text:
\section{Introduction}\label{1}

Let $G$ be a finite subgroup of $SL_n(\mathbb{C})$. The philosophy of the McKay correspondence is that the geometric properties of a crepant resolution of $\mathbb{C}^n/G$ should reflect the representation-theoretic properties of $G$ (cf. \cite{Re2}). The natural way to see the explicit correspondence is to realize a crepant resolution of $\mathbb{C}^n/G$ as the $G$-Hilbert scheme or, more generally, a moduli space of $G$-constellations (see e.g. \cite{Cr}, \cite{CCL}). In dimension two, it is well known that the unique crepant resolution is obtained as the $G$-Hilbert scheme. In dimension three, it was also proved by Craw and Ishii \cite{CI} that every projective crepant resolution is obtained as the moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for a suitable stability condition $\theta$ when $G$ is abelian. In general $\mathcal{M}_{\theta}$ for any finite subgroup $G\subset SL_3(\mathbb{C})$ is known to be a crepant resolution for any generic stability condition $\theta$ by the result of Bridgeland, King and Reid \cite{BKR}, and it is conjectured that the result by Craw and Ishii still holds for non-abelian $G$ \cite[Conjecture 1.4]{IINdC}.  The main result of this article is that this conjecture is affirmative:

\begin{thm}\label{main1}
Let $G\subset SL_3(\mathbb{C})$ be a finite subgroup. Then every projective crepant resolution of $\mathbb{C}^3/G$ is isomorphic to the moduli space of $\theta$-stable $G$-constellations for a generic stability condition $\theta$.
\end{thm}

In this paper we also give an explicit construction of an equivariant morphism from the spectrum of the Cox ring of a crepant resolution of $\mathbb{C}^3/G$ to (a certain quotient of) a space of $G$-constellations, as a generalization of the construction in Part I \cite{Y2} of the present paper. In \cite{Y2}, we gave a necessary and sufficient condition for a given (not-necessarily-projective) crepant resolution $X\to \mathbb{C}^n/G$ to admit a moduli description in terms of a natural $G$-constellation family when $G$ is abelian. The key idea to prove this result was to construct an equivariant morphism between certain toric varieties so that it descends to a birational morphism from $X$ to a candidate of a fine moduli space. In this paper we generalize this construction to non-abelian $G$ in dimension three so that we obtain an embedding $\mathcal{W}\hookrightarrow Y_\mathcal{W}$ of (a quotient of) a space of $G$-constellations into a toric variety such that it induces an embedding of $\mathcal{M}_\theta$ into a toric variety for any generic $\theta$ (Proposition \ref{embedding}). This embedding of $\mathcal{M}_\theta$ is so nice that it enables us to understand the behavior of $\mathcal{M}_\theta$ under variation of $\theta$ through the change of the ambient toric variety.  We will demonstrate this embedding and investigate the geometric structure of the moduli space explicitly via concrete examples in Section \ref{3.5}.

The embedding $\mathcal{W}\hookrightarrow Y_\mathcal{W}$ will be constructed by using the Cox ring $\mathrm{Cox}(X)$ of $X=\mathcal{M}_\theta$, whose generators play a similar role to homogeneous coordinates of a projective space. On the other hand $\mathcal{M}_\theta$ is regarded as a moduli space of certain representations of the McKay quiver of $G$, and semi- (or relative) invariants for these representations also play the role of homogeneous coordinates of $\mathcal{M}_\theta$. Then the equivariant morphism
$$\mathfrak{X}:=\mathrm{Spec}\,\mathrm{Cox}(X)\to \mathcal{W}$$
is obtained by assigning to each semi-invariant a global section of a line bundle on $X$. Moreover, as an application of Theorem \ref{main1}, we will show that generators of $\mathrm{Cox}(X)$ can be computed from the semi-invariants. More precisely, there is a ring homomorphism $\varphi$ from the ring $\mathbb{C}[\mathcal{W}]$ of semi-invariants to the (semi-)invariant ring $\mathbb{C}[\mathbb{C}^3]^{[G,G]}$ such that the ``associated elements" to the images of homogeneous generators of $\mathbb{C}[\mathcal{W}]$ under $\varphi$ give a generating system of $\mathrm{Cox}(X)$ (Proposition \ref{prop:generate}).

The strategy for the proof of Theorem \ref{main1} is based on the one given by Craw and Ishii for abelian cases. To explain this, let us recall that each GIT-chamber $C\subset \Theta$ gives a fine moduli space $X:=\mathfrak{M}_C$ and its universal family $\mathcal{U}_C$. By the result of \cite{BKR}, the Fourier-Mukai transform by $\mathcal{U}_C$ gives an equivalence between the bounded derived category of coherent sheaves on $X$ and that of $G$-equivariant sheaves on $\mathbb{C}^3$. This induces an isomorphism
$$\phi_C^*:\Theta\to F^1$$
from the space of stability conditions to the subspace $F^1$ inside the Grothendieck group $K(X)_\mathbb{R}$ generated by sheaves whose supports have codimension at least one. The proof of the theorem will be done by studying the behavior of $\phi_C^*(C)$ under crossing a wall of {\it Type 0} inside $\Theta$. Here a wall of Type 0 means a codimension-one face $W$ of $\overline{C}$ whose associated morphism $X\to \bar{\mathfrak{M}}_W$ induced by variation of GIT is an isomorphism. The significant fact to prove Theorem \ref{main1} is that, similarly to abelian cases, crossing such a wall gives a wall-crossing in $F^1$ as well, up to tensoring a line bundle (Proposition \ref{twist}). This enables us to find a wall $\tilde{B}$ which realizes the small contraction $X\to X_B$ for any given flop $X\to X_B\leftarrow X'$ after crossing walls of Type 0. Using the technical lemma (Lemma \ref{vertical}), we can confirm that crossing the wall $\tilde{B}$ gives the flop $X\dashrightarrow X'$. Since any two projective crepant resolutions are connected by a sequence of flops, this proves the theorem.

As an application of the construction of the equivariant map $\mathfrak{X}\to \mathcal{W}$, we will show that crepant resolutions of a certain four-dimensional quotient singularity $\mathbb{C}^4/T$ can be realized as a moduli space of $T$-constellations. More precisely, the group $T\subset Sp_4(\mathbb{C})$ is called the binary tetrahedral group, and $\mathbb{C}^4/T$ is one of symplectic quotient singularities having symplectic (or equivalently, crepant) resolutions. The existence of the resolutions was shown by the results of Bellamy and Namikawa \cite{B}, \cite{N}, and later the resolutions were explicitly constructed by Lehn and Sorger \cite{LS}. Although the abstract existence of an moduli interpretation of the resolutions is already known by the result of Bezrukavnikov and Kaledin \cite{BK}, we give an explicit description as moduli spaces with explicit stability conditions and also show that a component of the $T$-Hilbert scheme is a common blow-up of the two resolutions (Theorem \ref{main3}). The fact that the two resolutions are dominated  by a component of the $T$-Hilbert scheme was already observed in \cite{LS}. In the proof of Theorem \ref{main3}, we will describe how the parametrized $T$-constellations change under the birational contraction of the $T$-Hilbert scheme.

This paper is organized as follows. In Section \ref{2} we review the construction of a moduli space of $G$-constellations and semi-invariants of quiver representations. In Section \ref{3}, we introduce the Cox ring of a crepant resolution and relate it with the semi-invariants for representations of the McKay quiver of $G$ so that we obtain a nice embedding of a moduli space into  a toric variety (Proposition \ref{embedding}). We also give concrete examples in Subsection \ref{3.5}. In Section \ref{4} we give a proof of the main result (Theorem \ref{main2}). In Section \ref{5} we treat the symplectic quotient singularity $\mathbb{C}^4/T$ for the binary tetrahedral group $T$. In particular we show that their symplectic resolutions are obtained as moduli spaces of $T$-constellations and that a certain component of the $T$-Hilbert scheme is isomorphic to a common blowing-up of these resolutions.

\vspace{5mm}
\noindent{\underline{Conventions \& Notations}\\
In this paper all schemes are algebraic ones defined over $\mathbb{C}$. By a {\it point} of a scheme, we always mean a closed point. For an abelian group $A$, we denote by $A_\mathbb{R}$ its scalar extension $A\otimes_\mathbb{Z} \mathbb{R}$. Similarly, for an homomorphism $f$ of abelian groups, $f_\mathbb{R}$ denotes the scalar extension. For a finite dimensional $\mathbb{R}$-vector space $V$ and a subset $S\subset V$, we denote by $\mathrm{Cone}(S)$ the cone generated by $S$.

\vspace{5mm}
\noindent{\bf Acknowledgements}\\
The author is grateful to Alastair Craw and Akira Ishii for their useful comments to the draft of the present paper. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan, and by JSPS KAKENHI Grant Number JP19K14504.

\section{Moduli spaces of $G$-constellations}\label{2}
\subsection{GIT-construction}\label{2.1}
In this section we review the construction of a moduli space of $G$-constellations for an arbitrary finite subgroup $G\subset SL(V)$ with $V=\mathbb{C}^n$ (cf. \cite[Section 2]{CI}). See e.g. \cite{CMT} and \cite{Y2} for the special case where $G$ is abelian.

We first recall that a {\it $G$-constellation} $F$ is a $G$-equivariant $\mathbb{C}[V]$-module which is isomorphic to the regular representation $R=\bigoplus_{g\in G}\mathbb{C} g$ as a $\mathbb{C} G$-module where the $G$-action on $\mathbb{C}[V]$ comes from the inclusion $G\subset SL(V)$. Although $G$-constellations are defined as $G$-equivariant coherent sheaves on $V$ in \cite{CI}, we identify them with their global sections since $V$ is affine.

Let $\mathrm{Irr}(G)$ be the set of isomorphism classes of irreducible representations of $G$. As is well known, $R$ is isomorphic to $\bigoplus_{\rho\in \mathrm{Irr}(G)}\rho^{\oplus \dim \rho}$ as a $\mathbb{C} G$-module. We will construct a moduli space of $G$-constellations by taking a quotient of the affine scheme
\begin{equation}\label{N}
\mathcal{N}=\{B\in \mathrm{Hom}_{\mathbb{C} G}(V^*\otimes_\mathbb{C} R,R) \mid B\wedge B=0\}
\end{equation}
where $V^*$ is the dual representation of $V$ and $B\wedge B\in \mathrm{Hom}_{\mathbb{C}[G]}(V^*\otimes_\mathbb{C} V^*\otimes_\mathbb{C} R,R)$ is defined by
$$(B\wedge B)(x\otimes y\otimes a)=B(x\otimes B(y\otimes a))-B(y\otimes B(x\otimes a)).$$

We may regard $\mathcal{N}$ as a space parametrizing all $G$-constellations, and two elements $\mathcal{N}$ are isomorphic as $G$-equivariant $\mathbb{C}[V]$-modules if and only if they lie in the same $GL_R$-orbit where $GL_R:=\mathrm{Aut}_{\mathbb{C} G}(R)$ is the $G$-equivariant automorphism group of $R$, which acts on $\mathcal{N}$ by conjugation. By Shur's lemma, $GL_R$ is written as a direct product $\prod_{\rho\in\mathrm{Irr}(G)}GL_{\dim \rho}(\mathbb{C})$. Note that the diagonal subgroup $\mathbb{C}^*$ acts trivially on $\mathcal{N}$ and thus the action of $GL_R$ descends to that of $PGL_R:=GL_R/\mathbb{C}^*$.

Since the orbit space $\mathcal{N}/PGL_R$ does not admit a reasonable structure as a scheme, we apply the construction of geometric invariant theory (GIT). Concretely, we consider a $PGL_R$-linearization of the trivial line bundle of $\mathcal{N}$ and take the GIT-quotient with respect to it. Such a linearization is given by a character of $PGL_R$, and the group $\chi(PGL_R)$ of characters can be identified with the space of {\it stability conditions}
$$\bar{\Theta}:=\{\theta\in\mathrm{Hom}_\mathbb{Z}(R(G),\mathbb{Z})\mid \theta(R)=0\}$$
where $R(G)=\bigoplus_{\rho\in\mathrm{Irr}(G)} \mathbb{Z} \rho$ is the representation ring of $G$. Note that $\bar{\Theta}$ has rank $s-1$ where $s$ is the number of conjugacy classes in $G$. The identification is given by
$$\begin{aligned}
&\chi(PGL_R) & &\to & &\bar{\Theta}\\
\chi=(\bar{g}_{\rho}&\mapsto \mathrm{det}(g_{\rho})^{\theta_\rho},\;\forall \rho\in\mathrm{Irr}(G)) & &\mapsto & \theta_\chi=(\rho&\mapsto \theta_\rho,\;\forall \rho\in\mathrm{Irr}(G)).
\end{aligned}$$

For a character $\chi\in \chi(PGL_R)$ we consider the $\chi${\it-semistable locus}
$$\mathcal{N}^{\chi\text{-ss}}=\{p\in \mathcal{N} \mid f(p)\ne0\text{ for some }f\in A_{\chi^k}\text{ and }k\in\mathbb{N}\}$$
where $A_\chi$ is the $\mathbb{C}$-vector space consisting of regular functions $f$ on $\mathcal{N}$ such that $g\cdot f=\chi(g)f$ for any $g\in PGL_R$. As a smaller open subset of $\mathcal{N}^{\chi\text{-ss}}$, the $\chi${\it-stable locus} $\mathcal{N}^{\chi\text{-s}}$ is defined as the set of points $p\in\mathcal{N}^{\chi\text{-ss}}$ such that the orbit $PGL_R\cdot p$ is closed in $\mathcal{N}^{\chi\text{-ss}}$ and that the stabilizer subgroup of $p$ in $PGL_R$ is finite. We say that a character $\chi$ is {\it generic} if $\mathcal{N}^{\chi\text{-ss}}=\mathcal{N}^{\chi\text{-s}}$.

For any character $\chi$, we have a morphism
$$\pi_\chi:\mathcal{N}^{\chi\text{-ss}}\to \mathcal{N}/\!/_\chi PGL_R:=\mathrm{Proj}\left(\bigoplus_{k=0}^\infty A_{\chi^k}\right)$$
which gives a one-to-one correspondence between the set of closed $PGL_R$-orbits in $\mathcal{N}^{\chi\text{-ss}}$ and the set of closed points in $\mathcal{N}/\!/_\chi PGL_R$ (cf. \cite{MFK}). We call $\mathcal{N}/\!/_\chi PGL_R$ the {\it GIT-quotient} of $\mathcal{N}$ by $PGL_R$ with respect to $\chi$. If $\chi$ is generic, then $\mathcal{N}/\!/_\chi PGL_R$ is the orbit space of $\mathcal{N}^{\chi\text{-ss}}$, and in such a case $\mathcal{N}/\!/_\chi PGL_R$ is called the {\it geometric quotient}.

The stability of a point of $\mathcal{N}$ can be rephrased in representation-theoretic terms by the result of King:

\begin{prop}\label{King} \cite[Proposition 3.1]{Kin}
For a point $p\in\mathcal{N}$ and a character $\chi\in\chi(PGL_R)$, $p$ is in $\mathcal{N}^{\chi\text{-ss}}$ (resp. $\mathcal{N}^{\chi\text{-s}}$) if and only if the corresponding $G$-constellation $F_p$ is $\theta_\chi$-semistable (resp. $\theta_\chi$-stable), that is,  $\theta_\chi(M)\ge0$ (resp. $>0$) for any nonzero $G$-equivariant $\mathbb{C}[V]$-submodule $M\subsetneq F_p$.
\end{prop}

We will use the scaler extension $\Theta:=\bar{\Theta}\otimes_\mathbb{Z} \mathbb{R}$
as the space of stability conditions rather than $\bar{\Theta}$ itself in order to consider polyhedral cones inside $\Theta$ (or $\chi(PGL_R)_\mathbb{R}$). For $\theta\in\Theta$ and the corresponding character $\chi_\theta$, we denote $\mathcal{N}/\!/_{\chi_\theta} PGL_R$ also by $\mathcal{M}_\theta$. From King's result above, we see that $\Theta$ is divided into finitely many open chambers, on which the semistable locus is constant. We call such a chamber a {\it GIT-chamber} of $\Theta$. Since $G$-constellations are regular representations as $\mathbb{C} G$-modules, supporting hyperplanes of the GIT-chambers are of the form $\sum_{\rho\in \mathrm{Irr}(G)} a_\rho \theta_\rho=0$ with $a_\rho\in\{0,1,\dots,\dim \rho\}$.

Typical examples of $G$-constellations are given by the coordinate rings of  free $G$-orbits, that is, orbits $G\cdot x$ such that $\sharp(G\cdot x)=\sharp G$) in $\mathbb{C}^n$. Such $G$-constellations are simple $\mathbb{C}[V]$-modules and hence stable for any $\theta\in\Theta$. As a natural generalization of free $G$-orbits, $G$-constellations obtained as quotients of $\mathbb{C}[V]$ by its ideal is called $G$-{\it clusters}. Since $G$-clusters are characterized as the $G$-constellations generated by the trivial representation, the moduli space of $G$-clusters, called the $G$-{\it Hilbert scheme}, is realized as $\mathcal{M}_{\theta+}$ for $\theta_+\in\Theta$ satisfying $\theta_+(\rho)>0$ for any nontrivial $\rho\in \mathrm{Irr}(G)$.

By \cite[Proposition 2.2]{CI}, there exists an irreducible component of $\mathcal{M}_0$ which is isomorphic to $\mathbb{C}^n/G$ such that  its general points parametrize the coordinate rings of free $G$-orbits in $\mathbb{C}^n$. Accordingly, there is an irreducible component $\mathcal{V}\subset\mathcal{N}$ such that the restriction of the quotient map $\mathcal{N}\to \mathcal{M}_0$ to $\mathcal{V}$ factors into
\begin{itemize}
\item the morphism $\mathcal{V}\to\mathbb{C}^n/G$ which sends a $G$-constellation to its support, and
\item a closed immersion $\mathbb{C}^n/G\to \mathcal{M}_0$ onto an irreducible component of $\mathcal{M}_0$.\end{itemize}
In Subsection 3.3 we will give a more explicit description of $\mathcal{V}$ (cf. Proposition \ref{component V}). Note that the morphism $\mathfrak{M}_\theta:= \mathcal{V}/\!/_{\chi_\theta} PGL_R\to \mathbb{C}^n/G$ induced by the inclusion $\mathcal{V}\cap\mathcal{N}^{\chi_\theta\text{-ss}}\hookrightarrow \mathcal{V}$ is a birational morphism over $\mathbb{C}^n/G$ for any $\theta\in\Theta$. The irreducible component $\mathfrak{M}_\theta$ of $\mathcal{M}_\theta$ is sometimes called the {\it coherent component}, and we will adopt this terminology in this article. For a GIT-chamber $C\subset \Theta$, the moduli space $\mathfrak{M}_\theta$ is constant for $\theta\in C$ and thus we also denote it by $\mathfrak{M}_C$.

The main result of \cite{BKR} shows that the $G$-Hilbert scheme for $G\subset SL_3(\mathbb{C})$ is a crepant resolution of $\mathbb{C}^3/G$. More generally, the same proof of this result shows that the moduli space $\mathcal{M}_\theta$ of $\theta$-semistable $G$-constellations for any generic $\theta\in \Theta$ is a crepant resolution of $\mathbb{C}^3/G$. In particular $\mathcal{M}_\theta$ is irreducible and hence $\mathcal{M}_\theta=\mathfrak{M}_\theta$ in this case.

As explained in \cite[\S2.1]{CI}, the moduli space $\mathfrak{M}_C$ for a GIT-chamber $C$ admits a universal family $\mathcal{U}_C$ on $\mathfrak{M}_C \times \mathbb{C}^n$, and the associated tautological bundle $(p_{\mathfrak{M}_C})_*\mathcal{U}_C$ on $\mathfrak{M}_C$ admits a decomposition
$$(p_{\mathfrak{M}_C})_*\mathcal{U}_C=\bigoplus_{\rho \in \mathrm{Irr}(G)} \mathcal{R}_\rho\otimes \rho$$
into isotypic components where $p_{\mathfrak{M}_C}:\mathfrak{M}_C \times \mathbb{C}^n\to\mathfrak{M}_C$ is the projection. Note that each $\mathcal{R}_\rho$ is a vector bundle on $\mathfrak{M}_C$ of rank $\dim \rho$. Since tensoring a line bundle to $(p_{\mathfrak{M}_C})_*\mathcal{U}_C$ has no effect on the parametrized $G$-constellations, we assume that $\mathcal{R}_{\rho_0}\cong \mathcal{O}_{\mathfrak{M}_C}$ for the trivial representation $\rho_0$ whenever we talk about the universal family or the tautological bundle determined by $C$. Later we will identify $G$-constellations with representations of a quiver in Subsection \ref{2.3}. We will also use the associated tautological bundle $\mathcal{R}_C$ on $\mathfrak{M}_C$ as a moduli space of quiver representations rather than using the bundle $(p_{\mathfrak{M}_C})_*\mathcal{U}_C$.

