\begin{equation*}
z_\ell=\frac{z_1}{-ax_1+b+cz_1},
\end{equation*}
\begin{equation*}
t_\ell=\frac{1}{a}t_1(-ax_1+b+cz_1)^3.
\end{equation*}
and the relations with the coordinates in $C_2$ are given by
\begin{equation*}
y_\ell=\frac{y_2}{-ax_2+by_2+c},
\end{equation*}
\begin{equation*}
z_\ell=\frac{1}{-ax_2+by_2+c},
\end{equation*}
\begin{equation*}
t_\ell=\frac{1}{a}t_2(-ax_2+by_2+c)^3.
\end{equation*}
Note that
\begin{equation*}
dy_\ell\wedge dz_\ell\wedge dt_\ell=dy_0\wedge dz_0\wedge dt_0=dx_1\wedge dz_1\wedge dt_1=dx_2\wedge dt_2\wedge dy_2
\end{equation*}
on the intersections, hence the change of coordinates preserves the Calabi-Yau form.

Define the potential on $N_\ell$ by
\begin{equation*}
W_\ell=tr(y_\ell[z_\ell, t_\ell]).
\end{equation*}

\begin{lemma}
$i_\ell^{-1}(W_\ell)+I_{R_\ell, N_\ell}^2|_{R_\ell\cap R_0}=i_0^{-1}(W_0)+I_{R_0, N_0}^2|_{R_\ell\cap R_0}$.
\end{lemma}
\begin{proof}

Again, the argument is exactly the same as Lemma \ref{lemma10}, except for the computation of the difference between $W_0$ and $W_\ell$. Denote $a+by_0+cz_0$ by $u_0$. We abuse notation to denote the completion of $A_0$ at the ideal $I=(Id-u_0)$ also by $\widehat{A_0}$. Denoting the matrix of indeterminants $Id-u_0$ by $g_0$. Then $det(u_0)$ is invertible in $\widehat{A_0}$ whose inverse is given by
$$det(\sum_{n=0}^\infty g_0^n).$$

Then we have
\begin{equation*}
\begin{split}
W_0&=tr(y_0z_0t_0-y_0t_0z_0)\\
&=\frac{1}{b}(-g_0-cz_0+1-a)(z_0t_0-t_0z_0),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
W_\ell=&tr(y_\ell z_\ell t_\ell-y_\ell t_\ell z_\ell)\\
=&tr(\frac{1}{ab}(-g_0-cz_0+1-a)((1+g_0+g_0^2+...)z_0t_0(1-g_0)^2-\\
&t(1-g_0)^2z_0(1+g_0+g_0^2+...))).
\end{split}
\end{equation*}
We denote the image of
\begin{equation*}
(1+g_0+g_0^2+...)z_0t_0(1-g_0)^2-t_0(1-g_0)^2z_0(1+g_0+g_0^2+...),
\end{equation*}
in $A_0/{(g_0)^{n+1}}$ by $d_n$. Then for $n=1$, $d_n$ is given by:
\begin{equation*}
\begin{split}
d_1&=tr(-\frac{1}{ab}(-g_0^2z_0t_0+2g_0t_0g_0z_0-g_0^2t_0z_0))\\
&=tr(\frac{1}{ab}(g_0t_0-t_0g_0)(g_0z_0-z_0g_0))
\end{split}
\end{equation*}
and for $n\geq 2$
\begin{equation*}
\begin{split}
d_n=&tr(\frac{1-a}{ab}(-g_0^{n-2}(g_0^2t_0z_0-2g_0t_0g_0z_0+t_0g_0^2z_0))+\frac{1}{ab}(g_0^{n-1})(g_0^2t_0z_0-2g_0t_0g_0z_0+t_0g_0^2z_0))\\
&=tr(-\frac{1-a}{ab}g_0^{n-2}(g_0t_0-t_0g_0)(g_0z_0-z_0g_0)+\frac{1}{ab}g_0^{n-1}(g_0t_0-t_0g_0)(g_0z_0-z_0g_0)).
\end{split}
\end{equation*}
\end{proof}

\begin{lemma}
$i_\ell^{-1}(W_\ell)+I_{R_\ell, N_\ell}^2|_{R_\ell\cap R_1}=i_1^{-1}(W_1)+I_{R_1, N_1}^2|_{R_\ell\cap R_1}$.
\end{lemma}
\begin{proof}
We denote $-ax_1+b+cz_1$ by $u_1$, and the completion of $A_1$ at the ideal $I=(Id-u_1)$ by $\widehat{A_1}$. Denoting the matrix of indeterminants $Id-u_1$ by $g_1$. We have
\begin{equation*}
\begin{split}
W_1&=tr(x_1z_1t_1-x_1t_1z_1)\\
&=-\frac{1}{a}tr(-g_1-cz_1+1-b)(z_1t_1-t_1z_1),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
W_\ell&=tr(y_\ell z_\ell t_\ell-y_\ell t_\ell z_\ell)\\
&=(1/a)tr((1+g_1+g_1^2+...)^2z_1t_1(1-g_1)^3-(1+g_1+g_1^2+...)t_1(1-g_1)^2z_1).
\end{split}
\end{equation*}
We denote the image of $W_\ell-W_1$ in $A_1/{(g_1)^{n+1}}$ by $d_n$. Then $d_i=0$ for $i=0, 1$. For $n\geq 2$,
\begin{equation*}
\begin{split}
d_n&=(1/a)tr(-g_1^nt_1z_1+2g_1^{n-1}t_1g_1z_1-g_1^{n-2}t_1g_1^2z_1)\\
&=(1/a)tr(g_1^{n-2}(-g_1^2t_1z_1+2g_1t_1g_1z_1-t_1g_1^2z_1))\\
&=(1/a)tr(g_1^{n-2}(g_1t_1-t_1g_1)(g_1z_1-z_1g_1)).
\end{split}
\end{equation*}
\end{proof}