For later use, we define the notion of a {\it orbit cone} for an affine scheme acted on by an algebraic group, following \cite[Definition 2.1]{BH}:

\begin{dfn}\label{orbit cone}
Let $H$ be an algebraic group acting on an affine scheme $Z$. For a point $z\in Z$, the {\bf orbit cone} of $z$ is the polyhedral cone $C_z\subset\chi(H)_\mathbb{R}$ generated by characters $\chi$ such that $z$ is $\chi$-semistable.
\end{dfn}

\begin{rem}
The orbit cone of $z\in Z$ depends only on the $H$-orbit of $z$. If $Z$ is the affine scheme $\mathcal{N}$ acted on by $H=PGL_R$ as above, the orbit cone of a point of $\mathcal{N}$ corresponding to a $G$-constellation $F$ is identified with a cone in $\Theta$ consisting of stability conditions $\theta$ such that $F$ is $\theta$-semistable. Moreover, for a GIT-chamber $C\subset\Theta$, its closure $\overline{C}$ is equal to the intersection of all the orbit cones for points of $\mathcal{N}^{C\text{-ss}}$.
\end{rem}

\subsection{Semi-invariants for quiver representations}\label{2.2}

Let us consider the subgroup $SL_R:=\prod_{\rho\in\mathrm{Irr}(G)}SL_{\dim \rho}(\mathbb{C})$ of $GL_R$. Since $SL_R$ is reductive, there are finitely many generators of the invariant ring of the coordinate ring $\mathbb{C}[\mathcal{N}]$ of the affine scheme $\mathcal{N}$. Note that we have a decomposition of this ring into homogeneous components:
$$\mathbb{C}[\mathcal{N}]^{SL_R}=\bigoplus_{\chi\in \chi(PGL_R)} A_\chi.$$

As we will see in the next subsection, $\mathcal{N}$ is identified with a space of quiver representations (with relations). In general the ring of semi-invariants for representations of a quiver (in characteristic zero) are generated by {\it determinantal semi-invariants} (see Theorem \ref{determinant} below). In this subsection we explain how to obtain such semi-invariants.

Firstly, we briefly recall the concept of representations of a quiver. See e.g. \cite{DeW2},\cite{Kir} for more details. Let $Q$ be a quiver, that is, an oriented graph consisting of a finite set $I$ of vertices and a finite set $A$ of arrows between two vertices. Here we allow $Q$ to have loops and oriented cycles. For any map $\alpha:I\to \mathbb{Z}_{\ge0}$, we define a space $\mathrm{Rep}(\alpha)$ of representations of $Q$ with a dimension vector $\alpha$ as
$$\mathrm{Rep}(\alpha)=\bigoplus_{a\in A} \mathrm{Hom}_\mathbb{C}\left(\mathbb{C}^{\alpha(t_a)},\mathbb{C}^{\alpha(h_a)}\right)$$
where $t_a$ and $h_a$ denote the tail and the head of $a$ respectively. For a representation $H=(H_a)_{a\in A}\in \mathrm{Rep}(\alpha)$ and a path
$$p=a_1\cdots a_k\quad(a_i\in A,\,h_{a_i}=t_{a_{i+1}}),$$
we set $H_p=H_{a_k}\circ\cdots \circ H_{a_1}$. For the trivial path $e_v$ at a vertex $v$, we set $H_{e_v}=\mathrm{Id}_{\mathbb{C}^{\alpha(v)}}$. $\mathrm{Rep}(\alpha)$ is acted on by $GL_\alpha:=\prod_{v\in I} GL_{\alpha(v)}$ by conjugation.

Let $v_1,\dots,v_r,w_1,\dots,w_s$ be vertices of $Q$ with possible repetition such that
$$\sum_{i=1}^r \alpha(v_i)=\sum_{j=1}^s \alpha(w_j),$$
and let $p_{i,j}$ be a $\mathbb{C}$-linear combination of paths from $v_i$ to $w_j$. Then the function $f:\mathrm{Rep}(\alpha)\to \mathbb{C}$ defined by
\begin{equation}\label{eq:determinant}
H\mapsto \det
\begin{pmatrix}H_{p_{1,1}}&\cdots&H_{p_{1,s}}\\
\vdots& &\vdots\\
H_{p_{r,1}}&\cdots&H_{p_{r,s}}
\end{pmatrix}
\end{equation}
gives a semi-invariant satisfying $(g_v)_{v\in I}\cdot f=(\prod_{v\in I} \det g_v^{c_v})f$ for $(g_v)_{v\in I}\in GL_\alpha$ where
$$c_v=\alpha(v)(\sharp\{j \mid w_j=v\}-\sharp\{i \mid v_i=v\}).$$
A semi-invariant obtained in this way is called a {\it determinantal semi-invariant} of weight $(c_v)_{v\in I}$. Note that weights are identified with characters of $GL_\alpha$.

\begin{thm}(\cite{DeW1},\cite{DZ},\cite{SV})\label{determinant}
The ring $\mathbb{C}[\mathrm{Rep}(\alpha)]^{SL_\alpha}$ of semi-invariants on $\mathrm{Rep}(\alpha)$ with respect to the $GL_\alpha$-action is generated by determinantal semi-invariants.
\end{thm}

This was first proved by Derksen and Weyman \cite{DeW1} under the assumption that $Q$ has no oriented cycles, and generalized later by Domokos and Zubkov \cite{DZ} and by Shofield and van den Bergh \cite{SV}.

\subsection{$G$-constellations as representations of the McKay quiver}\label{2.3}

In this subsection we realize the affine scheme $\mathcal{N}$ in (\ref{N}) as a space of quiver representations.

The {\it McKay quiver} $Q_G$ of $G$ is the quiver having $\mathrm{Irr}(G)$ as the set of vertices and having $a_{\rho,\rho'}:=\dim_\mathbb{C}(\mathrm{Hom}_{\mathbb{C} G}(V^*\otimes_\mathbb{C} \rho,\rho'))$ arrows from $\rho$ to $\rho'$ for any $\rho,\rho'\in\mathrm{Irr}(G)$. Decomposing $R$ into irreducible summands, we see that $\mathrm{Hom}_{\mathbb{C} G}(V^*\otimes_\mathbb{C} R,R)$ is isomorphic to the representation space $\mathrm{Rep}(\alpha_G)$ of $Q_G$ for the dimension vector $\alpha_G:\rho\mapsto \dim \rho$ (cf. \cite{IN}). Thus, $\mathcal{N}$ can be realized as a $GL_R(=GL_{\alpha_G})$-invariant closed subscheme of $\mathrm{Rep}(\alpha_G)$.

Via the identification of a $G$-constellation with a representation of $Q_G$, the moduli space $\mathfrak{M}_C$ of $G$-constellations constructed in Subsection \ref{2.1} is also regarded as a moduli space of representations of $Q_G$. Then the tautological bundle $\mathcal{R}_C$ on $\mathfrak{M}_C$ as the latter moduli space admits a decomposition $\mathcal{R}_C=\bigoplus_{\rho\in\mathrm{Irr}(G)} \mathcal{R}_\rho$ into the isotypic components. Note that $\mathcal{R}_C$ is equipped with a $\mathbb{C}[V]$-action whose restriction to $\mathbb{C}[V]^G$ coincides with the $H^0(\mathcal{O}_{\mathfrak{M}_C})$-action as a vector bundle on $\mathfrak{M}_C$. Hereafter, by {\it the tautological bundle on} $\mathfrak{M}_C$, we mean the bundle $\mathcal{R}_C$.

\begin{ex}\label{relations}($D_5$-singularities)
We describe the space $\mathcal{N}$ explicitly for the case of the $D_5$-singularity. This singularity is given as a quotient of $\mathbb{C}^2$ by a binary dihedral group $G$ of order 12:
$$G=\left\langle
g_1=\begin{pmatrix}
\zeta&0\\
0&\zeta^{-1}
\end{pmatrix},
g_2=\begin{pmatrix}
0&1\\
-1&0
\end{pmatrix}
\right\rangle\subset SL_2(\mathbb{C})$$
where $\zeta$ is the sixth root of unity. $\mathrm{Irr}(G)$ consists of four 1-dimensional representations $\rho_k:G\to\mathbb{C}^*\,(k=0,1,2,3)$ defined by
$$\rho_k(g_j)=\begin{cases}
(-1)^k&\text{ if }j=1\\
i^k&\text{ if }j=2\end{cases},$$
and two 2-dimensional representations $V_1,V_2$; the first one is the inclusion $G\subset SL_2(\mathbb{C})$ presented above and the other is presented as $$g_1\mapsto\begin{pmatrix}
\zeta^2&0\\
0&\zeta^{-2}
\end{pmatrix},\quad
g_2\mapsto\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}.$$

Let $e_0,\dots,e_3$ be a basis of $\rho_0,\dots,\rho_3$ respectively. Also, we let $\{v_{11},v_{12}\}$ (resp. $\{v_{21},v_{22}\}$) be a basis of $V_1$ (resp. $V_2$) such that it gives the matrix presentation above. We moreover prepare their copies $\{v'_{11}, v'_{12}\}$ and $\{v'_{21},v'_{22}\}$ for the second direct summands of $V_1$ and $V_2$ in $R$ respectively. When we emphasize that $v_{11}$ and $v_{12}$ are elements of $V^*$, we instead denote them by $x$ and $y$ respectively.

We fix isomorphisms of $\mathbb{C} G$-modules as follows:
{\small
$$\begin{alignedat}{11}
V^*&\otimes \rho_0 & & \cong V_1, &\qquad V^*&\otimes \rho_1 & &\cong & & V_2,\; &\qquad V^*&\otimes \rho_2  &  &\cong & & V_1,\; &\qquad V^*&\otimes \rho_3 & &\cong & & V_2\\
x&\otimes e_0 & & \mapsto v_{11} & x&\otimes e_1 & &\mapsto & & v_{22} & x&\otimes e_2 &  &\mapsto & & v_{11} & x&\otimes e_3 & &\mapsto & & v_{22}\\
y&\otimes e_0& &\mapsto v_{12} & y&\otimes e_1& &\mapsto & i&v_{21} & y&\otimes e_2& &\mapsto -& & v_{12} & y&\otimes e_3& &\mapsto -i & & v_{21}
\end{alignedat}$$
$$\begin{alignedat}{11}
V^*&\otimes V_1 & & \cong & & \rho_0 & &\oplus \rho_2 \oplus & & V_2, &\qquad V^*&\otimes V_2 & & \cong & & \rho_1 & &\oplus  & & \rho_3 \oplus & & V_1,\\
x&\otimes v_{11} & & \mapsto & & & & & & v_{21} & x&\otimes v_{21} &  &\mapsto  & & e_1 & &+ & & e_3 & & \\
x&\otimes v_{12}& & \mapsto & & e_0 & & +e_2& & & x&\otimes v_{22}& & \mapsto & &  & & & & & & v_{12}\\
y&\otimes v_{11}& & \mapsto - & & e_0 & & +e_2& & & y&\otimes v_{21}& & \mapsto & &  & & & & &-& v_{11}\\
y&\otimes v_{12} & & \mapsto & & & & & & v_{22} & y&\otimes v_{22} &  &\mapsto -i & & e_1 & &+i & & e_3. & &
\end{alignedat}$$}
Thus the McKay quiver $Q_G$ is presented as
\[\xymatrix@=36pt{
& \rho_0 \ar@<-0.5ex>[d]_-{A_0}
& \rho_1 \ar@<-0.5ex>[d]_-{A_1}
& \\
\rho_2 \ar@<-0.5ex>[r]_-{A_2}
& V_1 \ar@<-0.5ex>[l]_-{B_2} \ar@<-0.5ex>[u]_-{B_0}\ar@[r]@<-0.5ex>[r]_-C
& V_2 \ar@<-0.5ex>[l]_-D \ar@<-0.5ex>[r]_-{B_3}  \ar@<-0.5ex>[u]_-{B_1}
& \rho_3 \ar@<-0.5ex>[l]_-{A_3}.
}\]

Using the isomorphisms above, matrices associated to the arrows in $Q_G$ defines an action of $\mathbb{C}[V]$ on the vector space
{\small$$\mathbb{C} e_0\oplus\cdots \oplus\mathbb{C} e_3 \oplus (\mathbb{C} v_{11}\oplus \mathbb{C} v_{12})\oplus (\mathbb{C} v'_{11}\oplus \mathbb{C} v'_{12})\oplus  (\mathbb{C} v_{21}\oplus \mathbb{C} v_{22})\oplus (\mathbb{C} v'_{21}\oplus \mathbb{C} v'_{22})\cong R.$$}
For instance, a matrix $\begin{pmatrix}a_1\\a_2\end{pmatrix}\,(a_i\in\mathbb{C})$ for $A_1$ lets $V^*$ act on $\mathbb{C} e_1= \rho_1$ as $x\cdot e_1=a_1 v_{22}+a_2 v'_{22}$ and $y\cdot e_1=a_1  iv_{21}+a_2 iv'_{21}$. Recall that the matrices must satisfy the condition $B\wedge B=0$. By writing down the conditions $x\cdot (y\cdot v)-y\cdot (x\cdot v)=0$ for each basis $v$ of $R$, one sees that this condition is expressed as the following matrix relations:
{\small
\begin{equation}\label{path}
B_0A_0=B_1A_1=B_2A_2=B_3A_3=A_0B_0-A_2B_2-DC=A_1B_1-A_3B_3-iCD=0.
\end{equation}}
\qed
\end{ex}

\begin{rem}
(1) Usually the quiver $Q_G$ itself can be obtained easily from the character table of $G$, without giving explicit $G$-equivariant homomorphisms $V^*\otimes \rho\to \rho'$ as above. However, this computation will be necessary in order to construct embeddings of moduli spaces into toric varieties as we will see in the next section.\\
(2) The existence of an ideal $I_G$ of the path algebra $\mathbb{C} Q_G$ such that
$$\mathcal{N}\cong \{H\in\mathrm{Rep}(\alpha_G) \mid H_p=0,\,p\in I_G\}$$
as in the above example is regarded as a consequence of the Morita equivalence between the smash product $G\rtimes\mathbb{C}[V]$ and a quotient of $\mathbb{C} Q_G$. For another method to calculate $I_G$, see \cite[\S3]{BTW}.
\qed
\end{rem}

\section{Embedding into toric varieties}\label{3}

In this section we explain how we embed moduli space $\mathfrak{M}_\theta$ into toric varieties such that the birational geometry of the embedded varieties is inherited from that of the ambient toric varieties.

\subsection{The Cox ring of a resolution of a quotient singularity}\label{3.1}

In this subsection we treat the Cox ring, denoted by $\mathrm{Cox}(X)$, of a crepant resolution $X$ of a quotient singularity $\mathbb{C}^n/G$ for $G\subset SL_n(\mathbb{C})$. $\mathrm{Cox}(X)$ is the direct sum $\bigoplus_{L\in \mathrm{Pic}(X)} H^0(X,L)$ as a $H^0(\mathcal{O}_{V/G})$-module and admits a multiplicative structure inherited from the tensor product of global sections of line bundles (see \cite{ADHL} for generalities regarding Cox rings). Then $\mathrm{Cox}(X)$ is a $\mathrm{Pic}(X)$-graded commutative ring and is known to be finitely generated by the fundamental result of Birker-Cascini-Hacon-McKernan \cite{BCHM}. The most important feature of it is that $X$ can be recovered from $\mathrm{Cox}(X)$ as a GIT-quotient of $\mathrm{Spec}\,\mathrm{Cox}(X)$ by an algebraic torus with respect to a generic stability parameter if $X\to\mathbb{C}^n/G$ is projective (see the next subsection). Note that the isomorphism class of $\mathrm{Cox}(X)$ is independent of the choice of $X$ since crepant resolutions are mutually isomorphic in codimension one. The structure of $\mathrm{Cox}(X)$ has been studied for special classes of $G$ in \cite{D}, \cite{DW}, \cite{G} and so on. See \cite{Y1} for a general treatment. Moreover, algorithms finding generators of $\mathrm{Cox}(X)$ are given in \cite{DK} and \cite{Y1}.

Now we review how to describe the structure of $\mathrm{Cox}(X)$ explicitly. We refer the reader to \cite{Y1} for details. We first recall the notion of {\it age}. For an element $g\in G$, we choose a basis $x_1,\dots,x_n$ of the dual space $V^*$ of $V=\mathbb{C}^n$ which are eigenvectors for the action of $g$. Then we can write $g\cdot x_j=e^{\frac{2\pi i a_j}{r}}$ with an integer $0\le a_j<r$ where $r$ is the order of $G$. We define the {\it age} of $G$ as $\frac{1}{r}\sum_{j=1}^n a_j$, which is an integer since $g\in SL_n(\mathbb{C})$. We call an element $g\in G$ {\it junior} if its age is one. Note that the age is invariant under conjugation. We also define a discrete valuation $\nu_g:\mathbb{C}(V)^*\to \mathbb{Z}$ by setting $\nu_g(x_j)=a_j$.

It is shown in \cite{IR} that there is a one-to-one correspondence between the set of conjugacy classes of junior elements in $G$ and the set of irreducible exceptional divisors of the resolution $\pi:X\to V/G$, such that the divisorial valuation $\nu_E:\mathbb{C}(X)^*\to \mathbb{Z}$ along the $\pi$-exceptional divisor $E$ corresponding to $g$ coincides with the restriction of $\frac{1}{r}\nu_g$ to $\mathbb{C}(X)$ (via the birational map $\pi$). Let $E_1,\dots,E_m$ be the set of $\pi$-exceptional irreducible divisors and let $g_k\,(k=1,\dots,m)$ be a representative of the junior conjugacy class corresponding to $E_k$. We also put $\nu_k=\nu_{g_k}$ and $r_k=\sharp\langle g_k\rangle$.

The Cox ring has an embedding into the Laurent polynomial ring $\mathbb{C}[V]^{[G,G]}[t_1^{\pm1},\dots,t_m^{\pm1}]$ with $m$ variables over the invariant ring for the commutator group of $G$ \cite[Lemma 4.2]{Y1}. Note that $\mathbb{C}[V]^{[G,G]}$ is naturally graded by the dual abelian group $Ab(G)^\vee$ of the abelianization $Ab(G)=G/[G,G]$ so that its degree-zero part is equal to $\mathbb{C}[V]^G$. For $Ab(G)^\vee$-homogeneous generators $f_1,\dots,f_\ell$ of $\mathbb{C}[V]^{[G,G]}$, we associate to them the following $\ell+m$ elements
\begin{equation}\label{gens}
f_1\prod_{k=1}^m t_k^{\nu_k(f_1)},\cdots,f_\ell\prod_{k=1}^m t_k^{\nu_k(f_\ell)}, t_1^{-r_1},\dots,t_m^{-r_m}
\end{equation}
of $\mathbb{C}[V]^{[G,G]}[t_1^{\pm1},\dots,t_m^{\pm1}]$, which lie in $\mathrm{Cox}(X)$. They do not give generators of $\mathrm{Cox}(X)$ in general. In fact they give a generating system if and only if $f_i$'s satisfy the {\it valuation lifting condition} cf. \cite[Theorem 2.2]{DG}. We do not explain this condition here as we will not use it later.

\begin{rem}\label{cox(V/G)}
The ring $\mathbb{C}[V]^{[G,G]}$ treated here should be understood as the Cox ring $\mathrm{Cox}(V/G)$ of $V/G$ (\cite[Theorem 3.1]{AG}), and the $Ab(G)^\vee$-grading also corresponds to the $\mathrm{Cl}(V/G)$-grading on $\mathrm{Cox}(V/G)$. Thus each $f_i$ above defines a Weil divisor $D_i$ on $V/G$, and the associated element $\tilde{f}_i:=f_i\prod_{k=1}^m t_k^{\nu_k(f_i)}$ to $f_i$ corresponds to a section defining the strict transform in $X$ of $D_i$ under the birational morphism $X\to V/G$.
\qed
\end{rem}

The $\mathrm{Pic}(X)$-grading of $\mathbb{C}[V]^{[G,G]}[t_1^{\pm1},\dots,t_m^{\pm1}]$ can be read off from the $\mathbb{Z}^m$-grading given by the multi-degrees for the variables $t_1\dots,t_m$. More precisely, the $\mathrm{Pic}(X)$-grading is given so that the generator $t_k^{-r_k}$ lies in the homogeneous component corresponding to the divisor class of $E_k$. Note that the divisor classes of $E_1,\dots,E_m$ are a $\mathbb{Q}$-basis (but usually not a $\mathbb{Z}$-basis) of $\mathrm{Pic}(X)$.

\subsection{Review of birational geometry of $X$}\label{3.2}

As mentioned in the previous subsection, any crepant resolution $X$ of $V/G$ has a finitely generated Cox ring. Such a variety is called a {\it Mori dream space}, and its birational contractions are controlled by polyhedral cones in $\mathrm{Pic}(X)_\mathbb{R}$ (cf. \cite{HK}). In this section we review this briefly.

Hereafter we assume $X\to V/G$ is projective. We first consider the polyhedral cone generated by the classes of divisors in $\mathrm{Pic}(X)_\mathbb{R}$ whose base loci are of codimension greater than one. This cone is called the {\it movable cone} and denoted by $\mathrm{Mov}(X)$. In our case, the movable cone is the union of the nef cones of all relative minimal models $X'\to V/G$ (cf. \cite{Y1}). Here, the nef cone $\mathrm{Nef}(X')\subset \mathrm{Pic}(X')_\mathbb{R}$ is regarded as a cone inside $\mathrm{Pic}(X)_\mathbb{R}$ via the birational map $X\dashrightarrow X'$ which is an isomorphism in codimension one. Note that, in dimension three, relative minimal models are nothing but projective crepant resolutions.

The nef cones $\mathrm{Nef}(X')\subset \mathrm{Pic}(X)_\mathbb{R}$ are full-dimensional and have disjoint interiors. If we take any $L$ in the interior $\mathrm{Amp}(X')$ of $\mathrm{Nef}(X')$, the complete linear system of a sufficiently divisible power of $L$ gives the birational map $X\dashrightarrow X'$. Set
$$T_X=\mathrm{Hom}(\mathrm{Pic}(X),\mathbb{C}^*)\cong(\mathbb{C}^*)^{\times m},$$
which naturally acts on $\mathfrak{X}:=\mathrm{Spec}\,\mathrm{Cox}(X)$. By \cite[Theorem 2.3]{HK}, this birational map is the same as the one $X\dashrightarrow \mathfrak{X}/\!/_{\chi} T_X$ between the GIT-quotients with the character $\chi\in \chi(T_X)$ corresponding to $L$ via the identification $\chi(T_X)_\mathbb{R}=\mathrm{Pic}(X)_\mathbb{R}$. Moreover, the similar results hold for any {\it rational contraction}, that is, a rational map $f:X\dashrightarrow Z$ which is the composite of a sequence of flops $X\dashrightarrow X'$ and a proper morphism $f':X'\to Z$ over $V/G$ such that
\begin{itemize}
\item $Z$ is a normal variety projective over $V/G$, and
\item $f'$ has connected fibers, i.e. $f'_*\mathcal{O}_{X'}=\mathcal{O}_Z$.
\end{itemize}
More precisely, for such $f$, there exists $L\in\mathrm{Mov}(X)$ whose linear system recovers $f$, and moreover $f$ equals the rational map $X\dashrightarrow \mathfrak{X}/\!/_{\chi} T_X$ induced by variation of GIT for the character $\chi$ corresponding to $L$. Note that general points of a facet (i.e. codimension-one face) of $\mathrm{Mov}(X)$ corresponds to a primitive divisorial rational contraction (i.e., a composite of a sequence of flops and a birational morphism contracting a single divisor which drops the Picard number by one). In contrast, small contractions are realized by taking divisors (or the corresponding characters) from the interior of $\mathrm{Mov}(X)$. In particular every flop $X\dashrightarrow X'$ corresponds to the common facet $\mathrm{Nef}(X)\cap \mathrm{Nef}(X')$.

\begin{rem}
(1) We can also consider divisors (or corresponding characters) outside the movable cone, but they again give rational contractions (cf. \cite[Proposition 1.11]{HK}) and thus considering the movable cone is essential.

(2) Classifying the GIT-quotients of $\mathfrak{X}$ (or equivalently, rational contractions of $X$) gives a fan structure on $\chi(T_X)_\mathbb{R}$ (or on $\mathrm{Pic}(X)_\mathbb{R}$). This fan is called the {\it GIT-fan}. When $X$ is toric, this is also called the {\it secondary fan}. See \cite[Ch. 14]{CLS} for a more detailed exposition of this fan.
\end{rem}

\begin{ex}\label{abel1}(Abelian group of type $\frac{1}{10}(1,3,6)$)\\
We exhibit how to compute the movable cone and nef cones for a crepant resolution of $\mathbb{C}^n/G$ for abelian $G$ through a particular example. Here we treat the case when $G\subset SL_3(\mathbb{C})$ is the cyclic group generated by
$$g=\begin{pmatrix}\zeta_{10}& 0& 0\\0& \zeta_{10}^3& 0\\ 0& 0& \zeta_{10}^6\end{pmatrix}$$
with $\zeta_{10}$ the primitive 10th root of unity. In this case $g,g^2,g^4,g^5,g^7$ are the junior elements, and the Cox ring of any crepant resolution $X$ is generated by the following 8 elements in $\mathbb{C}[x,y,z][t_1^{\pm1},\dots,t_5^{\pm1}]$:
\begin{equation}\label{ex:generator}
x t_1 t_2 t_3^2 t_4 t_5^7,\; y t_1^3 t_2^3 t_3 t_4 t_5,\; z t_1^6 t_2 t_3^2  t_5^2,\; t_1^{-10},\; t_2^{-5},\; t_3^{-5},\; t_4^{-2},\; t_5^{-10}
\end{equation}
where $x,y$ and $z$ are the standard coordinates of $\mathbb{C}^3$. Moreover, $X$ is a toric variety corresponding to a fan $\Sigma$ obtained as a triangulation of the cone $\sigma_0=\mathrm{Cone}(e_1,e_2,e_3)\subset\mathbb{R}^3$ such that the rays generated by $e_1,e_2,e_3,v_1,\dots,v_5$ is equal to the set of the rays of $\Sigma$ where $e_j$ is the standard basis of $\mathbb{R}^3$ and $v_k$'s are given as
$$v_1=\begin{pmatrix}1\\3\\6\end{pmatrix},\;v_2=\begin{pmatrix}1\\3\\1\end{pmatrix},\; v_3=\begin{pmatrix}2\\1\\2\end{pmatrix},\; v_4=\begin{pmatrix}1\\1\\0\end{pmatrix},\;v_5=\begin{pmatrix}7\\1\\2\end{pmatrix}$$
so that $v_k$ corresponds to $E_k$. See e.g. \cite[\S3]{Y2} for details. One can check that there are six crepant resolutions, which are in fact all projective. These resolutions $X_1,\dots,X_6$ fit into the following diagram of flops:\\
\[\xymatrix@=36pt{
&X_3 \ar@{<-->}[d]_-{}
&
&
&\\
X_2 \ar@{<-->}[r]_{}
&X_1 \ar@{<-->}[r]_{}
&X_4 \ar@{<-->}[r]_{}
&X_5 \ar@{<-->}[r]_{}
&X_6.
}\]
The cross sections of the fans $\Sigma_1$ and $\Sigma_6$ of $X_1$ and $X_6$ respectively are given in Figure \ref{fan1}. Among $X_i$'s, the $G$-Hilbert scheme corresponds to $X_1$ (cf. \cite{CR}).
\begin{figure}[h]
\begin{center}
 \includegraphics{figure.pdf}
\caption{The cross sections of the fans $\Sigma_1$ (left) and $\Sigma_6$ (right)}\label{fan1}
\end{center}
\end{figure}

As is well known, each $i$-dimensional cone $\tau\in\Sigma_1$ represents a torus orbit $O_\tau$ in $X_1$ of codimension $i$ (see e.g. \cite[\S3.2]{CLS}). $\tau$ also defines a cone $C_\tau\subset\mathrm{Pic}(X_1)_\mathbb{R}$ generated by line bundles having global sections which do not vanish on $O_\tau$. Under the identification of $\mathrm{Pic}(X_1)_\mathbb{R}$ with $\chi(T_X)_\mathbb{R}$, $C_\tau$ is the same as the orbit cone (see Definition \ref{orbit cone}) of some (and hence any) lift $\bar{x}\in\mathfrak{X}$ of some (and hence any) $x\in O_\tau$. $C_\tau$ is explicitly computed as the cone generated by the weights of the generators in (\ref{ex:generator}) whose corresponding rays are not contained in $\tau$. For example, $C_\tau$ for $\tau=\mathrm{Cone}(e_3,v_1)\in\Sigma_1$ is generated by the weights of the generators in (\ref{ex:generator}) with the third and fourth ones removed.

The movable cone is obtained as the intersection $\bigcap_{i=1}^5 C_{\tau_i}$ where $\tau_i=\mathbb{R}_{\ge0}v_i$. Similarly, the nef cone of $X_1$ is the intersection $\bigcap_{\tau\in\Sigma_1} C_\tau$. In our case, however, we do not have to compute $C_\tau$ for all $\tau\in \Sigma_1$. $\mathrm{Nef}(X_1)$ is given as the intersection of $\mathrm{Mov}(X)$ and just one cone $\mathrm{Cone}(v_2,v_4,v_5)$ since $\Sigma_1$ is clearly the only triangulation containing this three-dimensional cone.

The above computations of cones can be done, for instance by using the package ``Polyhedra" of Macaulay2 \cite{GS}. Then we see that $\mathrm{Mov}(X_1)$ has 9 rays and 7 facets. The fact that $\mathrm{Mov}(X_1)$ is non-simplicial comes from non-uniqueness of primitive rational contractions of exceptional divisors. For example, the divisor $E_3\subset X_1$ is mapped to a curve by a primitive contraction, while it is mapped to a point by a primitive contraction inside $X_6$. We also see that $\mathrm{Nef}(X_1)$ is a simplicial cone having facets corresponding to 2 primitive divisorial contractions for $E_1,E_3$ and 3 flops along the curves $E_2\cap E_4, E_2\cap E_5$, and $E_4\cap E_5$.

Later we will compare cones in $\mathrm{Mov}(X_1)$ and orbit cones in $\Theta$ for moduli spaces of $G$-constellations (see Example \ref{abel2}).
\qed
\end{ex}

\subsection{Construction of embeddings}\label{3.3}
In this subsection we give a construction of nice embeddings of moduli spaces $\mathfrak{M}_\theta$ into toric varieties.

Let us consider the homomorphism $\iota_X^*:\mathbb{C}[y_1,\dots,y_{\ell+m}]\to \mathrm{Cox}(X)$ sending $y_i$'s to homogeneous generators of $\mathrm{Cox}(X)$ of the form in (\ref{gens}) so that we obtain an embedding $\iota_\mathfrak{X}$ of $\mathfrak{X}=\mathrm{Spec}(\mathrm{Cox}(X))$ into the affine space $\mathbb{A}^{\ell+m}$. We endow $\mathbb{C}[y_1,\dots,y_{\ell+m}]$ with a $\mathbb{Z}^m$-grading so that $\iota_X^*$ preserves the grading. Then we see that a character $\chi\in \chi(T_X)_\mathbb{R} = \mathrm{Pic}(X)_\mathbb{R}\cong\mathbb{R}^m$ corresponding to an ample divisor on $X$ satisfies $\mathfrak{X}/\!/_\chi T_X=X$ by the GIT construction (see also the previous subsection). We can also consider the GIT-quotient $Y=\mathbb{A}^{\ell+m}/\!/_\chi T_X$ for the same character, which is a normal toric variety such that $\mathrm{Pic}(Y)\cong\chi(T_X)\cong\mathrm{Pic}(X)$ \cite{Co}, and $\iota_\mathfrak{X}$ induces an embedding $\iota_X:X\hookrightarrow Y$ between the quotients.

\begin{rem}
If the generators of the Cox ring are minimal, then $\iota_X$ is a {\it neat embedding} in the sense of \cite{Ro}. See \cite[\S2.3]{Ro} for details.
\qed
\end{rem}

This embedding has a nice property from the viewpoint of birational geometry. As explained in the previous subsection, every rational contraction $X\dashrightarrow X'$ is realized as the rational map $X\dashrightarrow \mathfrak{X}/\!/_{\chi'} T_X$ for some $\chi'$. The same character $\chi'$ gives a rational contraction $Y\dashrightarrow Y'$ as well, and $\iota_\mathfrak{X}$ again induces an embedding $X'\to Y'$. Note that GIT-chambers for $\mathbb{A}^{\ell+m}$ are finer than those for $\mathfrak{X}$ in general, and in particular $Y\dashrightarrow Y'$ may not be an isomorphism even if $X\dashrightarrow X'$ is an isomorphism. In other words, the inclusion $\iota_X^*(\mathrm{Nef}(Y))\subset\mathrm{Nef}(X)$ is strict in general. Note, however, that we have
$$\iota_X^*(\mathrm{Mov}(Y))=\mathrm{Mov}(X)$$
since every exceptional divisor of $Y\to \mathbb{A}^{\ell+m}/\!/_0 T_X$ restricts to an exceptional divisor of $X\to \mathbb{C}^n/G$ by the construction of $Y$.

Next we construct an embedding of a moduli space $\mathfrak{M}_\theta$ into a toric variety. Recall that $\mathfrak{M}_\theta$ is obtained as the GIT-quotient of the variety $\mathcal{V}$ by the action of $PGL_R$ with respect to the character $\chi_\theta\in\chi(PGL_R)$. Let $T_\Theta\subset PGL_R$ be the algebraic torus consisting of (the images of) the component-wise scalar matrices. Since the restriction map $\chi(PGL_R)\to \chi(T_\Theta)$ is an isomorphism, via this identification $\mathfrak{M}_\theta$ is isomorphic to $\mathcal{W}/\!/_{\chi_\theta} T_\Theta$ where $\mathcal{W}$ is the affine categorical quotient $\mathcal{V}/\!/SL_R$. Note that the points of $\mathcal{W}$ are in bijection with the closed $SL_R$-orbits in $\mathcal{V}$.

In the proof of \cite[Proposition 2.2]{CI}, it is shown that $\mathcal{M}_0=\mathcal{N}/\!/PGL_R$ admits an irreducible component which is isomorphic to $\mathbb{C}^n/G$ (see also Subsection \ref{2.1}) by constructing a morphism $V\to \mathcal{M}_0$ induced from a $G$-constellation family $\mathcal{F}$ over $V=\mathbb{C}^n$. We will explicitly describe $V\to \mathcal{M}_0$ by lifting it to a morphism $V\to \mathcal{V}$. To do this, let
$$\mu:\mathbb{C}[V]\to\mathbb{C}[G]\otimes_\mathbb{C} \mathbb{C}[V]$$
be the ring homomorphism corresponding to the action of $G\subset SL(V)$. We let $G$ act on $\mathbb{C}[G]$ as $(g\cdot f)(h)=f(g^{-1}h)$ for any $g,h\in G$ and $f\in \mathbb{C}[G]$. In terms of algebras, the $\mathbb{C}[V]$-action on each fiber $\mathcal{F}_p:=\mathbb{C}[G]\,(p\in V)$ of the family $\mathcal{F}$ is given by the morphism
$$\mathbb{C}[V]\otimes \mathbb{C}[G]\to\mathbb{C}[G];\quad x\otimes f\mapsto (\mu(x)f\;\mathrm{mod}\; \mathfrak{m}_p)$$
where $\mathfrak{m}_p\subset \mathbb{C}[V]$ is the maximal ideal for $p$. This is indeed $G$-equivariant since we have
$$(g\cdot(x\cdot f))(h)=(x\cdot f)(g^{-1}h)=(\mu(x)f\;\mathrm{mod}\; \mathfrak{m}_p)(g^{-1}h)=x(g^{-1}h\cdot p)f(g^{-1}h)$$
while
$$\begin{aligned}
((g\cdot x)\cdot(g\cdot f))(h)&=(\mu(g\cdot x)(g\cdot f)\;\mathrm{mod}\; \mathfrak{m}_p)(h)=((g\cdot x)(h\cdot p))((g\cdot f)(h))\\
&=x(g^{-1}h\cdot p)f(g^{-1}h)
\end{aligned}$$
for any $g,h\in G,\, x\in \mathbb{C}[V]$ and $f\in F_p=\mathbb{C}[G]$.

For each $\rho\in\mathrm{Irr}(G)$, we fix a $\mathbb{C}$-basis $v_1,\dots,v_{\dim \rho}$. The matrix presentation of the representation $\rho$ with respect to this basis gives a matrix $(f_{i,j})_{i,j}$ with entries in $\mathbb{C}[G]$, that is, $g\cdot v_j=\sum_{i=1}^{\dim \rho}f_{i,j}(g)v_i$ for all $g\in G$. Let $v_1^*,\dots,v_{\dim \rho}^*$ be the dual basis to $v_1,\dots,v_{\dim \rho}$. Then the assignment $f_{i,j}\mapsto v_i^*$ gives $G$-equivariant isomorphisms from the vector subspaces $V_{\rho,j}:=\bigoplus_{i=1}^{\dim \rho}\mathbb{C} f_{i,j}\subset\mathbb{C}[G]$ to the dual (or contragredient) representation $\rho^*$ of $\rho$. This can be checked by comparing the coefficients of $v_i$'s in
$$g^{-1}h\cdot v_j=\sum_{i=1}^{\dim \rho} f_{i,j}(g^{-1}h)v_i=\sum_{i=1}^{\dim \rho} (g\cdot f_{i,j})(h)v_i$$
and
$$g^{-1}\cdot (h\cdot v_j)=g^{-1}\cdot \left(\sum_{i=1}^{\dim \rho}f_{i,j}(h)v_i\right)=\sum_{i=1}^{\dim \rho}f_{i,j}(h)\left(\sum_{k=1}^{\dim \rho}f_{k,i}(g^{-1})v_k\right)$$
with $g,h\in G$. Since $\mathbb{C}[G]$ is a direct sum of $\{V_{\rho, j}\}_{\rho, j}$ by the character theory for finite groups, we obtain an explicit $G$-equivariant isomorphism $\mathbb{C}[G]\to\bigoplus_{\rho\in\mathrm{Irr}(G)}\rho^{\oplus \dim\rho}$. We will describe the family $\mathcal{F}$ as quiver representations with the matrix presentation with respect to the basis of $\mathbb{C}[G]$ obtained in this way. To this end, we also fix a basis of the $a_{\rho,\rho'}$-dimensional vector space $\mathrm{Hom}_{\mathbb{C} G}(V^*\otimes_\mathbb{C} \rho,\rho')$ for each pair $\rho,\rho'\in\mathrm{Irr}(G)$.

For each arrow $a\in A$ from $\rho$ to $\rho'$ in $Q_G$, we have an embedding $\rho'\to V^*\otimes\rho$. Regarding the fixed basis for $\rho$ and $\rho'$, we obtain a $(\dim\rho'\times\dim\rho)$-matrix $\tilde{H}_a$ with entries in $V^*$. Then we obtain a set of matrices $\{H_a^p:=(\tilde{H}_a\;\mathrm{mod}\; \mathfrak{m}_p)\}_{a\in A}$, which can be regarded as a set of representations of $Q_G$ parametrized by $V$ (see Subsection \ref{2.3}).

\begin{lem}
The representation of $Q_G$ corresponding to the $G$-constellation $\mathcal{F}_p$ is isomorphic to $\{H_a^p\}_{a\in A}$ for any $p\in V$.
\end{lem}

{\it Proof.} We take any arrow $a\in A$ from $\rho$ to $\rho'$ and set $m=\dim \rho$ and $n=\dim \rho'$. Let $(f_{i,j})_{1\le i,j\le m}$ (resp. $(g_{i,j})_{1\le i,j\le n}$) be the matrix presentation of $\rho^*$ (resp. $\rho'^*$) with respect to the fixed basis. We write the $n\times m$-matrix $\tilde{H}_a$ as $(h_{i,j})_{i,j}$ with $h_{i,j}\in V^*$. Regarded as a representation of $Q_G$, $H_a^p$ defines an action of $V^*$ on the vector space $V_{\rho^*,j}=\bigoplus_{i=1}^m\mathbb{C} f_{i,j}\subset\mathbb{C}[G]$, which is isomorphic to $\rho$ as a $\mathbb{C} G$-module, so that
\begin{equation}\label{Ha}
\begin{aligned}
\sum_{i=1}^m h_{1,i}\cdot f_{i,k} &=\sum_{j=1}^n h_{j,k}(p) g_{1,j}\\
\sum_{i=1}^m h_{2,i}\cdot f_{i,k} &=\sum_{j=1}^n h_{j,k}(p) g_{2,j}\\
&\vdots\\
\sum_{i=1}^m h_{n,i}\cdot f_{i,k} &=\sum_{j=1}^n h_{j,k}(p) g_{n,j}
\end{aligned}
\end{equation}
for each $k=1,\dots,m$. We show that this action coincides with the one for $\mathcal{F}_p$. By the definition of the $\mathbb{C}[V]$-module structure on $\mathcal{F}_p=\mathbb{C}[G]$, the LHS of the $j$-th row of (\ref{Ha}), as an element of $F_p$, is the function
$$g\mapsto h_{j,1}(g\cdot p)f_{1,k}(g)+h_{j,2}(g\cdot p)f_{2,k}(g)+\cdots+h_{j,m}(g\cdot p)f_{m,k}(g)$$
on $G$ for each $j$. Note that the value of this function at the identity $1\in G$ is $h_{j,k}(p)$ and this is also equal to the value of the RHS of the $j$-th row of (\ref{Ha}) (as a function on $G$) at $1\in G$. By the choice of $\tilde{H}_a$, the LHSs and the RHSs of (\ref{Ha}) give $G$-equivariantly isomorphic basis of $\rho'$. Noticing that $f(g)=g^{-1}\cdot f(1)$ for any $f\in\mathbb{C}[G]$ and $g\in G$, this implies that the two functions attain the same values for all $g\in G$, and hence the claim.
\qed

\vspace{3mm}

We will give concrete examples of the representations $\{H_a^p\}_{a\in A}$ in Subsection \ref{3.5}.