\begin{lemma}
$i_\ell^{-1}(W_\ell)+I_{R_\ell, N_\ell}^2|_{R_\ell\cap R_2}=i_2^{-1}(W_2)+I_{R_2, N_2}^2|_{R_\ell\cap R_2}$.
\end{lemma}
\begin{proof}
We denote $-ax_2+by_2+c$ by $u_2$, and the completion of $A_2$ at the ideal $I=(Id-u_2)$ by $\widehat{A_2}$. Denote the matrix of indeterminants $Id-u_2$ by $g_2$. Then we have
\begin{equation*}
\begin{split}
W_2&=tr(x_2t_2y_2-x_2y_2t_2)\\
&=-\frac{1}{a}tr(-g_2-by_2+1-c)(t_2y_2-y_2t_2),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
W_\ell&=tr(y_\ell z_\ell t_\ell-y_\ell t_\ell z_\ell)\\
&=(1/a)tr((1-g_2)^2y_2(1+g_2+g_2^2+...)t_2-(1-g_2)y_2t_2).
\end{split}
\end{equation*}
We denote the image of $W_\ell-W_2$ in $A_2/{(g_2)^{n+1}}$ by $d_n$. Then $d_i=0$ for $i=0, 1$. For $n\geq 2$,
\begin{equation*}
\begin{split}
d_n&=(1/a)tr(y_2g_2^nt_2-2g_2y_2g_2^{n-1}t_2+g_2^2y_2g_2^{n-2}t_2)\\
&=(1/a)tr(g_2^{n-2}(g_2^2t_2y_2-2g_2t_2g_2y_2+t_2g_2^2y_2))\\
&=(1/a)tr(-g_2^{n-2}(g_2t_2-t_2g_2)(g_2y_2-y_2g_2)).
\end{split}
\end{equation*}
\end{proof}

Since the $Hilb^n(C_\ell)$ cover $Hilb^n(\omega_S)$ and support pairwise compatible d-critical loci, we have proven
\begin{theorem}
\label{thm3.1}
\begin{sloppypar}
The charts $(Hilb^n(C_\ell), N_\ell, i_\ell, W_\ell)$ define an algebraic d-critical locus structure $(Hilb^n(\omega_S), s)$. The section $s$ is obtained by gluing the $W_\ell$'s.
\end{sloppypar}
\end{theorem}

subschemes isomorphic to $Hilb^n(X_{\sigma_j})$  for some $j$.  for which we can reduce the necessary compatibility to what was already checked in the first step using automorphisms of $\omega_{\mathbb{P}^2}$.

Then $Hilb^n(\omega_{\mathbb{P}^2})$ can be covered by the collection of Zariski open subsets $\{Hilb^n(C_\ell)\}_{\ell\subset\mathbb{P}^2}$. We  properly normalize the $W_\ell$'s and show that they glue to define a section in $H^0(\mathcal{S}_Y^0)$.   For

Note that $\Omega$ is invariant under

Let $U_\sigma$ be the affine toric variety associated to a maximal cone $\sigma$ of a fan for $\omega_S$.  consider 3-dimensional fans $\Sigma$ consisting In this section, we consider smooth toric Calabi-Yau threefolds $X_{\Sigma}$ associated to 3-dimensional fan $\Sigma$.  In the previous section, we saw that $Hilb^n(\mathbb{C}^3)$ can be globally realized as a degeneracy locus. In this section, we show that for  $\omega_S$ a local toric Calabi-Yau threefold, $Hilb^n(\omega_S)$ has a d-critical locus structure with critical charts isomorphic to

Since the sheaf $\mathcal{S}_{Hilb^n(\omega_S)}$ is intrinsic to $Hilb^n(\omega_S)$, it is enough to find a collection of critical charts covering $Hilb^n(\omega_S)$ such that the local sections satisfies the gluing condition in Definition \ref{dcrit}.  Before turning to this task, we

We let $N$ be a lattice and $\Sigma$ a fan in $N_\mathbb{R}=N\otimes\mathbb{R}$.  Let $v_1,\ldots, v_r$ be the primitive integral generators of the 1-dimensional cones in $\Sigma(1)$.

We describe $\omega_S$ as the quotient
\begin{equation*}
\omega_S=\left\{\left(x,y,z,p\right)\in \mathbb{C}^4\mid \left(x,y,z\right)\ne\left(0,0,0\right)
\right\}\big\slash\left(\left(x,y,z,p\right)\sim\left(\lambda x,\lambda y, \lambda z,\lambda^{-3}p\right)\right)
\end{equation*}
with $\lambda\in\mathbb{C}^*$.
Then
\begin{equation*}
\Omega=xdy\wedge dz\wedge dp-ydx\wedge dz \wedge dp\wedge dx+zdx\wedge dy\wedge dp - 3p dx \wedge dy \wedge dz
\end{equation*}
is a nowhere vanishing 3-form on $\omega_S$.  This 3-form arises from contracting the $\mathbb{C}^*$-invariant 4-form $dx\wedge dy\wedge dz \wedge dp$ with the Euler vector field $x\partial_x+y\partial_y+z\partial_z-3p\partial_p$.

We now let $\ell_x\subset\mathbb{P}^2$ be the line $x=0$, and we consider the chart $\mathbb{C}_{\ell_x}$ of $\omega_S$.  On $\mathbb{C}_{\ell_x}$ we introduce local affine coordinates
\begin{equation*}
(y_0,z_0,p_0)=(x^{-1}y,x^{-1}z,x^3p).
\end{equation*}
In these coordinates, we have $\Omega|_{\mathbb{C}_x}=dy_0\wedge dz_0\wedge dp_0$.  Using these local coordinates to identify $\mathbb{C}_{\ell_x}$ with $\mathbb{C}^3$, we get a critical chart $(Hilb^n(\mathbb{C}_{\ell_x}), NHilb^n(\mathbb{C}_{\ell_x}), W_{\ell_x}, i_{\ell_x})$ and a corresponding section of $s_{\ell_x}\in H^0(S_{Hilb^n(\mathbb{C}_{\ell_x})}^0)$.
This critical chart is independent of any determinant 1 change of linear coordinates, following the remark at the end of Section~\ref{subsec:hilb}.

Next, let $\ell_y\subset\mathbb{P}^2$ be the line $y=0$, and we consider the chart $\mathbb{C}_{\ell_y}$ of $\omega_S$.  On $\mathbb{C}_{\ell_y}$ we introduce local affine coordinates
\begin{equation*}
(z_1,x_1,p_1)=(y^{-1}z,xy^{-1},y^3p).
\end{equation*}
In these coordinates, we have $\Omega|_{\mathbb{C}_{\ell_y}}=dz_1\wedge dx_1\wedge dp_1$.
This coordinate system gives us another critical chart $(Hilb^n(\mathbb{C}_{\ell_1}), NHilb^n(\mathbb{C}_{\ell_1}), W_{\ell_1}, i_{\ell_1})$  and a corresponding section of $s_{\ell_y}\in H^0(S_{Hilb^n(\mathbb{C}_{\ell_y})}^0)$.

We now show that $s_{\ell_x}$ and $s_{\ell_y}$ agree on the intersection $Hilb^n(\mathbb{C}_{\ell_x})\cap Hilb^n(\mathbb{C}_{\ell_y})\subset Hilb^n(\omega_S)$.

Now let $\ell\subset\mathbb{P}^2$ be any line distinct from $\ell_x$.  We can find a determinant 1 linear transformation which fixes $x$ and send $\ell$ to the line $\ell_1$ defined by $y=0$.  On $\mathbb{C}_{\ell_0}$ we introduce local affine coordinates

We first note the change of coordinates
\begin{equation}
\label{equ3.1}
z_1=y_0^{-1}z_0,
x_1=y_0^{-1},
p_1=p_0y_0^3.
\end{equation}

\end{comment}

\section{Orientation on $(Hilb^n(\omega_S), s)$}\label{sec:orientation}

In this section, we understand $S$ to mean either $S=\mathbb{P}^2$ or $S=\mathbb{F}_n$.  We use the more generic notations $U_\alpha, V_\alpha, W_\alpha$, and $s_\alpha$ instead of using $U_\ell$ etc.\ for $S=\mathbb{P}^2$ and $U_{a,b,e}$ etc.\ for $S=\mathbb{F}_n$.
We first work out the virtual canonical bundle associated to the d-critical locus $(Hilb^n(\omega_S), s)$ \ccblue{following the construction in [Dav], see also \cite{Shi18}}.
\begin{proposition}
\label{prop8}
Let $F$ be the universal object on $Hilb^n(\omega_S)$ and let $\pi_1: Hilb^n(\omega_S)\times \omega_S\rightarrow Hilb^n(\omega_S)$ be the projection. Then we have
\begin{equation}
K_{Hilb^n(\omega_S), s}\simeq (det(\pi_{1*}F)^*)^2.
\end{equation}
\end{proposition}
\begin{proof}
We first prove condition (i) in Definition \ref{defK}. Let $(R, U, f, i) = (Hilb^n(V_\alpha), N_\alpha, i_\alpha, W_\alpha)$ be one of the critical charts of $(Hilb^n(\omega_S), s)$ constructed in the proof of Theorem~\ref{thm:dcrit}. It may be more clear if we use the quiver description of $U = N_\alpha$. It is well known that $NHilb^n(\mathbb{C}^3)$ is isomorphic to the moduli space of stable representations of the following quiver:
\begin{center}
\begin{tikzpicture}