The generic representation $\{\tilde{H}_a\}_a$ gives a morphism $V\to\mathcal{N}$, which factors through $V\to\mathcal{V}$ since $V$ is mapped onto $V/G\subset \mathcal{M}_0$. Let $f$ be a semi-invariant function on $\mathcal{N}$ and hence on $\mathcal{V}$. Then the (scheme-theoretic) zero locus of $f$ on $\mathcal{V}$ descends to a Weil divisor $D_f$ on $V/G$. Thus, by sending $f$ to the section of the divisorial sheaf $\mathcal{O}_{V/G}(D_f)$ defines a ring homomorphism
$$\varphi:\mathbb{C}[\mathcal{N}]^{SL_R}\to \mathrm{Cox}(V/G)\cong \mathbb{C}[V]^{[G,G]}$$
(cf. Remark \ref{cox(V/G)}). For a determinantal semi-invariant $f:H\mapsto \det (H_{p_{i,j}})_{i,j}$ defined by paths $p_{i,j}$ as in (\ref{eq:determinant}), $\varphi(f)$ is given as $\det \mathcal{H}\in\mathbb{C}[V]$ where $\mathcal{H}=(\tilde{H}_{p_{i,j}})_{i,j}$ is defined similarly to $\tilde{H}_a$. Note that if $f$ has weight $(c_\rho)_{\rho\in\mathrm{Irr}(G)}$, then $\varphi(f)$ is acted on by $G$ by the character
$$\det\left(\bigoplus_{\rho\in\mathrm{Irr}(G)} \rho^{\oplus c_\rho}\right):G\to \mathbb{C}^*.$$

Let us consider the $\bar{\Theta}$-graded algebra $\bigoplus_{\theta\in\bar{\Theta}}\mathbb{C}[V]^{[G,G]} t_{\theta}$ where $t_{\theta}$ is a formal variable indicating the component for $\theta\in\bar{\Theta}$. Using $\varphi$, we can give a description of the coordinate ring $\mathbb{C}[\mathcal{V}]^{SL_R}$ of $\mathcal{W}=\mathcal{V}/\!/SL_R$ similar to the description of $\mathrm{Cox}(X)$ in (\ref{gens}).

\begin{prop}\label{component V}
The algebra $\mathbb{C}[\mathcal{W}]=\mathbb{C}[\mathcal{V}]^{SL_R}$ is isomorphic to the graded subring $\mathbf{S}_\mathcal{W}$ of $\bigoplus_{\theta\in\bar{\Theta}}\mathbb{C}[V]^{[G,G]} t_{\theta}$ generated by $\varphi(h)t_\theta$ for semi-invariants $h\in \mathbb{C}[\mathcal{N}]^{SL_R}$ of weight $\theta$.
\end{prop}

{\it Proof.} It suffices to show that the kernel of the surjection $\mathbb{C}[\mathcal{N}]^{SL_R}\to\mathbf{S}_\mathcal{W}$ is the ideal for the subscheme $\mathcal{W}\subset \mathcal{N}/\!/SL_R$. Since we have $\mathbf{S}_\mathcal{W}^{T_\Theta}=\mathbb{C}[V]^G$, this follows form the fact that $\mathcal{V}\subset \mathcal{N}$ is just the inverse image of $V/G$ under the quotient map $\mathcal{N}\to \mathcal{M}_0$.
\qed

\vspace{3mm}

From now on we assume $n=3$, i.e. $V=\mathbb{C}^3$. We fix any GIT-chamber $C\subset \Theta$ and let $X:=\mathfrak{M}_C$ be the crepant resolution of $V/G$. Similar to the construction of $\varphi$, we define a ring homomorphism
$$\varphi_C:\mathbb{C}[\mathcal{N}]^{SL_R}\to \mathrm{Cox}(X)$$
whose composition with
$$\mathrm{Cox}(X)\hookrightarrow \mathbb{C}[V]^{[G,G]}[t_1,\dots,t_m]\to\mathbb{C}[V]^{[G,G]}$$
is equal to $\varphi$. This also induces a homomorphism $\psi_C:\bar{\Theta}\to\mathrm{Pic}(X)$ such that the degree of $\varphi_C(f)$ for a homogeneous element of weight $\theta:\rho\mapsto c_\rho$ is equal to $\psi_C(\theta)$.

Now we are ready to define an embedding of $\mathcal{W}$ into a toric variety. We first choose $Ab(G)^\vee$-homogeneous generators $f_1,\dots,f_\ell\in\mathbb{C}[V]^{[G,G]}$ such that the associated elements in (\ref{gens}) generate $\mathrm{Cox}(X)$. Let $\iota_\mathfrak{X}:\mathfrak{X}\to \mathbb{A}^{\ell+m}$ be the associated embedding constructed in the beginning of this subsection. Set $N_X=\mathbb{Z}^\ell\oplus\mathrm{Pic}(X)^\vee$ and let $\sigma_\mathbb{A}\subset (N_X)_\mathbb{R}\cong\mathbb{R}^{\ell+m}$ be the cone defining the affine toric variety $\mathbb{A}^{\ell+m}$ with the $T_X$-action. We also set $N_\mathcal{W}=\mathbb{Z}^\ell\oplus\bar{\Theta}^\vee$ and define $\sigma_\mathcal{W}\subset (N_\mathcal{W})_\mathbb{R}\cong\mathbb{R}^{\ell+s-1}$ as the cone generated by the set of cones $\{(\mathrm{id}_{\mathbb{Z}^\ell}\oplus \psi_C^*)_\mathbb{R}(\sigma_\mathbb{A})\}_X$ where $X=\mathfrak{M}_C$ runs for all GIT-chambers $C\subset\Theta$ and the choice of $f_i$'s is consistent regardless of the choice of $X$. Note that $\psi_C$ and its pullback $\psi_C^*:\mathrm{Pic}(X)^\vee\to \bar{\Theta}^\vee$ depend on $X$.

Let $\bar{Y}_\mathcal{W}$ be the affine normal toric variety defined by $\sigma_\mathcal{W}$. We show that the normalization $\bar{\mathcal{W}}$ of $\mathcal{W}$ has an embedding into $\bar{Y}_\mathcal{W}$. By the definition the coordinate ring of $\bar{Y}_\mathcal{W}$ is the semigroup algebra $\mathbb{C}[S_\mathcal{W}]$ for the semigroup $S_\mathcal{W}=M_\mathcal{W}\cap \sigma_\mathcal{W}^\vee$ where $M_\mathcal{W}=N_\mathcal{W}^\vee$ and $\sigma_\mathcal{W}^\vee$ is the dual cone
$$\{f\in (M_\mathcal{W})_\mathbb{R} \mid f(v)\ge0\text{ for all }v\in \sigma_\mathcal{W}\}$$ to $\sigma_\mathcal{W}$.
Since $f_i$'s generate $\mathbb{C}[V]^{[G,G]}$, we can choose homogeneous generators $\{h_j\}_j$ of $\mathbb{C}[\mathcal{W}]=\mathbb{C}[\mathcal{V}]^{SL_R}$ such that each $\varphi(h_j)$ is of the form $f_1^{a_1}\cdots f_\ell^{a_\ell}$. This presentation is not unique in general, but we fix it for each $\varphi(h_j)$. Then $\mathcal{W}$ admits a closed immersion into the spectrum $Y_\mathcal{W}$ of the semigroup algebra $\mathbb{C}[S'_\mathcal{W}]$ for the semigroup $S'_\mathcal{W}\subset M_\mathcal{W}$ generated by $\{(a_1,\dots,a_\ell,\theta_j)\}_j$ where $\theta_j\in\bar{\Theta}$ is the weight of $h_j$. This also induces a closed immersion of $\bar{\mathcal{W}}$ into the spectrum of the  integral closure of $\mathbb{C}[S'_\mathcal{W}]$.

\begin{lem}\label{integral closure}
The semigroup algebra $\mathbb{C}[S_\mathcal{W}]$ coincides with the integral closure of $\mathbb{C}[S'_\mathcal{W}]$.
\end{lem}

{\it Proof.} For each $f\in S_\mathcal{W}$, it defines a rational function $\bar{f}$ on $\bar{\mathcal{W}}$ via the above embedding of $\bar{\mathcal{W}}$. It suffices to show that $\bar{f}$ is regular on $\bar{\mathcal{W}}$. Since $\bar{\mathcal{W}}$ is normal it suffices to show that $\bar{f}$ is regular outside a locus of codimension greater than one. By the construction of $\sigma_\mathcal{W}$, $\bar{f}$ is regular at general points of any divisor $D\subset \bar{\mathcal{W}}$ which descends to a divisor on a resolution $\bar{\mathcal{W}}/\!/_\theta T_\Theta$ of $V/G$ for some generic stability condition $\theta$. Then the claim follows by the lemma below.
\qed

\vspace{3mm}

\begin{lem}
The complement of the open subset $\bigcup_{\theta\textup{:generic}} \mathcal{W}^{\theta\text{-ss}}$ in $\mathcal{W}$ is of codimension at least two.
\end{lem}

{\it Proof.} Assume that the complement contains an irreducible divisor $D\subset \mathcal{W}$ in order to deduce a contradiction. Let $C\subset \Theta$ be the cone generated by the weights of semi-invariants of $\mathcal{V}$ which do not vanish at general points of $D$. By the assumption, $C$ has positive codimension and, equivalently, general points of $D$ has a positive-dimensional stabilizer subgroup in $T_\Theta$. Then the image of $D$ under the quotient map $\mathcal{W}^{\theta\text{-ss}}\to \mathcal{W}/\!/_\theta T_\Theta$ for $\theta$ in the relative interior of $C$ has dimension greater than or equal to $n=3$. This is contrary to the irreducibility of $\mathcal{W}$ since the $T_\Theta$-orbits of general points of $\mathcal{W}$, which correspond to free $G$-orbits (see Subsection (\ref{2.1})), are closed and hence $\mathcal{W}/\!/_\theta T_\Theta$ has dimension three for any (non-generic) $\theta$.
\qed

\vspace{3mm}

Thus, we obtain the following commutative diagram of equivariant maps:
\begin{equation}\label{CD1}
\begin{CD}
\mathbb{A}^{n+m} @>>> \bar{Y}_\mathcal{W} @>>> Y_\mathcal{W}\\
@A \iota_\mathfrak{X} AA @AAA @AAA\\
\mathfrak{X}@>>> \bar{\mathcal{W}} @>>> \mathcal{W}\\
\end{CD}
\end{equation}
where vertical arrows are closed immersions. Note that $Y_\mathcal{W}$ is a possibly non-normal toric variety whose normalization is $\bar{Y}_\mathcal{W}$ by Lemma \ref{integral closure}. For any $\theta\in C$, we also obtain the following diagram of maps between quotients induced from (\ref{CD1}):
\begin{equation}\label{CD2}
\begin{CD}
Y @>>> \bar{Y}_\mathcal{W}/\!/_{\chi_\theta} T_\Theta @>>> Y_\mathcal{W}/\!/_{\chi_\theta} T_\Theta\\
@A \iota_X AA @AAA  @AAA\\
X @>>> \bar{\mathcal{W}}/\!/_{\chi_\theta} T_\Theta @>>> \mathcal{W}/\!/_{\chi_\theta} T_\Theta\\
\end{CD}
\end{equation}
where vertical arrows are again closed immersions and horizontal arrows are all isomorphisms. The results obtained so far are summarized as follows:

\begin{prop}\label{embedding}
The irreducible component $\mathcal{W}$ of $\mathcal{N}/\!/SL_R$ such that $\mathcal{W}/\!/T_\Theta\cong \mathbb{C}^3/G$ admits a $T_\Theta$-equivariant closed immersion $\iota_\mathcal{W}: \mathcal{W}\to Y_\mathcal{W}$ into a possibly non-normal toric variety whose torus contains $T_\Theta$, satisfying the following property:
\begin{itemize}
\item for any generic $\theta\in\Theta$, the induced morphism $\mathcal{W}/\!/_{\chi_\theta} T_\Theta\to Y_\mathcal{W}/\!/_{\chi_\theta} T_\Theta$ between the GIT-quotients is identified with a closed immersion $\iota_X: X\to Y$ of the resolution $X:=\mathfrak{M}_\theta$ into a normal toric variety such that the restriction $\iota_X^*: \mathrm{Pic}(Y)_\mathbb{R}\to \mathrm{Pic}(X)_\mathbb{R}$ is an isomorphism and $\mathrm{Mov}(X)=\iota_X^*(\mathrm{Mov}(Y))$.
\end{itemize}
\end{prop}

In the next subsection we will explain how the (semi)stable locus of $\bar{Y}_\mathcal{W}$ for a given stability condition is determined in terms of the fan $\sigma_\mathcal{W}$.

\begin{rem}\label{vphi}
(1) Let $v_1,\dots,v_m\in \sigma_\mathcal{W}$ be the primitive generators of the rays corresponding to the exceptional divisors $E_k$ of $X=\mathfrak{M}_C\to V/G$. Recall that the order of zeros (or poles) of a rational function $f\in M_\mathcal{W}$ along the corresponding divisor to $v_k$ is given by $f(v_k)$, by toric geometry. Thus, the image of a homogeneous semi-invariant $h:=f_1^{a_1}\cdots f_\ell^{a_\ell}t_{\theta}\,(a_i\in \mathbb{Z})$ under $\varphi_C$ is given by
\begin{equation}\label{phi(h)}
\tilde{f}_1^{a_1}\cdots \tilde{f}_\ell^{a_\ell}\prod_{k=1}^m t_k^{-\mathbf{a}(v_k)r_k}
\end{equation}
(see Remark \ref{cox(V/G)}) where $\mathbf{a}=(a_1,\dots,a_\ell,\theta)\in \mathbb{Z}^\ell\oplus \bar{\Theta}$ is identified with an element of $M_\mathcal{W}$. In particular, $\psi_C(\theta)\in \mathrm{Pic}(X)$ is given by the multi-degree of $t_k$'s in (\ref{phi(h)}). Note that $\psi_C$ only depends on $v_k$'s and therefore $\psi_{C'}$ is equal to $\psi_C$ as long as we take a GIT-chamber $C'$ inside the intersection of the orbit cones of $G$-constellations corresponding to general points of the divisors $E_1,\dots,E_m$.

(2) The homomorphism $\varphi_C:\mathbb{C}[\mathcal{N}]^{SL_R}\to \mathrm{Cox}(X)$ is never surjective unless $G$ is trivial since each $t_k^{-r_k}\in\mathrm{Cox}(X)$ is clearly outside of the image. However, we will show that, for homogeneous generators $\{h_i\}$ of $\mathbb{C}[\mathcal{N}]^{SL_R}$, the associated elements to the images $\varphi(h_i)$ generate $\mathrm{Cox}(X)$ (Proposition \ref{prop:generate}).
\qed
\end{rem}

\subsection{GIT-quotients of a toric variety by a subtorus-action}\label{3.4}

As we saw in the previous subsection, every crepant resolution of the form $\mathfrak{M}_\theta$ is embedded in a quotient of the toric variety $\bar{Y}_\mathcal{W}$ (or $Y_\mathcal{W}$) by the algebraic torus $T_\Theta$. In this subsection we show how to compute the $\theta$-semistable locus of $\bar{Y}_\mathcal{W}$ for $\theta\in \Theta$ in terms of the fan of $\bar{Y}_\mathcal{W}$. This also determines the semistable locus of $\bar{\mathcal{W}}$ since we have
$$\bar{Y}_\mathcal{W}^{\theta\text{-ss}}\cap \bar{\mathcal{W}} =\bar{\mathcal{W}}^{\theta\text{-ss}}$$
by the construction of $\bar{Y}_\mathcal{W}$, If $\theta$ is generic, $\bar{\mathcal{W}}^{\theta\text{-ss}}$ is also identified with $\mathcal{W}^{\theta\text{-ss}}$ since the geometric quotient of $\mathcal{W}^{\theta\text{-ss}}$ is smooth.

To consider quotients of $\bar{Y}_\mathcal{W}$, first note that $T_\Theta$ is regarded as a subtorus of the {\it big torus} $T_\mathcal{W}:=\mathrm{Hom}_\mathbb{Z}(N_\mathcal{W},\mathbb{C}^*)\subset Y_\mathcal{W}$. Quotients of a toric variety by an action of a subtorus of the big torus are studied in \cite{KSZ},\cite{AH},\cite{Hu} and so on. When we consider GIT-quotients (or more generally good quotients), the quotient map naturally induces a one-to-one correspondence between the sets of the maximal cones for the toric variety and its quotient (see e.g. \cite[Proposition 3.2]{AH}). To state this more precisely for our situation, recall that the toric variety $\bar{Y}_\mathcal{W}$ is defined by the cone $\sigma_\mathcal{W}$ in $(N_\mathcal{W})_\mathbb{R}$. For any $\theta\in\Theta$, its semistable locus $\bar{Y}_\mathcal{W}^{\theta\text{-ss}}$ is $T_\mathcal{W}$-invariant since functions on $\bar{\mathcal{W}}$ defining the semistable locus can be taken as homogeneous ones with respect to $T_\mathcal{W}$. Thus, $\bar{Y}_\mathcal{W}^{\theta\text{-ss}}$ is also a toric variety defined by a subfan $\Sigma_\theta$ of $\Sigma_\mathcal{W}$ where $\Sigma_\mathcal{W}$ is the fan consisting of all faces of $\sigma_\mathcal{W}$. Then the GIT-quotient $Y_\theta:=\bar{Y}_\mathcal{W}/\!/_\theta T_{\Theta}$ is a toric variety defined by a fan $\bar{\Sigma}_\theta$, and the quotient map $\bar{Y}_\mathcal{W}^{\theta\text{-ss}}\to Y_\theta$ is a toric morphism induced by the projection
$$p:(N_\mathcal{W})_\mathbb{R}\to(N_\mathcal{W}/\bar{\Theta}^\vee)_\mathbb{R}=\mathbb{R}^\ell$$
satisfying the following condition ($*$):
\begin{itemize}\item For any cone $\sigma\in\bar{\Sigma}_\theta$, the cone $p^{-1}(\sigma)\cap|\Sigma_\theta|$ belongs to $\Sigma_\theta$ where $|\Sigma_\theta|\subset \mathbb{R}^\ell$ is the support of the fan $\Sigma_\theta$. Moreover, if $\theta$ is generic, $p$ induces an isomorphism of $\Sigma_\theta$ and $\bar{\Sigma}_\theta$.\end{itemize}

Now we give an explicit method to compute $\Sigma_\theta$ and hence $\bar{Y}_\mathcal{W}^{\theta\text{-ss}}$. To this end, it is sufficient to determine a general point of $p^{-1}(v)\cap|\Sigma_\theta|$ for each $v\in \mathbb{R}^\ell$ by the condition ($*$). Let $q:(N_\mathcal{W})_\mathbb{R}\cong \mathbb{R}^\ell\times\Theta^\vee \to \Theta^\vee$ be the second projection to the dual vector space of $\Theta\cong\mathbb{R}^{s-1}$ (while $p$ is regarded as the first projection). Then we set $P_v=q(p^{-1}(v)\cap|\Sigma_\mathcal{W}|)\subset\Theta^\vee$, which is a rational convex polyhedron since $\sigma_\mathcal{W}$ is rational and strongly convex. Note that $\theta$ is a linear function on $P_v$. The following result is regarded as a generalization of \cite[Theorem 7.2]{CMT} to the case where $G$ is non-abelian, see Example \ref{3.5.1}.

\begin{prop}\label{minimize}
For any $\theta\in\Theta$, a face $F$ of $\sigma_\mathcal{W}$ belongs to $\Sigma_\theta$ if and only if $q(F)\cap P_{p(\tilde{v})}$ equals the set of points of $P_{p(\tilde{v})}$ for which the function $\theta:P_{p(\tilde{v})}\to \mathbb{R}$ is minimized for some (and hence any) point $\tilde{v}$ of the relative interior $\mathrm{relint}(F)$ of $F$.
\end{prop}

Although this result can be proved e.g. by using a similar argument to the the proof of \cite[Theorem 7.2]{CMT}, we give another proof using the Hilbert-Mumford numerical criterion in order to understand the result in terms of one-parameter subgroups. The numerical criterion states that a point $x\in \bar{Y}_\mathcal{W}$ is $\theta$-semistable if and only if we have $\lambda\cdot\theta\ge0$ for any 1-parameter subgroup (1-PS) $\lambda:\mathbb{C}^*\to T_\Theta$ such that $\lim_{t\to0}(\lambda(t)\cdot x)$ exists, where $\lambda\cdot\theta$ means the natural paring between a 1-PS and a character of $T_\Theta$.