\node [circle, minimum size=0.1pt,inner sep=0pt,outer sep=0pt] (a) at (0,0) {$s_\infty$};
\node[right=2cm of a, circle, minimum size=0.1pt,inner sep=0pt,outer sep=0pt] (b) {$s_0$} edge [in=-50,out=-120,loop, scale=3] node[midway, fill=white, scale=0.8]{x} ();
\node[right=2cm of a,circle,  minimum size=0.1pt,inner sep=0pt,outer sep=0pt] (b) {$s_0$} edge [in=-10,out=50,loop, scale=3] node[midway, fill=white, scale=0.8]{y} ();
\node[right=2cm of a, circle,  minimum size=0.1pt,inner sep=0pt,outer sep=0pt] (b) {$s_0$} edge [in=130,out=70,loop, scale=3] node[midway, fill=white, scale=0.8]{z} ();

\draw [->] (a.east) -- (b.west) node[midway, fill=white, scale=0.8]{a} ;

\end{tikzpicture}
\end{center}
with $V_\infty$ of dimension one and $V_0$ of dimension $n$.
Here the quiver stability condition is equivalent to the condition that for any nonzero $v\in V_\infty$, $a(v)$ generates $V_0$ under the action of $x$, $y$ and $z$.

We abuse notation slightly, also denoting by $V_0$ and $V_\infty$ the corresponding universal bundles on $N_\ell$.  We see that the cotangent bundle of $N_\alpha$ is given by
\begin{equation}
\label{equcot}
\begin{split}
\Omega_{N_\alpha} &\simeq (V_0^*\otimes V_0)\oplus (V_0^*\otimes V_0)\oplus (V_0^*\otimes V_0)\oplus (V_0^*\otimes V_\infty).\\
\end{split}
\end{equation}
The bundle $(V_0)|_{Hilb^n(V_\alpha)}$ is isomorphic to $\pi_{1*}F$ by the construction of the Hilbert scheme.  Also, the global section 1 of $\pi_{1*}F$ corresponds to a trivialization of $(V_\infty)|_{Hilb^n(\mathbb{C}_\ell)}$.
Hence we have a canonical isomorphism induced by (\ref{equcot}):
\begin{equation*}
K_{N_\alpha}^2|_{Hilb^n(V_\alpha)}\simeq det\left(\left(V_0^*\otimes V_\infty\right)|_{Hilb^n(V_\alpha)}\right)^2\simeq (det(\pi_{1*}F)^*|_{Hilb^n(V_\alpha)})^2.
\end{equation*}
This shows condition (i).

It is easy to see condition (ii) is satisfied by checking the restriction of $(det(\pi_{1*}F)^*)^2$ to the critical charts $(Hilb^n(V_\alpha), N_\alpha, i_\alpha, W_\alpha)$ and $(Hilb^n(V_\alpha)', N_\alpha', i_\alpha', W_\alpha')$, where $N_\alpha'$ is a Zariski open subset of $N_\alpha$ and $Hilb^n(V_\alpha)'=Hilb^n(V_\alpha)\cap N_\alpha'$.
\end{proof}

By Proposition \ref{prop8}, we see that $det(\pi_{1*}F)^*$ defines an orientation on $(Hilb^n(\omega_S), s)$.
\begin{comment}
Actually we obtain a collection of orientation data here. We use $p$ to denote the map
\begin{equation*}
p:Hilb^n(\omega_S)\rightarrow Sym^n(\omega_S),
\end{equation*}
and $q$ to denote the map
\begin{equation*}
q:(\omega_S)^n\rightarrow Sym^n(\omega_S).
\end{equation*}
Let $L$ be a square root line bundle of $O_{\omega_S}$. Then $\widetilde{L}:=p^*(q_*(L\boxtimes ...\boxtimes L)^{S_n})$ is a square root line bundle of $O_{Hilb^n(\omega_S)}$. Hence $\widetilde{L}\otimes det(\pi_{1*}F)^*$ also defines an orientation data for $(Hilb^n(\omega_S), s)$.
\end{comment}

\section{Motivic DT invariants for $Hilb^n(\omega_S)$}

In this section we use the d-critical locus structure from Section~\ref{sec:dcrit} and the orientation  from Section~\ref{sec:orientation} to compute the motivic DT invariants of $Hilb^n(\omega_S)$, showing that it agrees with the computation in \cite{BBS13} for $S=\mathbb{P}^2$ and $S=\mathbb{F}_n$.

We first recall some definations used in \cite{BBS13}. Let $X$ be a quasi-projective threefold. Then $Hilb^n(X)$ admits a stratification
$$Hilb^n(X)=\coprod_{\gamma\vdash n}Hilb^n_\gamma(X),$$
where $\gamma$ is a partition of $n$, and $Hilb^n_\gamma(X)$ is the locally closed subscheme of $Hilb^n(X)$ parametrizing length $n$ subschemes whose support multiplicities are given by $\gamma$. In particular, there are $\gamma_i$ length $i$ clusters in the length $n$ subscheme.

First consider the case of $Hilb^n(\mathbb{C}^3)$. Define the relative motivic class of $Hilb^n(\mathbb{C}^3)$ in $\mathcal{M}_{Hilb^n(\mathbb{C}^3)}^{\hat{\mu}}$ by:
\begin{equation}
[Hilb^n(\mathbb{C}^3)]_{relvir}=\mathbb{L}^{-dim(NHilb^n(\mathbb{C}^3)/2)}\odot MF_{NHilb^n(\mathbb{C}^3), W}^\phi.
\end{equation}
Comparing to Theorem \ref{thm1.1}, we see that this is the motivic invariant associated to $(Hilb^n(\mathbb{C}^3), W)$ without the contribution of a principal $\mathbb{Z}_2$ bundle from the orientation.
Let $[Hilb^n_\gamma(\mathbb{C}^3)]_{relvir}$ to be the pullback of $[Hilb^n(\mathbb{C}^3)]_{relvir}$ to $Hilb^n_\gamma(\mathbb{C}^3)$. On the deepest strata, there is an embedding
\begin{equation}\label{eq:translation}
\{0\}\times Hilb^n(\mathbb{C}^3)_0\subset \mathbb{C}^3\times Hilb^n(\mathbb{C}^3)_0\simeq Hilb^n_{(n)}(\mathbb{C}^3)
\end{equation}
where $Hilb^n_{(n)}(\mathbb{C}^3)$ is the punctual Hilbert scheme.   The isomorphism
in (\ref{eq:translation}) is given by sending $(p,Z)\in \mathbb{C}^3\times Hilb^n(\mathbb{C}^3)_0$ to the subscheme $Z+p\subset\mathbb{C}^3$ obtained by translating $Z$ by $p$.

Let $[Hilb^n(\mathbb{C}^3)_0]_{relvir}$ be the pullback of $[Hilb^n_{(n)}(\mathbb{C}^3)]_{relvir}$ to $Hilb^n(\mathbb{C}^3)_0$. Also define the absolute motivic classes to be the pushforwards of the corresponding relative motivic classes to a point, and denote them by $[Hilb^n(\mathbb{C}^3)]$, $[Hilb^n_\gamma(\mathbb{C}^3)]$ and $[Hilb^n(\mathbb{C}^3)_0]$ respectively.

Recall the following proposition and definition in \cite{BBS13}:
\begin{proposition} (\cite{BBS13} Proposition 2.6)
\label{prop13}
(1) The absolute motivic class $[Hilb^n_\gamma(\mathbb{C}^3)]$ and $[Hilb^n_\gamma(\mathbb{C}^3)_0]$ live in the subring $\mathcal{M}_\mathbb{C}\subset \mathcal{M}_\mathbb{C}^{\hat{\mu}}$.

(2) On the closed stratum,
\begin{equation*}
[Hilb^n_{(n)}(\mathbb{C}^3)]=\mathbb{L}^3\cdot[Hilb^n(\mathbb{C}^3)_0]\in \mathcal{M}_\mathbb{C}.
\end{equation*}

(3) More generally, for a general stratum,
\begin{equation*}
[Hilb^n_\gamma(\mathbb{C}^3)]=\pi_{G_\gamma}([\prod_i(\mathbb{C}^3)^{\gamma_i}\setminus\Delta]\cdot\prod_i[Hilb^i(\mathbb{C}^3)_0^{\gamma_i}]).
\end{equation*}
where the map $\pi_{G_\gamma}$ is defined by taking the orbit space on generators.
\end{proposition}

\begin{definition}(\cite{BBS13} definition 3.1)
\label{def4}
We define motivic classes
$[Hilb^n_\gamma(X)]\in\mathcal{M}_\mathbb{C}$ and $[Hilb^n(X)]\in\mathcal{M}_\mathbb{C} $
as follows.

(1) on the deepest stratum,
\begin{equation*}
[Hilb^n_{(n)}(X)]=[X]\cdot[Hilb^n(\mathbb{C}^3)_0].
\end{equation*}

(2) More generally, on all strata,
\begin{equation*}
[Hilb^n_\gamma(X)]=\pi_{G_\gamma}([\prod_i X^{\gamma_i}\setminus\Delta]\cdot\prod_i[Hilb^i(\mathbb{C}^3)_0^{\gamma_i}]).
\end{equation*}

(3) Finally
\begin{equation*}
[Hilb^n(X)]=\sum_\gamma[Hilb^n_\gamma(X)].
\end{equation*}