To apply this criterion to toric varieties, let us recall that faces $F$ of $\sigma_\mathcal{W}$ and $T_\mathcal{W}$-orbits of $\mathcal{W}$ are in one-to-one correspondence which assigns to $F$ the orbit $O(F)$ of $x_F:=\lim_{t\to0}\lambda^{\tilde{v}}(t)$ for the 1-PS $\lambda^{\tilde{v}}:\mathbb{C}^*\to T_\mathcal{W}$ corresponding to $\tilde{v}\in\mathrm{relint}(F)$. Moreover, the closure of $O(F)$ is equal to $\bigcup_{F\subset F'\in\Sigma_\mathcal{W}} O(F')$, and we have $\lim_{t\to0}(\lambda^u(t)\cdot x_F)\in O(F')$ for any $u\in\mathrm{relint}(F')$ where $\lambda^u$ is the  1-PS corresponding to $u$. See \cite[Proposition 3.2.2]{CLS} and its proof for these standard facts. We also observe that the limit $\lim_{t\to0}(\lambda^u(t)\cdot x_F)$ for $u\in N_\mathcal{W}$ exists if and only if there exist (sufficiently small) $h>0$ and a face $F'$ of $\sigma_\mathcal{W}$ containing $F$ such that $u_1+hu$ lies in $F'$ for some (and hence any) $u_1\in\mathrm{relint}(F)$.

\vspace{3mm}

{\it Proof of Proposition \ref{minimize}.} \\
According to the observation above, for a 1-PS $\lambda^w$ of $T_\Theta$ corresponding to a vector $w\in\bar{\Theta}^\vee$ and any point $x$ of the orbit $O(F)$, the limit $\lim_{t\to0} (\lambda^w(t)\cdot x)$ exists if and only if there exist $h>0$ and a face $F'$ containing $F$ such that $u_1+hu\in F$ for some $u_1\in\mathrm{relint}(F)$.

Let us assume that $q(F)\cap P_{p(\tilde{v})}$ does not minimize $\theta$. Then there is another face $F'$ of $\sigma_\mathcal{W}$ such that $q(F')\cap P_{p(\tilde{v})}$ minimizes $\theta$, and $(w_1-q(\tilde{v}))\cdot \theta<0$ for any $w_1\in \mathrm{relint}(q(F')\cap P_{p(\tilde{v})})$. Since $q(\tilde{v})+1\cdot (w_1-q(\tilde{v}))=w_1$ is inside $P_{p(\tilde{v})}$, the convexity of $P_{p(\tilde{v})}$ implies that the limit for the 1-PS corresponding to $w_1$ exists and thus $F$ is not in $\Sigma_\theta$ by the numerical criterion.

Conversely, if $q(F)\cap P_{p(\tilde{v})}$ minimizes $\theta$, then $q(F)\cap P_{p(\tilde{v})}$ is a face of $P_{p(\tilde{v})}$ having $\{\theta=(\text{the minimal value})\}\subset\Theta$ as the supporting hyperplane. Therefore, $(w-q(\tilde{v}))\cdot \theta\ge0$ for any $w\in P_{p(\tilde{v})}$ and $F$ is in $\Sigma_\theta$ by the numerical criterion again.
\qed

\vspace{3mm}

\begin{rem}
Whenever we are given an affine toric variety and a subtorus of the big torus, the similar statement to the above proposition applies to compute the semistable locus of the given toric variety with respect to a character of the subtorus.
\end{rem}

\subsection{Examples}\label{3.5}
In this subsection demonstrate the construction of the toric embedding and the homomorphism $\varphi_C$ in the previous subsection via concrete examples.

\subsubsection{Abelian cases}\label{3.5.1}
We first consider the case where $G\subset SL_3(\mathbb{C})$ is a finite abelian subgroup of order $r$. In this case all irreducible representations of $G$ are 1-dimensional, which particularly implies that $SL_R$ is trivial and hence $\mathcal{V}=\mathcal{W}$. Then there is a natural choice of generators of the Cox ring of a creapnt resolution $X$ and generators of the coordinate ring of $\mathcal{V}$ as follows.

Once we fix a basis of $V=\mathbb{C}^3$ such that $G$ is a diagonal group, we obtain generators of $\mathbb{C}[V]^{[G,G]}=\mathbb{C}[V]$ as the standard basis $x_1,x_2,x_3$ of $V^*$ and also standard generators of $\mathrm{Cox}(X)$ (cf. \cite[Proposition 3.5]{Y2}. As explained in \cite{CMT} (see also \cite{Y2}), the coordinate ring of $\mathcal{V}$ is generated by $3r$ variables $\{x_{i,j}\}_{0\le i\le r-1,\,1\le j\le 3}$ with certain relations where $r$ is the order of $G$. In terms of the quiver representation, the variable $x_{i,j}$ is a homogeneous semi-invariant corresponding to an arrow $\rho_i$ to $\rho_i\otimes \chi_j$ of the McKay quiver of $G$ where $\mathrm{Irr}(G)=\{\rho_0,\dots,\rho_{r-1}\}$ and we regard the character $\chi_j:G\to\mathbb{C}^*$ defined by $g\cdot x_j=\chi_j(g)x_j$ as a 1-dimensional representation of $G$. Then $\mathcal{V}$ is already a possibly non-normal toric variety \cite[Theorem 3.10]{CMT}, and the ambient toric variety $Y_\mathcal{V}$ constructed from these generators is nothing but $\mathcal{V}$ itself.

For each orbit $O\subset \mathcal{V}$ by the big torus, it has the {\it distinguished point} $v\in O$ satisfying $x_{i,j}(v)=1$ for all $i,j$ such that $x_{i,j}(O)\ne\{0\}$. For any $\theta$, the fan $\Sigma_\theta$ of the $\theta$-semistable locus of $\mathcal{V}$ is computed using Proposition \ref{minimize}, and one obtains the distinguished point corresponding to a cone $\sigma\in \Sigma_\theta$ by setting
$$x_{i,j}(v)=\begin{cases}1&\text{ if }\sigma\subset H_{i,j}\\0&\text{ if }\sigma\not\subset H_{i,j}\end{cases}$$
where $H_{i,j}\subset (N_\mathcal{V})_\mathbb{R}$ is the hyperplane defined by $x_{i,j}\in M_\mathcal{V}$. This is essentially the same procedure as the one given in \cite[Theorem 7.2]{CMT}, which enables us to obtain the distinguished point corresponding to the cone $\sigma_w\in \Sigma_\theta$ such that $p(\sigma_w)$ contains a given $w\in\mathbb{R}^3$ in its relative interior. Note that one can also consider distinguished points of $\bar{Y}_ \mathcal{W}$ for non-abelian $G$ but these points usually do not lie inside $\bar{\mathcal{W}}$.

In \cite{Y2}, an explicit description of the homomorphism $\varphi_C:\mathbb{C}[\mathcal{V}]\to\mathrm{Cox}(X)$ is given in terms of the notion of a {\it $G$-nat family} introduced by Logvineko \cite{Lo}. To put it shortly, a $G$-nat family on a crepant resolution $X\to \mathbb{C}^3/G$ is a family of $G$-constellations which extends to $X$ the family over the open subset of $\mathbb{C}^3/G$ parametrizing the free $G$-orbits. To give such a family is equivalent to choose rays of $\sigma_\mathcal{V}$ such that their images under $p:(N_\mathcal{V})_\mathbb{R}\to\mathbb{R}^3$ are the rays of the toric fan of $X$. The main result of \cite{Y2} for $n=3$ states that $X$ is realized as a fine moduli space of $G$-constellations (not-necessarily of the form $\mathfrak{M}_\theta$) if and only if  $X$ admits a $G$-nat family $\mathcal{F}$ such that the maximal cones of the ``associated" subfan $\Sigma_{X,\mathcal{F}}$ of $\Sigma_\mathcal{V}$ has the expected dimension $3$ (see \cite[\S4]{Y2} for details). Note that the results in \cite{Y2} work for arbitrary $n$ and abelian $G\subset SL_n(\mathbb{C})$.

\begin{rem}
If we add redundant homogeneous generators (like $x_j+x_{j'}$ if $\chi_j=\chi_{j'}$) to the generators $x_1,\dots,x_n$ of $\mathbb{C}[V]$, we obtain a strictly bigger toric variety $\bar{Y}_\mathcal{V}$ than $\bar{\mathcal{V}}$.
\qed
\end{rem}

\begin{ex}\label{abel2}(Continuation of Example \ref{abel1})\\
We consider again the abelian $G$ treated in Example \ref{abel1}. In order to describe $\bar{\mathcal{V}}$ concretely, we fix coordinates $\theta_0,\dots,\theta_9$ of $\theta\in\Theta$ so that $\theta_i=\theta(\rho^{\otimes i})\in\mathbb{R}$ where $\rho:G\to \mathbb{C}^*$ is the one-dimensional representation defined by $\rho(g)=\zeta_{10}$. We identify $\Theta$ with $\mathbb{R}^9$ via $\theta\mapsto(\theta_1,\dots,\theta_9)$ from now on. Note that $\sum_{i=0}^9 \theta_i=0$. Then we see that the cone $\sigma_\mathcal{V}$ admits 65 rays, whose images under $p:\mathbb{R}^3\times\mathbb{R}^9\to \mathbb{R}^3$ are equal to the rays $\mathbb{R}_{\ge0}e_j\,(j=1,2,3)$ and $\mathbb{R}_{\ge0}v_k\,(k=1,\dots,5)$, as expected.

In order to determine which rays of $\sigma_\mathcal{V}$ correspond to exceptional divisors of $X_1$, recall that the $G$-Hilbert scheme is realized as $\mathcal{V}/\!/_{\theta_+} T_\Theta$ for any $\theta_+\in\Theta$ satisfying $(\theta_+)_i>0$ for all $i>0$. Note that GIT-chamber $C_1$ for the $G$-Hilbert scheme contains all such $\theta_+$. Using Proposition \ref{minimize}, we see that the rays generated by the following vectors in $\mathbb{R}^3\times\mathbb{R}^9$ are the rays in $\Sigma_{\theta_+}$:
{\small $$\begin{aligned}
w_1&=(1,3,6,-1,-2,-3,-4,-5,-6,-7,-8,-9)\\
w_2&=(1,2,3,-1,-2,-3,-4,-5,-1,-2,-3,-4)\\
w_3&=(2,1,2,-2,-4,-1,-3,-5,-2,-4,-6,-3)\\
w_4&=(1,1,0,-1,0,-1,0,-1,0,-1,0,-1)\\
w_5&=(7,1,2,-7,-4,-1,-8,-5,-2,-9,-6,-3).
\end{aligned}$$}
One can also determine $w_k$'s by using the fact that the universal family for the $G$-Hilbert scheme is the {\it maximal shift family}, see \cite[\S3.5]{Lo} for details.

Now we can explicitly describe $\varphi_{C_1}$ as explained in Remark \ref{vphi}(1). If we embed $\mathrm{Pic}(X_1)$ into $\mathbb{Z}^5$ as we have been doing so far (cf. (\ref{ex:generator})), then the map $(\psi_{C_1})_\mathbb{R}:\mathbb{R}^9\to\mathbb{R}^5$ is given by the matrix
{\small $$\begin{pmatrix}1&2&3&4&5&6&7&8&9\\
1&2&3&4&5&1&2&3&4\\
2&4&1&3&5&2&4&6&3\\
1&0&1&0&1&0&1&0&1\\
7&4&1&8&5&2&9&6&3
\end{pmatrix}.$$}
One can check that we have $(\psi_{C_1})_\mathbb{R}(\overline{C}_1)=\mathrm{Nef}(X_1)$.

Now we can explicitly confirm that every crepant resolution of $\mathbb{C}^3/G$ is realized as $\mathfrak{M}_\theta$ for some generic $\theta$, which is guaranteed by the main result of \cite{CI}. As mentioned in Remark \ref{vphi}(1), the map $\psi_C$ for a GIT-chamber $C$ coincides with $\psi_{C_1}$ as long as $C$ is contained in the intersection $C_+:=\bigcap_{k=1}^5 C_{w_k}$ of the orbit cones
$$C_{w_k}=\mathrm{Cone}(\{\mathrm{wt}(x_{i,j})\in\Theta\mid w_k\subset\{x_{i,j}=0\}\}).$$
One sees that we have $(\psi_{C_1})_\mathbb{R}(C_+)=\bigcup_{i=1}^5 \mathrm{Nef}(X_i)$, which implies that the interior of
$$\overline{C}_i:=(\psi_{C_1})_\mathbb{R}^{-1}(\mathrm{Nef}(X_i))\cap C_+ \quad(i=1,\dots,5)$$
is a GIT-chamber $C_i$ whose associated moduli space equals $X_i$. The resolution $X_6$ is also achieved as a moduli space by crossing a boundary of $C_+$. More precisely, one can check that the hyperplane $\{\theta_0+\theta_1+\theta_3=0\}$ forms facets of both $C_+$ and $C_3$ and that the adjacent chamber $C'_3$ to $C_3$ regarding this facet corresponds to  the ray of $\sigma_\mathcal{V}$ generated by
$$w'_3=(2,1,2,-2,1,-1,2,0,3,1,-1,2).$$
Then the counterpart $C'_+$ of $C_+$ obtained by replacing $C_3$ with $C'_3$ satisfies
$$(\psi_{C'_3})_\mathbb{R}(C'_+)\supset\mathrm{Nef}(X_6).$$
Therefore, we obtain a GIT-chamber $C_6\subset C'_+$ such that $\mathfrak{M}_{C_6}\cong X_6$.
\qed
\end{ex}

\subsubsection{$D_5$-singularity, continuation of Example \ref{relations}} \label{3.5.2}
We next consider the case of $D_5$-singularity again (see Example \ref{relations}). Note that the construction of embeddings in Subsection \ref{3.3} is valid in dimension two since the moduli spaces for any generic $\theta$ are crepant (or minimal) resolutions of $\mathbb{C}^2/G$. We fix a basis of the regular representation of $G$ determined by the matrix presentations in Example \ref{relations} (see Subsection \ref{3.3}).

If we use $x,y$ as the standard coordinates of $V=\mathbb{C}^2$ again, the generic matrices for the $G$-constellation family over $\mathbb{C}^2$ are given as
{\small
$$\begin{aligned}
&A_0=\begin{pmatrix}x\\y\end{pmatrix},B_0=\begin{pmatrix}-y&x\end{pmatrix},
A_1=\begin{pmatrix}-i y\\x\end{pmatrix},B_1=\begin{pmatrix}x&i y\end{pmatrix},
A_2=\begin{pmatrix}x\\-y\end{pmatrix},B_2=\begin{pmatrix}y&x\end{pmatrix},\\
&A_3=\begin{pmatrix}i y\\x\end{pmatrix},B_3=\begin{pmatrix}x&-i y\end{pmatrix},
C=\begin{pmatrix}x&0\\0&y\end{pmatrix},D=\begin{pmatrix}-y&0\\0&x\end{pmatrix}
\end{aligned}$$
}for the choice of basis of the vector spaces $\mathrm{Hom}_{\mathbb{C} G}(V^*\otimes_\mathbb{C} \rho,\rho')$ in Example \ref{relations}. Here we regard the arrows of $Q_G$ as the corresponding matrices by abuse of notation. Note that these matrices satisfy the condition (\ref{path}). We take $Ab(G)^\vee$-homogeneous elements
$$f_1=x^3+iy^3,\quad f_2=xy, \quad f_3=x^3-iy^3$$
so that $\mathbb{C}[V]^{[G,G]}=\mathbb{C}[f_1,f_2,f_3]$. Note that $g\in G$ acts on $f_k$ via the character $\rho_k$. As is shown in \cite[\S6]{D} and \cite[Example 1]{Y1}, these $f_k$'s give rise to generators of the Cox ring of the minimal resolution $X$ of $V/G$. We will consider the embedding of $\mathcal{W}$ into a toric variety with respect to these $f_k$'s.

Let us consider the semi-invariants obtained as the determinants of the following 42 square (block) matrices:
{\small
$$\begin{aligned}
&C,D, B_2A_0,B_0A_2,B_3A_1,B_1A_3, \begin{pmatrix}A_0&A_2\end{pmatrix},\begin{pmatrix}B_0\\B_2\end{pmatrix}, \begin{pmatrix}A_1&A_3\end{pmatrix},\begin{pmatrix}B_1\\B_3\end{pmatrix}, B_1CA_0, B_3CA_0, \\
&B_1CA_2, B_3CA_2, B_0DA_1, B_2DA_1, B_0DA_3, B_0DA_3, \begin{pmatrix}CA_0&A_1\end{pmatrix},\begin{pmatrix}CA_0&A_3\end{pmatrix},\\
&\begin{pmatrix}CA_2&A_1\end{pmatrix},\begin{pmatrix}CA_2&A_3\end{pmatrix},\begin{pmatrix}A_0&DA_1\end{pmatrix},\begin{pmatrix}A_0&DA_3\end{pmatrix},\begin{pmatrix}A_2&DA_1\end{pmatrix},\begin{pmatrix}A_2&DA_3\end{pmatrix},\\
&\begin{pmatrix}B_0\\B_1C\end{pmatrix}, \begin{pmatrix}B_0\\B_3C\end{pmatrix},\begin{pmatrix}B_2\\B_1C\end{pmatrix},\begin{pmatrix}B_2\\B_3C\end{pmatrix},\begin{pmatrix}B_0D\\B_1\end{pmatrix},\begin{pmatrix}B_0D\\B_3\end{pmatrix},\begin{pmatrix}B_2D\\B_1\end{pmatrix}, \begin{pmatrix}B_2D\\B_3\end{pmatrix},\begin{pmatrix}A_0&D\\0&B_1\end{pmatrix}, \\
&\begin{pmatrix}A_0&D\\0&B_3\end{pmatrix}, \begin{pmatrix}A_2&D\\0&B_1\end{pmatrix}, \begin{pmatrix}A_2&D\\0&B_3\end{pmatrix},\begin{pmatrix}B_0&0\\mathbb{C}&A_1\end{pmatrix}, \begin{pmatrix}B_0&0\\mathbb{C}&A_1\end{pmatrix}, \begin{pmatrix}B_2&0\\mathbb{C}&A_3\end{pmatrix}, \begin{pmatrix}B_2&0\\mathbb{C}&A_3\end{pmatrix}.
\end{aligned}$$
}We will see that these semi-invariants give a generating system of $\mathbb{C}[\mathcal{W}]=\mathbb{C}[\mathcal{V}]^{SL_R}$. Although it might be possible to check this directly (with help of a computer) by using algorithms such as the one in \cite[4.1]{DeK}, we adopt another approach.

In order to construct $Y_\mathcal{W}$ explicitly, we use coordinates $\theta_0,\dots,\theta_5$ of $\theta\in\Theta$ such that $\theta_i=\theta(\rho_i)$ for $i=0,1,2,3$ and $\theta_i=\theta(V_{i-3})$ for $i=4,5$. We then identify $\Theta$ with $\mathbb{R}^5$ via $\theta\mapsto (\theta_1,\dots,\theta_5)$. Note that we have $\theta_0+\theta_1+\theta_2+\theta_3+2\theta_4+2\theta_5=0$ in this case. Each of the 42 semi-invariants above is mapped to one of $f_i$'s under $\varphi$ up to constant multiplication, and we construct an embedding $\mathcal{W}\hookrightarrow Y_\mathcal{W}$ with respect to $f_i$'s. For example, the associated vectors in $M_\mathcal{W}\subset\mathbb{Z}^{3+5}$ to the semi-invariants for the matrices $C$ and $\begin{pmatrix}B_2&0\\mathbb{C}&A_3\end{pmatrix}$ are given as $(0,1,0,0,0,0,-1,1)$ and $(1,0,0,0,1,-1,-1,1)$ respectively.

Let $\sigma'\subset \mathbb{R}^{3+5}$ be the dual cone to the vectors in $M_\mathcal{W}$ associated to the above semi-invariants. We will show that $\sigma'$ equals the cone $\sigma_\mathcal{W}\subset \mathbb{R}^8$ defining $\bar{Y}_\mathcal{W}$. Let $\mathcal{W}'$ be the spectrum of the subalgebra of $\mathbb{C}[\mathcal{W}]$ generated by the 42 semi-invariants above. Then we obtain  an embedding of $\mathcal{W}'$ into a toric variety $Y'_\mathcal{W}$ whose normalization is defined by $\sigma'$, similar to the construction of $\mathcal{W}\hookrightarrow Y_\mathcal{W}$. In fact one can check that $Y'_\mathcal{W}$ is normal by comparing $\sigma'^\vee\cap M_\mathcal{W}$ and the semi-group generated by the elements of $M_\mathcal{W}$ associated to the 42 semi-invariants. This computation can be done e.g. by using the package ``Normaliz" of Macaulay2 \cite{GS}.

We see, by calculation, that $\sigma'$ admits 165 rays. The primitive generators in $\mathbb{Z}^8$ of these rays include the first three coordinate vectors $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$, and the other 162 vectors are divided into five types by looking at their first three coordinates:
\begin{equation}\label{rays}
v_1=(1,2,1),\quad v_2=(1,1,1),\quad v_3=(3,2,3),\quad v_4=(3,2,5), \text{ or }v_5=(5,2,3).
\end{equation}
These five types correspond to the prime exceptional divisors $E_1,\dots,E_5$ of the minimal resolution $X$ defined by the conjugacy classes of $g_1,g_1^2,g_1^3,g_2,g_1g_2$ respectively (cf. \cite[Example 1]{Y1}). More precisely, the rays for $E_1,\dots,E_5$ in the fan of the ambient toric variety $Y$ of $X$ (with respect to $f_i$'s) are generated by the vectors in (\ref{rays}).

One can check that the following condition holds:
\begin{equation}\label{*}
\text{the polyhedral cone }p^{-1}(\mathbb{R}_{\ge0}v_k)\cap\sigma'\text{ is generated by rays of }\sigma'\text{ for each }k.
\end{equation}
This implies that $Y'_\mathcal{W}/\!/_\theta T_\Theta$ is isomorphic to the toric variety $Y$ for any generic $\theta\in\Theta$. Therefore, we have $\sigma_\mathcal{W}=\sigma'$, and the natural toric morphism $\bar{Y}_\mathcal{W}\to{Y}'_\mathcal{W}$ is an isomorphism. This also implies that the 42 semi-invariants generate $\mathbb{C}[\mathcal{W}]$.

Since that the $G$-Hilbert scheme is given by a stability condition $\theta_+$ satisfying $(\theta_+)_i>0$ for all $i>0$, we see that generators of the rays of $\sigma_\mathcal{W}$ corresponding the exceptional divisors $E_1,\dots,E_5$ of the $G$-Hilbert scheme are given by
{\small $$\begin{aligned}
&w_1=(1,2,1,-1,-2,-1,-2,-2),\quad &w_2=(1,1,1,-1,-1,-1,-2,-2), \\
&w_3=(3,2,3,-3,-2,-3,-4,-6),\quad &w_4=(3,2,5,-3,-2,-5,-4,-6),\\
&w_5=(5,2,3,-5,-2,-3,-4,-6)\end{aligned}$$
}respectively.  From these data, one can also explicitly compute the map $(\psi_C)_\mathbb{R}:\mathbb{R}^5\to\mathbb{R}^5$ associated to the GIT-chamber $C$ containing $\theta_+$.

\begin{rem}
In \cite[\S2]{Le}, the $G$-Hilbert schemes $X$ for binary dihedral groups are studied, and in particular the author introduced distinguished $G$-clusters which give rise to open affine covering of $X$. For example, the distinguished $G$-cluster $F$ of {\it type B} is defined by an homogeneous ideal of $\mathbb{C}[x,y]$ and is presented as the following $\mathbb{C}$-vector subspace:
$$\mathbb{C}\oplus(\mathbb{C} x\oplus\mathbb{C} y)\oplus\mathbb{C} xy\oplus(\mathbb{C} x^2\oplus\mathbb{C} y^2)\oplus\mathbb{C} x^3 \oplus \mathbb{C} y^3\oplus(\mathbb{C} x^4\oplus\mathbb{C} y^4)\oplus(\mathbb{C} x^5\oplus\mathbb{C} y^5).$$
Its structure as a $\mathbb{C}[V]$-module is also presented as in the following diagram:
\[\xymatrix@=24pt{
\rho_0 \ar[d]_-{}
&
&\rho_1 \ar[dr]_{}
&
&\\
V_1 \ar[d]_{} \ar[r]_{}
&V_2 \ar[ur]_{} \ar[dr]_{}
&
&V_2 \ar[r]_{}
&V_1\\
\rho_2
&
&\rho_3 \ar[ur]_-{}
&
&
}\]
where each arrow $\rho\to \rho'$ represents a nontrivial action of $V^*$ on $\rho$ into $\rho'$ similarly to the McKay quiver. One sees that, among the 42 semi-invariants, the ones $h_1,\dots,h_9$ for the matrices
{\footnotesize$$B_1CA_0, B_2A_0,B_3CA_0,\begin{pmatrix}CA_0&A_1\end{pmatrix},\begin{pmatrix}CA_0&A_3\end{pmatrix},\begin{pmatrix}A_0&DA_1\end{pmatrix},\begin{pmatrix}A_0&DA_3\end{pmatrix},\begin{pmatrix}A_0&D\\0&B_1\end{pmatrix}, \begin{pmatrix}A_0&D\\0&B_3\end{pmatrix}$$
}do not vanish at $F$, and thus the face of $\sigma_\mathcal{W}$ corresponding to $F$ is given as the intersection $\sigma_\mathcal{W}\cap\{w_{h_1}=\dots =w_{h_9}=0\}$ where $w_{h_i}\in M_\mathcal{W}$ is the associated element to $h_i$. This is equal to $\mathrm{Cone}(w_1,w_2 )$, and thus $F$ corresponds to the the intersection point of the curves $E_1$ and $E_2$.
\qed
\end{rem}

\subsubsection{Trihedral group of order 21}\label{3.5.3}
Here we deal with the following three-dimensional example:
{\small $$G=\left\langle
g_1=\begin{pmatrix}
\zeta_7&0&0\\
0&\zeta_7^2&0\\
0&0&\zeta_7^4
\end{pmatrix},
g_2=\begin{pmatrix}
0&0&1\\
1&0&0\\
0&1&0\end{pmatrix}
\right\rangle\subset SL_3(\mathbb{C})$$
}where $\zeta_7$ is the seventh root of unity. Then $\mathrm{Irr}(G)$ consists of three 1-dimensional representations $\rho_k:G\to\mathbb{C}^*\,(k=0,1,2)$ defined by $\rho_k(g_1)=1,\,\rho_k(g_2)=\omega^k$ where $\omega$ is the third root of unity, and two 3-dimensional representations $V_1,V_2$, the first one is the inclusion $G\subset SL_3(\mathbb{C})$ presented above and the other is presented as
{\small $$g_1\mapsto\begin{pmatrix}
\zeta_7^3&0&0\\
0&\zeta_7^6&0\\
0&0&\zeta_7^5
\end{pmatrix},\;
g_2\mapsto\begin{pmatrix}
0&0&1\\
1&0&0\\
0&1&0
\end{pmatrix}.$$
}We again fix the basis obtained in this way for the subsequent computations.

The McKay quiver $Q_G$ for this case is as follows:

{\large
\[\xymatrix@=60pt{
\rho_0 \ar[d]_-{A_0}
& \rho_1 \ar[ld]_(.6){A_1}
&\rho_2 \ar[lld]_(.15){A_2}\\
V_1\ar@(l,d)[]_{L_1} \ar@<0.5ex>[rr]^-{C_i}
&
&V_2\ar@(r,d)[]^{L_2} \ar[lu]_(.35){B_1} \ar[u]_-{B_2} \ar@<0.5ex>[ll]^-{D} \ar[llu]^(.3){B_0}
}\]
}
where $C_i$ stands for two arrows $C_1$ and $C_2$.

With the standard coordinates $x,y,z$ of $V=\mathbb{C}^3$, the generic matrices are given as
{\small $$\begin{aligned}
&A_0=\begin{pmatrix}x\\y\\z\end{pmatrix},B_0=\begin{pmatrix}z&x&y\end{pmatrix},
A_1=\begin{pmatrix}x\\\omega y\\\omega^2 z\end{pmatrix},B_1=\begin{pmatrix}\omega z&x&\omega^2 y\end{pmatrix},\\
&A_2=\begin{pmatrix}x\\\omega^2 y\\\omega z\end{pmatrix}, B_2=\begin{pmatrix}\omega^2 z&x&\omega y\end{pmatrix}, C_1=\begin{pmatrix}0&x&0\\0&0&y\\z&0&0\end{pmatrix}, C_2=\begin{pmatrix}y&0&0\\0&z&0\\0&0&x\end{pmatrix}, \\
&D=\begin{pmatrix}0&y&0\\0&0&z\\x&0&0\end{pmatrix}, L_1=\begin{pmatrix}0&0&z\\x&0&0\\0&y&0\end{pmatrix},L_2=\begin{pmatrix}0&z&0\\0&0&x\\y&0&0\end{pmatrix}
\end{aligned}$$
}for some choice of basis $\mathrm{Hom}_{\mathbb{C} G}(\rho\otimes_\mathbb{C} V^*,\rho')$ with $\rho,\rho'\in\mathrm{Irr}(G)$.

In this case, it seems hard to obtain generators of semi-invariants directly using the algorithm in \cite[Ch. 4]{DeK} since the input data is large. Instead we give a candidate of a system of generators and then confirm the correctness as we did in the previous example.

As computed in \cite[\S5.1]{DG}, there are $Ab(G)^\vee$-homogeneous polynomials $f_1,\dots,f_{13}\in\mathbb{C}[V]^{[G,G]}$ such that the associated elements
$$\tilde{f}_1,\dots,\tilde{f}_{13}, t_1^{-7}, t_2^{-3}, t_3^{-3}\in \mathbb{C}[V]^{[G,G]}[t_1^{\pm1}, t_2^{\pm1}, t_3^{\pm1}]$$
generate the Cox ring of a crepant  resolution $X$ of $V/G$, where we set the variables $t_1, t_2, t_3$ so that they correspond to junior elements $g_1, g_2, g_2^2$ respectively (see Subsection \ref{3.1}).

Similarly to the previous examples, we realize the cone $\sigma_\mathcal{W}$ inside $\mathbb{R}^{13+5-1}$ by ordering the elements of $\mathrm{Irr}(G)$ as $\rho_0,\rho_1,\rho_2,V_1,V_2$. Regarding the embedding $\iota_X:X\hookrightarrow Y$ with respect to $f_1,\dots, f_{13}$, the generators of the rays in $\mathbb{R}^{13}$ corresponding to the exceptional divisors $E_1, E_2, E_3$ are given as
$$\begin{aligned}
&\mathbf{v}_1=(1,1,1,1,1,1,1,1,1,1,1,1,1),\\
&\mathbf{v}_2=(0,0,2,1,0,2,1,3,2,1,3,2,4),\\
&\mathbf{v}_3=(0,0,1,2,0,1,2,3,1,2,3,4,2)
\end{aligned}$$
respectively.
Then we define $\sigma'\subset\mathbb{R}^{17}$ as the cone generated by the coordinate vectors $\mathbf{e}_1, \dots, \mathbf{e}_{13}$ and the following 16 vectors:
$$\begin{aligned}
&\mathbf{v}_{1,1}=(\mathbf{v}_1, -1,-1,-1,-2),\; \mathbf{v}_{1,2}=(\mathbf{v}_1, 0,-1,-1,-2),\;\mathbf{v}_{1,3}=(\mathbf{v}_1, -1,0,-1,-2),\\
&\mathbf{v}_{1,4}=(\mathbf{v}_1, 0,0,-1,-2),\;\mathbf{v}_{1,5}=(\mathbf{v}_1, 0,0,-1,0),\;\mathbf{v}_{1,6}=(\mathbf{v}_1, 0,0,0,1),\\
&\mathbf{v}_{1,7}=(\mathbf{v}_1, 0,0,2,1),\; \mathbf{v}_{1,8}=(\mathbf{v}_1, 0,1,2,1),\;\mathbf{v}_{1,9}=(\mathbf{v}_1, 1,0,2,1),\; \mathbf{v}_{1,10}=(\mathbf{v}_1, 1,1,2,1),\\
&\mathbf{v}_{2,1}=(\mathbf{v}_2,-1,-2,-3,-3),\;\mathbf{v}_{2,2}=(\mathbf{v}_2,-1,1,0,0),\;\mathbf{v}_{2,3}=(\mathbf{v}_2,2,1,3,3),\\
&\mathbf{v}_{3,1}=(\mathbf{v}_3,-2,-1,-3,-3),\;\mathbf{v}_{3,2}=(\mathbf{v}_3,1,-1,0,0),\;\mathbf{v}_{3,3}=(\mathbf{v}_3,1,2,3,3).
\end{aligned}$$

By computing $\sigma'$ explicitly (with help of Macaulay2 \cite{GS}), it has 400 facets and accordingly we obtain 400 primitive elements of $M_\mathcal{W}$ defining these facets. In fact we can take determinantal semi-invariants corresponding to these primitive elements. One can also check that the toric variety $Y'_\mathcal{W}$ constructed from these elements is already normal again. Since the condition (\ref{*}) holds also in this case, we have that $\sigma'=\sigma_\mathcal{W}$ and that the associated 400 semi-invariants generate $\mathbb{C}[\mathcal{W}]$.

\begin{rem}
In general, a candidate $\sigma'$ of $\sigma_\mathcal{W}$ can be given by computing determinantal semi-invariants $h$ (from ones with $\varphi(h)$ having smaller degrees) until the dual cone
$$\sigma':=\bigcap_h\{v\in\mathbb{R}^\ell\oplus \Theta^\vee\mid w_h(v)\ge0\}$$
satisfies the condition (\ref{*}). This is achieved by computing relatively small number of semi-invariants (compared to the number of generators of $\mathbb{C}[\mathcal{W}]$). In the above case, such $\sigma'$ is obtained from determinantal semi-invariants coming from $3\times3$-matrices only, while the 400 generators of $\mathbb{C}[\mathcal{W}]$ include ones coming from $6\times6$-matrices. One may also verify that $\sigma_\mathcal{W}=\sigma'$ by checking the existence of $G$-constellations corresponding to rays of $\sigma'$, as will  be done below.
\qed
\end{rem}

Once we know the cone $\sigma_\mathcal{W}$, one can investigate the structures of $G$-constellations presented by faces of $\sigma_\mathcal{W}$. As an demonstration, we take the ray generated by $\mathbf{v}_{1,1}$. If we take the generators $\mathbf{v}_{1,1},\dots,\mathbf{v}_{1,10}$ of $\sigma'_\mathcal{W}$ whose former coordinates is $\mathbf{v}_1$, the orbit cone $C_{\mathbf{v}_{1,j}}\subset\Theta$ of the corresponding divisor to $\mathbf{v}_{1,j}$ is given as the cone
$$\{\theta\in\Theta \mid p(\mathbf{v}_{1,j})\cdot\theta=\underset{j=1,\dots,10}{\mathrm{min}}p(\mathbf{v}_{1,j})\cdot\theta\}$$
by Proposition \ref{minimize}. Note that $\{C_{\mathbf{v}_{1,j}}\}_{j=1,\dots,10}$ give a subdivision of $\Theta$ into polyhedral cones. The orbit cone $C_{\mathbf{v}_{1,1}}\subset\Theta$ is defined by the following 5 inequalities:
$$\theta_1\ge0,\;\theta_2\ge0,\;\theta_1+\theta_2+2\theta_4\ge0,\;\theta_1+\theta_2+\theta_3+3\theta_4\ge0,\;\theta_1+\theta_2 +3\theta_3  +3\theta_4\ge0.$$
Notice that the coefficients of these inequalities are obtained as the direction vectors $\overrightarrow{\mathbf{v}_{1,1}\mathbf{v}_{1,j}}$ for $j$ such that the line segment $\overline{\mathbf{v}_{1,1}\mathbf{v}_{1,j}}$ forms an edge of the convex hull $\mathrm{Conv}(\mathbf{v}_{1,1},\dots,\mathbf{v}_{1,10})$.  One can infer the structure of the $G$-constellation $F$ of a general point of $E_1$ from these inequalities. In this case the structure of $F$ is presented as the following diagram for some decomposition $F=\rho_0\oplus\rho_1\oplus\rho_2\oplus V_1^{\oplus3}\oplus V_2^{\oplus3}$:

\[\xymatrix@=48pt{
\rho_0 \ar[r]_-{}
&V_1 \ar[r]_{} \ar[d]_{}
&V_2 \ar[d]_{}
&V_2 \ar[r]_{} \ar[dr]_{}
&\rho_1 \\
&V_1 \ar[r]_{}
&V_1 \ar[r]_{} \ar[ur]_{}
&V_2 \ar[r]_{} \ar[ur]_{}
&\rho_2.
}\]
As another example, the orbit cone $C_{\mathbf{v}_{1,6}}$ for $\mathbf{v}_{1,6}=(\mathbf{v}_1, 0,0,0,1)$ is defined by
$$\theta_3\ge0,\;\theta_1+2\theta_3\ge0,\;\theta_2+2\theta_3\ge0,\;\theta_1+\theta_2+2\theta_3\ge0,$$
$$\theta_3+\theta_4\le0,\;\theta_1+\theta_3+3\theta_4\le0,\; \theta_2+\theta_3 +3\theta_4\le0,\;\theta_1+\theta_2+\theta_3+3\theta_4\le0$$
and the structure of the $G$-constellation $F$ for $\mathbf{v}_{1,6}$ is presented as

\[\xymatrix@=36pt{
V_2 \ar[r]_{} \ar[d]_{} \ar[r]_{}
&V_2 \ar[d]_{}
&\rho_0 \ar[r]_{} \ar[dr]_{}
&V_1 \\
V_1 \ar[r]_{}
&V_2 \ar[r]_{} \ar[ur]_{} \ar[dr]_{}
&\rho_1 \ar[r]_{} \ar[ur]_{}
&V_1. \\
&
&\rho_2 \ar[ur]_{} \ar[uur]_{}
&
}\]
As a more explicit description, $F$ can be realized as a quotient of the ideal of $\mathbb{C}[V]$ generated by $xy,yz$ and $zx$. Note that $\mathbb{C} xy\oplus\mathbb{C} yz\oplus\mathbb{C} zx$ is isomorphic to $V_2$ as a $\mathbb{C} G$-module.

Finally, we compute the homomorphism $\psi_\theta$ for certain $\theta$ and see that every projective crepant resolution of $V/G$ is obtained as a moduli space of $G$-constellations. Let $C_{i,j}\subset \Theta$ be the orbit cones corresponding to the rays of $\sigma_\mathcal{W}$ generated by $\mathbf{v}_{i,j}$. There are 42 choices of a triple $(i,j,k)$ such that $\overline{C}_{1,i}\cap \overline{C}_{2,j}\cap \overline{C}_{3,k}$ has the full dimension 4. Among them, $C_+:=\overline{C}_{1,1}\cap \overline{C}_{2,1}\cap \overline{C}_{3,1}$ is the one containing the GIT-chamber $C_{++}$ for the $G$-Hilbert scheme. $C_{++}$ is in fact defined by the inequalities $\theta_i>0\,(i=1,2,3,4)$.