\end{definition}

Now consider the case when $X=\omega_S$ for $S=\mathbb{P}^2$ or $\mathbb{F}_n$. Let $[Hilb^n(\omega_S)]$ be defined as in Definition \ref{def4}.  Then we have
\begin{comment}
By the discussion at the end of section 2.2, we can choose $det(\pi_{1*}F)^*$ as a square root of $K_{Hilb^n(\omega_S),s}$.
\end{comment}
\begin{theorem}
\label{thm3.2}
Given the d-critical locus $(Hilb^n(\omega_S), s)$ and the choice of orientation $det(\pi_{1*}F)^*$, then
the absolute motivic class of the motive given by Theorem \ref{thm1.1} concides with
$[Hilb^n(\omega_S)]$.
\end{theorem}

\begin{proof}
Since we are given a d-critical locus structure with a choice of orientation, by Theorem \ref{thm1.1} there is a unique globally well defined motivic DT invariant $MF_{Hilb^n{\omega_S}, s}$.  We continue our practice of using a generic index $\gamma$ to be common shorthand for the indexing of either $\mathbb{P}^2$ or $\mathbb{F}_n$.  Furthermore, for any critical chart $(R_\alpha, U_\alpha, f_\alpha, i_\alpha)=(Hilb^n(V_\alpha), N_\alpha, W_\alpha, i_\alpha)$ this invariant satisfies:
\begin{equation*}
MF_{Hilb^n(\omega_S),s}|_{Hilb^n(V_\alpha)}=i_\alpha^*(\mathbb{L}^{-dim{N_\alpha}/2}\odot MF^\phi_{N_\alpha,W_\alpha})\odot\Upsilon(Q_{Hilb^n(V_\alpha), N_\alpha, W_\alpha, i_\alpha})\in\overline{\mathcal{M}}_{Hilb^n(V_\alpha)}^{\hat{\mu}}.
\end{equation*}

By the proof of Proposition \ref{prop8} we have a canonical isomorphism
\begin{equation*}
\label{equ101}
det(\pi_{1*}F)^*|_{Hilb^n(V_\alpha)^{red}}\simeq K_{N_\alpha}|_{Hilb^n(V_\alpha)^{red}}.
\end{equation*}
Hence $Q_{Hilb^n(V_\alpha), N_\alpha, W_\alpha, i_\alpha}$ and therefore $\Upsilon(Q_{Hilb^n(V_\alpha), N_\alpha, W_\alpha, i_\alpha})$ are trivial, implying
\begin{equation*}
\label{equMF}
MF_{Hilb^n(\omega_S),s}|_{Hilb^n(V_\alpha)}=[Hilb^n(V_\alpha)]_{relvir}.
\end{equation*}
By Proposition \ref{prop13}, $[Hilb^n(V_\alpha)]_{relvir}\in \mathcal{M}_{Hilb^n(V_\alpha)}$.
Since the $Hilb^n(V_\alpha)$ cover $Hilb^n(\omega_S)$,
it follows that $MF_{Hilb^n(\omega_S),s}\in \mathcal{M}_{Hilb^n(\omega_S)}$.

Hence we have
\begin{equation}
\label{decom}
MF_{Hilb^n(\omega_S),s}=\sum_\gamma MF_{Hilb^n(\omega_S),s}|_{Hilb^n_\gamma(\omega_S)}.
\end{equation}

Since
\begin{equation*}
Hilb^n_{(n)}(V_\alpha)\simeq V_\alpha\times Hilb^n(\mathbb{C}^3)_0
\end{equation*}
and $W$ is translation invariant in the sense that
\begin{equation*}
tr([X,Y]Z)=tr([X+xId,Y+yId](Z+zId))
\end{equation*}
for any $(x,y,z)\in \mathbb{C}^3$, we see that
\begin{equation*}
[Hilb^n_{(n)}(V_\alpha)]_{relvir}=p_2^*[Hilb^n(\mathbb{C}^3)_0]_{relvir},
\end{equation*}
where $p_2:V_\alpha\times Hilb^n(\mathbb{C}^3)_0\to Hilb^n(\mathbb{C}^3)_0$ is the projection to the second factor.
\begin{comment}
Hence
\begin{equation*}
[Hilb^n_{(n)}(C_\ell)]_{relvir}=[C_\ell]\cdot[Hilb^n(\mathbb{C}^3)_0]_{relvir}.
\end{equation*}
\end{comment}
Then on the deepest strata, we have
\begin{equation*}
\label{equn}
MF_{Hilb^n(\omega_S),s}|_{Hilb^n_{(n)}(V_\alpha)}=[Hilb^n_{(n)}(V_\alpha)]_{relvir}=[V_\alpha\times Hilb^n(\mathbb{C}^3)_0]\cdot_{Hilb^n(\mathbb{C}^3)_0}[Hilb^n(\mathbb{C}^3)_0]_{relvir}.
\end{equation*}
We use the subscript to indicate the scheme over which the product of relative motivic classes takes place. Since the motivic invariant $MF_{Hilb^n(\omega_S),s}$ is uniquely determined by its restriction to the $Hilb^n(V_\alpha)$, we have
\begin{equation*}
MF_{Hilb^n(\omega_S),s}|_{Hilb^n_{(n)}(\omega_S)}=[\omega_S\times Hilb^n(\mathbb{C}^3)_0]\cdot_{Hilb^n(\mathbb{C}^3)_0}[Hilb^n(\mathbb{C}^3)_0]_{relvir}.
\end{equation*}
Now consider a general stratum. Let $Y_\gamma\subset\prod_iHilb^i_{(i)}(\mathbb{C}^3)^{\gamma_i}$ be the open subset on which the clusters have distinct supports.
We have
\begin{equation*}
\label{equa}
\begin{split}
MF_{Hilb^n(\omega_S),s}|_{Hilb^n_\gamma(V_\alpha)}=&[Hilb^n_\gamma(V_\alpha)]_{relvir}\\
=&\pi_{G_\gamma}(\prod_i[Hilb^i_{(i)}(\mathbb{C}^3)^{\gamma_i}]_{relvir}|_{Y_\gamma}).
\end{split}
\end{equation*}
We also have a fiber product expression
\begin{equation*}
\prod_i[Hilb^i_{(i)}(\mathbb{C}^3)^{\gamma_i}]_{relvir}=\prod_i[\left(\mathbb{C}^3\times Hilb^i(\mathbb{C}^3)_0\right)^{\gamma_i}]\cdot_{Hilb^n_{\gamma,0}(\mathbb{C}^3)}[(Hilb^n_{\gamma,0}(\mathbb{C}^3)]_{relvir}.
\end{equation*}
where we have put $Hilb^n_{\gamma,0}(\mathbb{C}^3)=\prod_i \left(Hilb^i(\mathbb{C}^3)_0\right)^{\gamma_i}$.