Similarly to the abelian case in Example \ref{abel2}, one can compute $\psi_{C_{++}}$ and obtain $\psi_{C_{++}}(C_+)=\mathrm{Mov}(X)$. Thus, every projective crepant resolution of $V/G$ is realized as $\mathfrak{M}_\theta$ for some generic $\theta\in C_+$. In this case the movable cone $\mathrm{Mov}(X)$ is a three-dimensional simplicial cone divided into four nef cones (cf. \cite[Proposition 5.2]{DG}). The nef cone for the $G$-Hilbert scheme is given as $\psi_{C_{++}}(\overline{C}_{++})$ and we see that it is the central one which can pass to the three remaining (corner) ones. Note that not all of $\overline{C}_{1,i}\cap \overline{C}_{2,j}\cap \overline{C}_{3,k}$ of dimension 4 realize all the resolutions. For example, $\psi_{C'}(C)$ is strictly smaller than $\mathrm{Mov}(X)$ for the cone $C=\overline{C}_{1,2}\cap \overline{C}_{2,1}\cap \overline{C}_{3,1}$ adjacent to $C_+$ and any GIT-chamber $C'\subset C$.

\section{Proof of the main result}\label{4}

In this section we give a proof of the following theorem:

\begin{thm}\label{main2}(=Theorem \ref{main1})
Let $G\subset SL_3(\mathbb{C})$ be a finite subgroup and $X\to\mathbb{C}^3/G$ a projective crepant resolution. Then there is a generic stability condition $\theta\in\Theta$ such that $X\cong \mathfrak{M}_\theta$.
\end{thm}

To prove Theorem \ref{main2}, we will use a similar argument to the one in \cite[\S8]{CI}. To this end, we should know how the or the tautological bundle $\mathcal{R}_C$ of $\mathfrak{M}_C$ changes when we cross a wall in $\Theta$. Here we say that the intersection $W=\overline{C}\cap\overline{C'}$ for GIT-chambers $C, C'\subset \Theta$ is a {\it wall} (of $C$) if $W$ is a codimension-one face of $C$ (and hence of $C'$).

For any given GIT-chamber $C\subset \Theta$, we choose any general point $\theta_0$ of a wall $W\subset C$. Then we have $\mathcal{W}^{C\text{-ss}}\subset\mathcal{W}^{\theta_0\text{-ss}}$ and this inclusion induces a morphism $\alpha:\mathfrak{M}_C\to\bar{\mathfrak{M}}_{\theta_0}$ over $\mathfrak{M}_0=\mathbb{C}^3/G$ where $\bar{}$ denotes the normalization. Similarly to the case of $\mathfrak{M}_C$, we denote the moduli space $\bar{\mathfrak{M}}_{\theta_0}$ by $\bar{\mathfrak{M}}_W$. We also denote the crepant resolution $\mathfrak{M}_C$ by $X$ as in previous sections.

Recall that each line bundle $L\in\mathrm{Pic}(X)$ defines a character $\chi_L\in \chi(T_X)$ and its associated GIT-quotient $X_L:=\mathfrak{X}/\!/_{\chi_L^k}T_X$ with a sufficiently divisible $k>0$. As observed in Subsection \ref{3.2}, $X_L$ is the same as the image of the morphism obtained by the complete linear system of (some power of) $L$. Note that $X_0=\mathbb{C}^3/G$ and that $X_L$ makes sense for $L\in\mathrm{Pic}(X)_\mathbb{R}$. One can determine $\alpha:\mathfrak{M}_{C}\to\bar{\mathfrak{M}}_W$ by looking at the homomorphism $\psi_C:\Theta\to \mathrm{Pic}(X)_\mathbb{R}$ constructed in Subsection \ref{3.3}:

\begin{lem}\label{descent}(cf. \cite[Lemma 3.3]{CI})
For any point $\theta\in \overline{C}$, the normalized GIT-quotient $\bar{\mathfrak{M}}_\theta$ is isomorphic to $X_{\psi_C(\theta)}$.
\end{lem}

{\it Proof.} Recall that $X=\mathfrak{M}_C$ is regarded as a GIT-quotient of $\mathfrak{X}=\mathrm{Spec}\,\mathrm{Cox}(X)$ via the restriction $\mathfrak{X}^{\psi_C(C)\text{-ss}}\to \mathcal{W}^{C\text{-ss}}$ of the equivariant map $\mathfrak{X}\to \mathcal{W}$ obtained from $\varphi_C$ (see Subsection \ref{3.3}). Then the line bundle $L:=\psi_C(\theta)$ is just the pullback of the ample line bundle $\mathcal{O}_{\bar{\mathfrak{M}}_\theta}(1)$ under the morphism $X\to\mathfrak{M}_\theta$ induced by GIT. Then the claim follows since $X\to\bar{\mathfrak{M}}_\theta$ is identified with the morphism coming from the linear system of $L$.
\qed

\vspace{3mm}

\begin{lem}\label{vertical}
Let $F\subset\Theta$ be a face of an orbit cone $\tilde{C}_x\subset\Theta$ of a $G$-constellation corresponding to a point $x\in\mathfrak{M}_C$. If the relative interior $\mathrm{relint}(F)$ of $F$ intersects with $\overline{C}$ and if $\psi_C(F)\subset\mathrm{Pic}(X)_\mathbb{R}$ is not of full-dimension, then $\psi_C(F)$ is on the boundary of the orbit cone $C_x\subset \mathrm{Pic}(X)_\mathbb{R}$ of a point of $\mathfrak{X}$ which is a lift of $x\in X$.
\end{lem}

{\it Proof.} We first choose homogeneous generators $h_1,\dots,h_\ell$ of $\mathbb{C}[\mathcal{W}]=\mathbb{C}[\mathcal{V}]^{SL_R}$ so that we have an associated embedding $\mathcal{W}\hookrightarrow Y_\mathcal{W}$ and its quotient $X\hookrightarrow Y$ as in Proposition \ref{embedding}. Let $\bar{x}\in\mathcal{W}$ be any lift of $x$. By reordering $h_i$'s, we may assume that\\
(1) there exists $\ell_1<\ell$ such that $h_1,\dots,h_{\ell_1}$ do not vanish at $\bar{x}$ and their weights generate the cone $F$ and that\\
(2) for any $h_k$ with $k>\ell_1$, either $h_k$ vanishes at $\bar{x}$ or $\mathrm{wt}(h_k)$ is not in $F$.\\
The locally closed subset of $\mathcal{W}$ consisting of points having $F$ as their orbit cones is given by
$$\mathcal{W}_F:=\mathcal{W}\cap \{h_1 h_2\cdots h_{\ell_1}\ne0\}\cap \{h_{\ell_1+1}=\cdots=h_{\ell}=0\}.$$
Note that torus-orbits of the toric varieties $Y,\,\mathbb{A}^{\ell+m}$, and $Y_\mathcal{W}$ restrict to stratification of $X, \mathfrak{X}$, and $\mathcal{W}$ respectively, and that $\mathcal{W}_F$ is such a stratum of $\mathcal{W}$.

By construction, the equivariant map $\varphi_C^*:\mathfrak{X}\to \mathcal{W}$ associated to $C$ is a restriction of a toric morphism $\mathbb{A}^{\ell+m}\to Y_\mathcal{W}$ which is obtained by sending the rays $\tau_k$ of the fan of $\mathbb{A}^{\ell+m}$ corresponding to the exceptional divisors $E_k$ to the rays $\tau_{C,k}$ of $\sigma_\mathcal{W}$ corresponding to $E_k$ in $\mathfrak{M}_C$. By Lemma \ref{descent}, $\bar{\mathfrak{M}}_{\theta_0}$ is isomorphic to $X_{\psi_C(\theta_0)}$ for $\theta_0\in \mathrm{relint}(F) \cap \overline{C}$. We ``perturb" the map $\varphi_C^*$ to obtain another equivariant map $\varphi'^*:\mathfrak{X}\to \mathcal{W}$ such that it still induces an isomorphism $X_{\psi_C(\theta_0)}\to \bar{\mathfrak{M}}_{\theta_0}$ and that every point of $\mathcal{W}_F$ is inside a $T_\Theta$-orbit of a point in the image of $\varphi'^*$. Such $\varphi'^*$ is explicitly constructed as follows. For each $k=1,\dots,m$, let $\sigma_k$ be the face of $\sigma_\mathcal{W}$ generated by $\tau_{C,k}$ and the rays $\tau$ of $\sigma_\mathcal{W}$ (if any) such that $\tau$ is projected to $\tau_k$ under $p:(N_\mathcal{W})_\mathbb{R}\to (N_\mathcal{W}/\bar{\Theta}^\vee)_\mathbb{R}$ (see Subsection \ref{3.4}) and that the orbit cone of $\tau$ contains $F$ as a face. Then $\varphi'^*$ is obtained as a restriction of the toric morphism $\mathbb{A}^{\ell+m}\to Y_\mathcal{W}$ obtained by sending each ray $\tau_k$ to a ray generated by a general point of $\sigma_k$. Note that $\varphi'^*$ gives an associated homomorphism $\psi': \Theta\to \mathrm{Pic}(X)_\mathbb{R}$ similarly to $\psi_C$. More concretely, for a homogeneous semi-invariant $f\in\mathbb{C}[\mathcal{W}]$ of weight $c\in \Theta$, $\psi'(c)$ is the class of the sum of the strict transform of the image in $V/G$ of $\{f=0\}$ and $\sum_k v_k(f)E_k$ where $v_k$ is the primitive element in $\sigma_k$ defining $\varphi'^*$ (cf. Remarks \ref{cox(V/G)} and \ref{vphi}(1)).

One sees that $\psi'|_F=\psi_C|_F$ and in particular $\psi'(F)$ is not full-dimensional by the assumption. Let $\varphi': \mathbb{C}[\mathcal{W}]\to \mathbb{C}[\mathfrak{X}]$ be the ring homomorphism associated to $\varphi'^*$. The equivariant map $\varphi'^*$ restricts to a map from the nonempty (affine) locally closed subset
$$\mathfrak{X}_F:=(\varphi'^*)^{-1}(\mathcal{W}_F)=\mathfrak{X}\cap \{\varphi'(h_1 h_2\cdots h_{\ell_1})\ne0\}\cap \{\varphi' (h_{\ell_1+1})=\cdots=\varphi_C(h_{\ell})=0\}$$
of $\mathfrak{X}$ to $\mathcal{W}_F$, which descends to an isomorphism between the quotients by $T_X$ and $T_\Theta$.

We assume that $\psi'(F)$ does not lie on the boundary of $C_x$ to deduce a contradiction. Then the quotients of $\mathfrak{X}_F$ and $\mathcal{W}_F$ are isomorphic to the stratum $S_x$ of $X$ containing $x$. If $\mathfrak{X}_F$ itself is a stratum of $\mathfrak{X}$, then it must be the set $\mathfrak{X}_x$ of points of $\mathfrak{X}$ whose orbit cones is $C_x$. The fact that $\psi'(F)$ is not of full-dimension implies that every point of $\mathfrak{X}_F$ has a positive-dimensional stabilizer in $T_X$, which is a contradiction since $C_x$ is full-dimensional. If there exists a stratum $\mathfrak{X}_{x'}\subsetneq\mathfrak{X}_F$ not equal to $\mathfrak{X}_x$, then this is again a contradiction since the orbit cone of such $\mathfrak{X}_{x'}$ is a proper face of $C_x$ and in particular $\mathfrak{X}_{x'}$ cannot be mapped to $S_x$ under the quotient map.
\qed

\vspace{3mm}

If a wall $W=\overline{C}\cap\overline{C'}$ is not contained in the pullback of a hyperplane of $\mathrm{Pic}(X)_\mathbb{R}$ under $\psi_C$, then the morphism $\mathfrak{M}_C\to\bar{\mathfrak{M}}_{W}$ is an isomorphism by Lemma \ref{descent}, and we call such a wall of {\it Type 0} following \cite{CI}. Conversely, for any wall $W$ whose image under $\psi_C$ is not of full dimension, the morphism $\mathfrak{M}_C\to\bar{\mathfrak{M}}_{W}$ is a non-trivial contraction of either a divisor or a curve by Lemma \ref{vertical}. For a wall $W$ of Type 0, the two moduli spaces $\mathfrak{M}_C$ and $\mathfrak{M}_{C'}$ are isomorphic but the tautological bundle changes. In fact $\mathcal{R}_{C'}$ is a modification of $\mathcal{R}_C$ by tensoring a line bundle to a subbundle of $\mathcal{R}_C$ similar to the abelian cases \cite[Corollary 4.2]{CI}, as we will see below.

In order to study what happens when we cross a wall $W$ of Type 0, we first see that the {\it unstable locus}
$$D_W=\{x\in X\mid \text{any lift }\bar{x}\in\mathcal{W}\text{ of }x\text{ is not }W\text{-semistable}\}$$
is a reduced compact divisor. Note that $D_W$ has a natural scheme structure as the set of zeros of sections of line bundles. Later we will also see that $D_W$ is connected (cf. Proposition \ref{twist}).

\begin{lem}\label{divisor}
For any given GIT-chamber $C\subset\Theta$ and its wall $W$ of Type 0, the  unstable locus $D_W$ is a reduced compact divisor.
\end{lem}

{\it Proof.} We first show that $D_W$ is reduced. Note that $D_W$ is a geometric quotient of (an open subset of) a closed subscheme $\tilde{D}_W$ of $\mathcal{W}$. By the definition of the stability, $\tilde{D}_W$ is the zero locus of the semi-invariants which vanish at the closed points of $\tilde{D}_W$. Since semi-invariants are nothing but regular functions on the affine variety $\mathcal{W}$, the subscheme $\tilde{D}_W$ is reduced and so is $D_W$.

Next we show that $D_W$ is a divisor. We fix an embedding $\mathcal{W}\hookrightarrow Y_\mathcal{W}$ again, and let $\Sigma_\theta$ be the subfan of $\sigma_\mathcal{W}$ defining $Y_\mathcal{W}^{\theta\text{-ss}}$ for $\theta\in \Theta$ (see Subsection \ref{3.4}). We also consider the ambient toric variety $Y$ of $X$ and let $\sigma_x$ be the cone of the fan $\Sigma_X$ of $Y$ corresponding to a point $x\in X\subset Y$.

For any point $x\in D_W$, there exists a face $\tilde{\sigma}_x\in \Sigma_\theta$ for $\theta\in C$ such that $p(\tilde{\sigma}_x)=\sigma_x$. Since $\mathfrak{M}_C\to\bar{\mathfrak{M}}_W$ is an isomorphism, we have a cone $\bar{\sigma}_x\in\Sigma_{\theta_0}$ containing $\tilde{\sigma}_x$ (as a facet) such that it satisfies $p(\bar{\sigma}_x)=\sigma_x$ where $\theta_0$ is a general point of $W$. Recall that the maximal cones of the fan of $\mathcal{W}^{W\text{-ss}}$ are projected under $p$ to the maximal cones of $\Sigma_X$ (see Subsection \ref{3.4}). Since the image of any ray of $\sigma_\mathcal{W}$ under $p$ is again a ray of $\Sigma_X$, there must be a ray $\tau$ of $\tilde{\sigma}_x$ and a face $\bar{\tau}\subset \bar{\sigma}_x$ such that $\bar{\tau}\supsetneq\tau$ and $p(\tau)=p(\bar{\tau})$. This implies that $x$ is contained in the unstable divisor of $X$ corresponding to $p(\bar{\tau})$. Thus, $D_W$ is of pure codimension one.

For the compactness of $D_W$, one may apply exactly the same argument in \cite[Proposition 4.4]{CI}.
\qed

\vspace{3mm}

Now we describe how $G$-constellations over unstable points change when we cross the wall $W$. For convenience, we consider corresponding representations of $Q_G$ rather than $G$-constellations. Let $F$ be the representation of $Q_G$ corresponding to any point $x\in D_W \subset X=\mathfrak{M}_C$. Then there exists a unique maximal $\mathbb{C}[V]$-submodule $S\subset F$ such that $\theta_0(S)=0$ for any $\theta_0$ of $W$, and let $Q$ be the quotient of $F$ by $S$. Since the regular representation $R$ is primitive as an element of $R(G)$, at least one of $S$ and $Q$ is primitive as well. We assume $Q$ is primitive since the other case can be treated similarly by switching the roles of $C$ and $C'$. Then $Q$ is indecomposable since it is $\theta_0$-semistable for general $\theta_0\in W$. Note that $S$ and $Q$ as $\mathbb{C} G$-modules may a priori vary on different connected components of $D_W$ if $S$ is imprimitive in $R(G)$, but the primitiveness of the quotient $Q$ is independent of the choice of $x$.

We take the 1-PS $\lambda: \mathbb{C}^*\to GL_R$ defined by $Q$ (for a fixed connected component of $D_W$), that is, $\lambda$ is positive on $C$ and zero on $W$ as a function on $\Theta\cong\chi(GL_R)_\mathbb{R}$. Then $\lambda$ induces a filtration of $F$ by its submodules (cf. \cite[\S3]{Kin}) such that $Q$ is one of its associated graded component. This works for a family of representations of $Q_G$ and thus we obtain an exact sequence
\begin{equation}\label{exact1}
0\to\mathcal{S}\to \mathcal{R}_C|_{D_W}\to \mathcal{Q}\to0
\end{equation}
of flat families $\mathcal{S}$ and $\mathcal{Q}$ on $\mathbb{C}^3$ corresponding to $S$ and $Q$ respectively. Note that this sequence splits as vector bundles on $D_W$, whereas it is a nontrivial extension as $\mathbb{C}[V]$-modules. The ranks of $\mathcal{S}$ and $\mathcal{Q}$ as vector bundles on $D_W$ a priori vary on different connected components of $D_W$ but it will be shown that $D_W$ is connected in Proposition \ref{twist}. Let $\mathcal{K}$ be the kernel of the natural surjection $\mathcal{R}_C\to \mathcal{Q}$. Then, applying the inverse of the elementary transformation (see \cite{M}) to $\mathcal{K}$, we obtain an exact sequence
\begin{equation}\label{exact2}
0\to\mathcal{Q}\otimes \mathcal{O}_X(-D_W) \to \mathcal{K}|_{D_W}\to \mathcal{S}\to0
\end{equation}
of flat families on $\mathbb{C}^3$. Note that $\mathcal{K}$ parametrizes the same representations of $Q_G$ as $\mathcal{R}_C$ outside $D_W$ while sub and quotient modules are switched over $D_W$.

\begin{lem}\label{tautological bundle}
The family $\mathcal{K}$ coincides with the tautological bundle $\mathcal{R}_{C'}$ for the GIT-chamber $C'$ up to tensoring a line bundle on $X$.
\end{lem}

{\it Proof.} First recall that the bundle $\mathcal{R}_{C'}=\bigoplus_{\rho\in\mathrm{Irr}(G)} \mathcal{R}'_\rho$ is normalized so that its component $\mathcal{R}'_{\rho_0}$ becomes the trivial line bundle. In order to see that $\mathcal{K}$ parametrizes $\theta'$-(semi)stable representations of $Q_G$ for $\theta'\in C'$, we only have to show that the fibers of $\mathcal{K}|_{D_W}$ are indecomposable as $\mathbb{C}[V]$-modules. To do this, it suffices to show that the fibers of $\mathcal{K}|_{D_W}$ are indecomposable over a Zariski open dense subset of $D_W$ since $\mathcal{K}$ and $\mathcal{R}_{C'}$ are vector bundles on a smooth variety.

Let $x\in D_W$ be a general point contained in a single irreducible component $D\subset D_W$ and let $F$ be the corresponding representation of $Q_G$ in $\mathfrak{M}_C$. We first show that the fiber $S$ of $\mathcal{S}$ at $x$ is indecomposable. We assume otherwise to deduce a contradiction. Then one can take a 1-PS $\lambda$ of $GL_R$ such that $\lim_{t\to0}(\lambda(t)\cdot F)$ admits exactly two direct irreducible summands one of which is a nontrivial irreducible direct summand $S_1$ of $S$. Since having a nontrivial direct summand (of a fixed $\mathbb{C}[V]$-module structure) is a closed condition, the limits exist for all points of $D$ by the genericity of $x$. Therefore, similarly to (\ref{exact1}), we have a exact sequence
$$0\to\mathcal{S}_1\to \mathcal{R}_C|_D\to \mathcal{Q}_1\to0$$
of families corresponding to the sequence $0\to S_1\to F\to Q_1\to0$. Then applying the inverse of the elementary transformation again, we obtain a representation $F'$ of $Q_G$ fitting into the exact sequence $0\to Q_1\to F_1\to S_1\to0$. By construction $F_1$ admits sub and quotient modules whose values for $\theta_0\in W$ are zero, and hence $F_1$ is strictly $\theta_0$-semistable. Since every strictly semistable point is a fixed point for a nontrivial 1-PS of $PGL_R$ by the Hilbert-Mumford numerical criterion, $F_1$ must be decomposable. However, this is a contradiction since the same argument as in the proof of \cite[Proposition 4.1]{CI} shows that $F_1$ is a nontrivial extension of $Q_1$ by $S_1$ and hence indecomposable.