Then we get
\begin{equation}\label{eq:mfstrata}
MF_{Hilb^n(\omega_S),s}|_{Hilb^n_\gamma(\omega_S)}=\pi_{G_\gamma}\left(\left[\left(\prod_i \omega_S^{\gamma_i}-\Delta\right)\times Hilb^n_{\gamma,0}(\mathbb{C}^3)\right]\cdot[(Hilb^n_{\gamma,0}(\mathbb{C}^3)]_{relvir}\right),
\end{equation}
where $\Delta\subset\prod_i \omega_S^{\gamma_i}$ is the big diagonal.
\begin{comment}
If we let
\begin{equation*}
MF_{Hilb^n(\omega_S),s}^{(n)}=[\omega_S]\cdot[Hilb^n(\mathbb{C}^3)_0]_{relvir},
\end{equation*}
\begin{equation*}
MF_{Hilb^n(\omega_S),s}^\gamma=\pi_{G_\alpha}([\prod_i(\omega_S)^{\alpha_i}\setminus\Delta]\cdot\prod_i[Hilb^i(\mathbb{C}^3)_0^{\alpha_i}]_{relvir}).
\end{equation*}
and
$$
MF_{Hilb^n(\omega_S),s}=MF_{Hilb^n(\omega_S),s}^{(n)}+\sum_\alpha MF_{Hilb^n(\omega_S),s}^\alpha,
$$
\end{comment}

Taking the absolute motivic class of (\ref{decom}) and using (\ref{eq:mfstrata}), we see that the absolute motive of $MF_{Hilb^n(\omega_S),s}$ matches the motivic class of \cite{BBS13} from Definition \ref{def4}, completing the proof.
\end{proof}

These absolute motives can be combined into a generating function
\begin{equation*}
Z_{\omega_S}(t)=\sum_{n=0}^\infty MF_{Hilb^n(\omega_S),s}t^n.
\end{equation*}
Let $\mathrm{Exp}$ denote the plethystic exponential.

\begin{corollary}
$$Z_{\omega_S}(t)=
\left\{
\begin{array}{cc}
\mathrm{Exp}\left(
\frac{\mathbb{L}^{-1/2}(\mathbb{L}^2+\mathbb{L}+1)t}{\left(1-\mathbb{L}^{1/2}t\right)\left(1-\mathbb{L}^{-1/2}t\right)}
\right)& S=\mathbb{P}^2\\
\mathrm{Exp}\left(
\frac{\mathbb{L}^{-1/2}(\mathbb{L}^2+2\mathbb{L}+1)t}{\left(1-\mathbb{L}^{1/2}t\right)\left(1-\mathbb{L}^{-1/2}t\right)}
\right)& S=\mathbb{F}_n.
\end{array}
\right.
$$

\begin{proof}
Follows immediately from \cite[Theorem 4.3]{BBS13}, Theorem~\ref{thm3.2},  $[\omega_S]=\mathbb{L}[S]$ for any $S$, $[\mathbb{P}^2]=\mathbb{L}^2+\mathbb{L}+1$, and $[\mathbb{F}_n]=\mathbb{L}^2+2\mathbb{L}+1$.
\end{proof}
\end{corollary}

\bigskip\noindent
{\bf Remark.} We expect that our methods are adaptable to other local toric Calabi-Yaus.  We leave this investigation for future work.

redpajama_set_name: RedPajamaArXiv


With 2017 holiday sales expected to increase by up to 4% YOY, retailers looking to see similar sales bumps should consider the effects that the ever-changing NRF retail calendar will have on seasonal foot traffic. Here are ShopperTrak's 3 key 2017 retail calendar shifts to bear in mind this holiday season.
2017 marks the first time since 2012 that four December Saturdays precede Christmas Day. Naturally, Saturdays in December are always some of the busiest days of the entire year, and they become increasingly busy as the month progresses. In fact, ShopperTrak's historical data indicates that shopper visits increase each Saturday, from week to week, as Christmas Day approaches. Each of these pre-Christmas Saturdays in December is on ShopperTrak's Top 10 Busiest Days of 2017 list.
The fourth and final Saturday before the holiday, commonly known as Super Saturday, Dec. 23, is predicted to be the second busiest shopping day after Black Friday. Because Super Saturday falls within 48 hours of Christmas this year, shoppers are likely to hit stores at the last minute.
Considering the seven days before and after Christmas by their proximity to the holiday often yields informative insights that can greatly affect staffing decisions.This year, for example, pay attention to the Friday, the 15th; Monday, the 18th; and Friday, the 22nd. These days each serve as an extension of December weekends that are in close proximity to Christmas Day, meaning they'll likely see higher traffic numbers than typical December Fridays and Mondays. Friday, the 22nd, in fact, is projected to be the 6th busiest day of the year.
However, prudent retailers would be wise to also pay close attention to the calendar alignment of the week after Christmas, as the nuances of the 2017 retail calendar shifts do not end when Christmas is over. Saturday, December 30th, is another full shopping day that precedes New Year's Eve. Shoppers will likely enjoy the extra days before the new year to exchange items, use gift cards or take advantage of the post-holiday sales.
Generally, ShopperTrak finds that the recent trend of retailers opening on Thanksgiving Day has little effect on overall holiday performance. Indeed, Thanksgiving Day openings merely redistribute traffic numbers from the Wednesday before Thanksgiving, as well as the Saturday and Sunday that follow. Historically, traffic activity erosion on those days is negatively correlated with the increase in Thanksgiving Day activity (-94%).Overall, Black Friday has remained remarkably consistent, mirroring overall YOY traffic trend. This pattern serves to substantiate the fact that the key shopping days for Thanksgiving week are always Black Friday, Saturday, and Sunday.
Looking for more holiday advice? Check out ShopperTrak's tips on forecasting holiday traffic, what 2017's busiest days are, or how to prep cross-functionally for the holiday.


Howard Bamsey is an Honorary Professor at RegNet and is Adjunct Professor at the ANU Climate Change Institute and at the University of Sydney. As of January 2019, he is the appointed Chair of the Global Water Partnership (GWP). He was Executive Director at the Green Climate Fund from January 2017 until July 2018. Previously he was Director General of the Global Green Growth Institute for a year until early 2014.
He has served as Special Adviser on Sustainable Development to the Australian Department of Foreign Affairs and Trade and Special Adviser on Green Growth to AusAID. Earlier he was Australia's Special Envoy on Climate Change, a Deputy Secretary in the Australian Public Service and CEO of the Australian Greenhouse Office. He held several appointments at ambassadorial level during over twenty five years in the Australian foreign service. He is one of Australia's most experienced negotiators, playing a key role in international climate change negotiations.
With a federal election looming, Australia's top mandarins will once again be turning their minds to the incoming government briefs, the so-called blue book if the Coalition is returned and the red book if Labor is elected.
What role will India play in the climate negotiations?
Adjunct Professor, Howard Bamsey, has been awarded an AM in the 2013 Order of Australia Honours announced on 26 January.


How to eat your kale and move on!
Have you heard all the alarmist news stories about how kale is bad for your health? Well, kale is goitrogenic and thus should not be consumed in ridiculously vast quantities, especially if you already have thyroid dysfunction. For most people, we are talking having a big green juice in the morning made of mostly kale, followed by a massaged kale salad for lunch and maybe some kale chips for snacking on. However, there are two ways to eat your kale and stay abundantly nourished and vibrantly healthy.
Most people are iodine deficient. Many types of radiation are taken in by the thyroid in search of iodine, so make sure your thyroid is fully topped up with iodine. Our family enjoys iodine rich foods in the form of seaweed, especially kelp and also supplementing with Nascent Iodine product. Kelp is extremely rich in iodine, in fact many times higher than most other types of readily available seaweeds. It is also relatively inexpensive. Kelp is something that should be eaten in small quantities regularly. If you want to mix it up a little, our Sacred Strawberry brand Tahini Dill Kale Chips contain kelp. It's a perfect marriage!
By regularly switching up the greens you eat, you avoid any build up of plant alkaloids or anti-nutrients that one particular green may have. Kale, collards and mustard greens contain goitrogens; spinach and beet greens have oxalic acid; lettuce has traces of opioid like sedative compounds, etc. Eating too many of one thing only is not eating optimally- we need variety! By eating only one type of green that you find tastiest, you also miss out on the benefits that other greens have- different vitamins, minerals and phytonutrients. Eating a variety of greens over time is what we want to strive for.
photo by KeepCalmShop on Etsy 🙂 I can totally see this framed and in the kitchen!


What do pony club parents do?
VOLUNTEER! The parents of Pony Club members play a big part in making PC events and activities happen. The USPC functions because of its dedicated volunteers. Not only do parent volunteers make this a unique experience for the kids but it is a great opportunity to gain knowledge and understanding of their child's passion.


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