Thus the fiber $S$ of $\mathcal{S}$ at $x$ is indecomposable, and the same argument as above shows that the fiber of $\mathcal{K}$ at $x$ is indecomposable as well.
\qed

\vspace{3mm}

\begin{rem}
In abelian cases every irreducible representation of $G$ has multiplicity one in $R$ and thus the modules $S$ and $Q$ are clearly indecomposable. When $G$ is non-abelian and $S$ is non-primitive in $R(G)$, the above lemma shows that $S$ and $Q$ are still indecomposable at least for general points $x\in D_W$. However, the $\theta_0$-polystable representation $\bar{F}$ of $Q_G$ (i.e. the direct sum of $\theta_0$-stable modules) corresponding to the image of $x$ under $\mathfrak{M}_C\to\mathfrak{M}_W$ is not isomorphic to the direct sum $S\oplus Q$ in general. $S$ may degenerate in $\bar{F}$ to decompose into a direct sum $\bigoplus_i S_i$ satisfying $\theta_0 (S_i)=0$.
\qed
\end{rem}

Next we describe the behavior of the Grothendieck group $K(X)$ of locally free sheaves on $X$ under the wall-crossing. As we will see below, it is basically the same as the abelian cases and this enables us to proceed the proof of the main result in a similar way to \cite{CI}. To this end we briefly review the necessary notions. See \cite[\S2.4, \S5, and \S7.1]{CI} and also \cite[\S9.2]{BKR} for details.

The main result of \cite{BKR} states that the Fourier-Mukai transform by the universal family $\mathcal{U}_C$ gives an equivalence $\Phi_C: D(X)\cong D^G(\mathbb{C}^3)$ whose inverse $\Phi^{-1}_C: D^G(\mathbb{C}^3)\to D(X)$ is given by $\mathcal{O}_{\mathbb{C}^3}\otimes\rho\mapsto \mathcal{R}^\vee_\rho$. $\Phi_C$ induces an isomorphism between $K(X)$ and the Grothendieck group $K^G(\mathbb{C}^3)$ of $D^G(\mathbb{C}^3)$. Let $K_0(X)$ be the subgroup of $K(X)$ generated by the classes of complexes whose supports are contained in the fiber $\pi^{-1}(0)$ for $\pi:X \to \mathbb{C}^3/G$. Then $K_0(X)$ is identified with the dual of $K(X)$ via the perfect pairing $\chi(-, -): K(X)\times K_0(X)\to\mathbb{Z}$ defined by
$$(\alpha, \beta)\mapsto \sum_i (-1)^i \dim \mathrm{Ext}^i(\alpha, \beta),$$
which is skew-symmetric on $K_0(X)$. Via the equivalence $\Phi_C$ we also have a perfect pairing $\chi^G(-,-)$ between $K^G(\mathbb{C}^3)$ and the Grothendieck group $K^G_0(\mathbb{C}^3)$ of $G$-equivariant sheaves on $\mathbb{C}^3$ supported at the origin. With respect to this pairing, the classes of $\{\mathcal{O}_{\mathbb{C}^3}\otimes \rho\}_{\rho\in\mathrm{Irr}(G)}$ and $\{\mathcal{O}_0 \otimes \rho\}_{\rho\in\mathrm{Irr}(G)}$ give mutually dual basis of $K(\mathbb{C}^3)$ and $K_0(\mathbb{C}^3)$ respectively where $\mathcal{O}_0$ is the skyscraper sheaf at the origin of $\mathbb{C}^3$.

The equivalence $\Phi_C$ also induces an isomorphism $\phi_C: K_0(X)\to K^G_0(\mathbb{C}^3)$ and its pullback $\phi_C^*: K^G(\mathbb{C}^3)\to K(X)$. The vector space $\Theta$ is regarded as a subspace of $K^G(\mathbb{C}^3)_\mathbb{R}$, and the restriction of $\phi_C^*$ to $\Theta$ gives an isomorphism $\Theta\to F^1$ where we have a filtration
$$K(X)_\mathbb{R}=F^0\supset F^1\supset F^2\supset F^3=0$$
with $F^i$ being the subspace generated by sheaves whose supports are of codimension at least $i$. The homomorphism $\phi_C^*|_\Theta:\Theta \to F^1$ is explicitly given as
$$\theta=(\theta_\rho)_\rho\mapsto \sum_{\rho\in\mathrm{Irr}(G)} \theta_\rho[\mathcal{R}_\rho^\vee]$$
where $[\mathcal{R}_\rho^\vee]$ denotes the class of the dual sheaf $\mathcal{R}_\rho^\vee$ in $F^1$.
Then we obtain a homomorphism $L_C:\Theta \to \mathrm{Pic}(X)_\mathbb{R}$ such that
the following diagram of vector spaces is commutative:
\begin{equation}\label{Pic}
\begin{CD}
\Theta @> \phi_C^*|_\Theta >> F^1\\
@V L_C VV @V \mathrm{pr} VV\\
\mathrm{Pic}(X)_\mathbb{R} @<< \det^{-1} < F^1/F^2\\
\end{CD}
\end{equation}
where $\mathrm{pr}$ is the projection and the lower horizontal arrow denotes the isomorphism $F^1/F^2\cong \mathrm{Pic}(X)_\mathbb{R}$ given by sending the class $[\mathcal{F}]\in F^1$ to $\det(\mathcal{F})^{-1}$.

\begin{lem}\label{commute}
$L_C$ is equal to the homomorphism $\psi_C:\Theta\to\mathrm{Pic}(X)_\mathbb{R}$ defined in Subsection \ref{3.3}.
\end{lem}

{\it Proof.} By definition $L_C$ is given as
$$\theta=(\theta_\rho)_\rho\mapsto \det\left(\bigoplus_{\rho\in\mathrm{Irr}(G)} \mathcal{R}_\rho^{\oplus \theta_\rho}\right).$$

For a determinantal semi-invariant $f$ having $\theta=(\theta_\rho)_\rho$ as its weight, $\varphi_C(f)$ is regarded as a section of a line bundle $L_C(\theta)$ by noticing that each arrow $a_{\rho,\rho'}$ in $Q_G$ is regarded as a section of $\mathcal{R}_\rho^\vee\otimes \mathcal{R}_{\rho'}$. Thus, $L_C(\theta)$ is nothing but the line bundle $\psi_C(\theta)$. Since $\Theta$ is spanned by the weights of determinantal semi-invariants by Theorem \ref{determinant}, $L_C$ and $\psi_C$ the same on the whole $\Theta$.
\qed

\vspace{3mm}

When we cross a wall from a GIT-chamber $C$ to an adjacent one $C'$, the corresponding chambers $\phi_C^*(C)$ and $\phi_{C'}^*(C')$ in $F^1$ may not be adjacent in general. For example, in dimension two, the automorphism $\phi_{C'}^*\circ (\phi_{C}^*)^{-1}$ of $F^1\cong \mathrm{Pic}(X)$ is a reflection along the hyperplane containing $\phi_{C}^*(W)$ for every wall $W=\overline{C}\cap\overline{C'}$. In the proof of the main result of \cite{CI}, namely the abelian case of Theorem \ref{main2}, the fact that, for a wall-crossing of Type 0, the chambers $\phi_C^*(C)$ and $\phi_{C'}^*(C')$ in $F^1$ are adjacent up to tensoring a line bundle plays a crucial role. We will show that the same property holds for non-abelian groups as well (cf. Proposition \ref{twist}).

\begin{lem}\label{structure}
Let $W=\overline{C}\cap\overline{C'}$ be a wall of Type 0 and consider the exact sequence on the unstable locus $D_W$ as in (\ref{exact1}). Then, for each connected component $D\subset D_W$, exactly one of the following two situations occurs:\\
(1) Either $\mathcal{S}$ or $\mathcal{Q}$ parametrizes mutually isomorphic $G$-equivariant $\mathbb{C}[V]$-modules on $D$.\\
(2) $D$ is a product of two curves $C_1, C_2$ where each $C_i$ is a tree of rational curves of Dynkin type, that is, $C_i$ is isomorphic to the (reduced) exceptional locus of the minimal resolution of an ADE surface singularity.
\end{lem}

{\it Proof.} Let $R_1\subset R$ (resp. $R_2\subset R$) be the $\mathbb{C} G$-module obtained by forgetting the structure as $\mathbb{C}[V]$-modules on the $G$-equivariant modules parametrized by $\mathcal{S}_x$ (resp. $\mathcal{Q}_x$) of $x\in D$. Since the morphism $\alpha: \mathfrak{M}_C\to \bar{\mathfrak{M}}_W$ is an isomorphism, we may regard the divisor $D$ as a moduli space parametrizing pairs of isomorphism types of structures as $\mathbb{C}[V]$-modules on $R_1$ and $R_2$. We construct moduli spaces $\mathfrak{M}_{\mathcal{S}}$ (resp. $\mathfrak{M}_{\mathcal{Q}}$) parametrizing $\mathbb{C}[V]$-modules appearing as fibers of $\mathcal{S}|_D$ (resp. $\mathcal{Q}|_D$), as GIT-quotients of certain affine schemes $\mathcal{V}_{\mathcal{S}}, \mathcal{V}_{\mathcal{Q}}$ by the $GL_{R_i}$-actions ($i=1,2$) in a similar way to the case of $\mathfrak{M}_\theta$ (see Section {2}). The stability condition for $\mathfrak{M}_{\mathcal{S}}$ (resp. $\mathfrak{M}_{\mathcal{Q}}$) is chosen so that its associated semi-stable locus in $\mathcal{V}_{\mathcal{S}}$ (resp. $\mathcal{V}_{\mathcal{Q}}$) contains the points corresponding to fibers of $\mathcal{S}|_D$ (resp. $\mathcal{Q}|_D$) and is minimal satisfying this property. Then we have a natural equivariant morphism $\mathcal{V}_{\mathcal{S}}\times\mathcal{V}_{\mathcal{Q}}\to \mathcal{V}$ which descends to an isomorphism $\mathfrak{M}_{\mathcal{S}}\times \mathfrak{M}_{\mathcal{Q}}\to \bar{D}\subset \mathfrak{M}_W$ where $\bar{D}$ is the image of $D$ under the morphism $\mathfrak{M}_C\to \mathfrak{M}_W$.

Since $D_W$ is a surface by Lemma \ref{divisor}, we only have to show that $\mathfrak{M}_{\mathcal{S}}$ and $\mathfrak{M}_{\mathcal{Q}}$ are curves of Dynkin type when they are one-dimensional. Indeed, this particularly implies that each irreducible component of $\bar{D}$ is smooth and thus $D\to \bar{D}$ is an isomorphism. Note that $\mathfrak{M}_C\to \mathfrak{M}_W$ has connected fibers since each fiber of the quotient map $\mathcal{V}^{W\text{-ss}}\to \mathfrak{M}_W$ is the closure in $\mathcal{V}^{W\text{-ss}}$ of a $GL_R$-orbit.

Let $E'_1,\dots, E'_{m'}$ be the irreducible components of $D$ and $C_{E'_k}\subset \Theta$ their orbit cones. We also let $H\subset \Theta$ be the hyperplane containing $W$. Then, by moving from the relative interior of $\bigcap_{k}C_{E'_k}\cap H$ to its smaller faces, we obtain $\theta\in\Theta$ and its associated birational map $\alpha_\theta: \mathfrak{M}_C\dashrightarrow \mathfrak{M}_\theta$ such that $\alpha_\theta$ is defined on $D$ and that $\alpha_\theta$ only collapses the first factor $\mathfrak{M}_{\mathcal{S}}$ of $D$ and in particular the image $\alpha_\theta(D)$ is the curve $\mathfrak{M}_{\mathcal{Q}}$. This is indeed possible since, if we encounter a wall corresponding to a contraction of the second factor $\mathfrak{M}_{\mathcal{Q}}$ during the process, then we just have to cross it. Note that moving from a face to its codimension-one face gives an isomorphism or a contraction of relative Picard number one between the corresponding (normalized) moduli spaces, and in particular each process does not contract the two factors at the same time.

Therefore, the curve $\mathfrak{M}_{\mathcal{S}}$ is identified with the fiber, under $\alpha_\theta$, over a general point of the one-dimensional singular locus of $\bar{\mathfrak{M}}_\theta$. Then it must be a tree of rational curves of Dynkin type since $\bar{\mathfrak{M}}_\theta$ has only canonical singularities and $\alpha_\theta$ is crepant (cf. \cite[Corollary 1.14]{Re1}). The same argument works for $\mathfrak{M}_{\mathcal{Q}}$ as well.
\qed

\vspace{3mm}

For given $\mathcal{E}\in K_0(X)$, we consider the automorphism $\mathcal{T}_\mathcal{E}$ of $K(X)$ defined by
$$\mathcal{T}_\mathcal{E}(\xi)=\xi-\chi(\xi, \mathcal{E})\cdot \mathcal{E}.$$
Note that $\mathcal{T}_\mathcal{E}$ has positive determinant and fixes the hyperplane
$$\mathcal{E}^\perp:=\{\xi\in K(X) \mid \chi(\xi, \mathcal{E})=0\}$$
pointwise. We denote by $\mathcal{T}'_\mathcal{E}$ the inverse of $\mathcal{T}_\mathcal{E}$. We will show that the automorphism $\phi_{C'}^*\circ (\phi_{C}^*)^{-1}$ of $F^1$ coincides with $\mathcal{T}_\mathcal{E}$ or $\mathcal{T}'_\mathcal{E}$ for some $\mathcal{E}\in K_0(X)$ up to an automorphism obtained by tensoring a line bundle of $X$ (Proposition \ref{twist}).

For the families $\mathcal{S}$ and $\mathcal{Q}$ in (\ref{exact1}) associated to the wall $W$, let $\mathcal{S}_\rho$ and $\mathcal{Q}_\rho$ be their $\rho$-isotypic components respectively.

\begin{lem}\label{cohomology}
For any $\rho, \sigma\in\mathrm{Irr}(G)$, we have $H^i(D_W, \mathcal{S}_\rho^\vee\otimes \mathcal{Q}_\sigma)=0$ for all $i$.
\end{lem}

{\it Proof.} We first note that the $G$-equivariant bundle $\bigoplus_{\rho\in \mathrm{Irr}(G)} \mathcal{R}^\vee_\rho \otimes \rho^*$ on $X\times \mathbb{C}^3$ can be regarded as the universal family of the moduli space $\mathfrak{M}_{\theta'}$ for $\theta'\in \Theta$ defined by $\theta'(\rho)=-\theta(\rho^*)$ with $\theta\in C$ by \cite[Lemma 2.6(ii)]{CI}. Similarly, $\mathcal{K}^\vee$ is regarded as a tautological bundle for a moduli space obtained from $\mathfrak{M}_{\theta'}$ by crossing a wall of Type 0. Then we see that the dual bundles are related as
\begin{equation}\label{dual}
0\to \mathcal{R}_C^\vee\to \mathcal{K}^\vee\to \mathcal{Q}^\vee\otimes \mathcal{O}_X(D_W)\to 0
\end{equation}
noticing that $\mathcal{R}_C$ and $\mathcal{K}$ are related by the exact sequences
\begin{equation}\label{K}
0\to \mathcal{K}\to \mathcal{R}_C\to \mathcal{Q}\to 0 \quad \text{and} \quad 0\to \mathcal{R}_C(-D_W)\to \mathcal{K}\to \mathcal{S}\to 0.
\end{equation}
From (\ref{dual}) we obtain a long exact sequence
\begin{equation}\label{long}
\begin{aligned}
\cdots &\to \mathrm{Ext}^i_{\mathcal{O}_X}(\mathcal{R}_C^\vee, \mathcal{R}_C^\vee)\to \mathrm{Ext}^i_{\mathcal{O}_X}(\mathcal{R}_C^\vee, \mathcal{K}^\vee)\to \mathrm{Ext}^i_{\mathcal{O}_X}(\mathcal{R}_C^\vee, \mathcal{Q}^\vee \otimes \mathcal{O}_X(D_W))\\
&\to \mathrm{Ext}^{i+1}_{\mathcal{O}_X}(\mathcal{R}_C^\vee, \mathcal{R}_C^\vee)\to \cdots
\end{aligned}
\end{equation}
since $\mathcal{R}_C^\vee$ is locally free. Applying the equivalence $\Phi_C$, we have
\begin{equation}\label{Ext}
\mathrm{Ext}^i_{\mathcal{O}_X}(\mathcal{R}_C^\vee, \mathcal{R}_C^\vee)=\mathrm{Ext}^i_{\mathcal{O}_{\mathbb{C}^3}}(\mathcal{O}_{\mathbb{C}^3}\otimes_\mathbb{C} R, \mathcal{O}_{\mathbb{C}^3}\otimes_\mathbb{C} R)^G=
\begin{cases}
\mathbb{C}[V] &\text{ if }i=0\\
0 &\text{ if }i\ne0
\end{cases}
\end{equation}
(cf. \cite[Lemma 5.4]{CI}). Since the kernel of the map from the second term to the third in (\ref{long}) is isomorphic to
$$H^i(D_W, \mathcal{R}_C|_{D_W}\otimes \mathcal{S}^\vee)=H^i(D_W, (\mathcal{S}\oplus\mathcal{Q})\otimes \mathcal{S}^\vee),$$
we obtain $H^i(D_W, \mathcal{S}^\vee\otimes \mathcal{Q})=0$ for $i>0$ by (\ref{Ext}). Since elements of $\mathbb{C}[V]$ of positive degree are zero on $\mathcal{R}_C^\vee|_{D_W}$, we also see that $H^0(D_W, \mathcal{S}^\vee\otimes \mathcal{Q})=0$. Decomposing $\mathcal{S}^\vee\otimes \mathcal{Q}$ into isotypic components, we obtain the claim.
\qed

\vspace{3mm}

\begin{prop}\label{twist}
Let $W=\overline{C}\cap\overline{C'}$ be a wall of Type 0 with the associated sequence (\ref{exact1}) on the unstable locus $D_W$. Then $D_W$ is connected. Moreover, if we let $R_1$ and $R_2$ be the underlying $\mathbb{C} G$-modules of fibers of $\mathcal{S}$ and $\mathcal{Q}$ respectively, then there exists $\mathcal{E}\in K_0(X)$ such that\\
(i) the image $\phi_C(\mathcal{E})\in K_0(\mathbb{C}^3)\cong R(G)$ is equal to either $R_1$ or $R_2$\\
(ii) the automorphism of $K(X)$ induced by the autoequivalence $\Phi_{C'}^{-1}\circ \Phi_{C}$ of $D(X)$ is equal to one of the four automorphisms: (a) $\mathcal{T}_\mathcal{E}$, (b) $\mathcal{T}'_\mathcal{E}$, (c) $\xi\mapsto \mathcal{T}_\mathcal{E}(\xi)\otimes \mathcal{O}(-D_W)$, and (d) $\xi\mapsto \mathcal{T}'_\mathcal{E}(\xi\otimes \mathcal{O}(D_W))$.\\
In particular, $\phi^*_{C'}(C')$ is adjacent to $\phi^*_C(C)\otimes L$ for some $L$ in $\mathrm{Pic}^c(X)$, the subgroup of $\mathrm{Pic}(X)$ generated by the classes of compact divisors.
\end{prop}

{\it Proof.} Let $D_W=\bigcup_i D_i$ be a decomposition into connected components and let $R_{1,i}$ (resp. $R_{2,i}$) be the $\mathbb{C} G$-modules for fibers of $\mathcal{S}|_{D_i}$ (resp. $\mathcal{Q}|_{D_i}$). Note that all $R_{1,i}$ (resp. $R_{2,i}$) are proportional in $R(G)$ since $D_i$'s are defined by the same wall $W$.

We first show that we can take $\mathcal{E}$ for each $D_i$ such that $\phi_C^*(\mathcal{E})$ is proportional to $R_{1,i}$ or $R_{2,i}$. We consider the two cases (1), (2) in Lemma \ref{structure} separately. For Case (1), we first assume that $\mathcal{Q}|_{D_i}$ is a trivial family of $G$-equivariant $\mathbb{C}[V]$-modules. Since every fiber of $\mathcal{Q}$ is indecomposable, there exists a line bundle $L$ on $D_i$ such that $\mathcal{Q}_\rho|_{D_i}\cong L^{\oplus \mathrm{rank}(\mathcal{Q}_\rho|_{D_i})}$ for all $\rho$. Note that $R_{2,i}=\bigoplus_{\rho\in\mathrm{Irr}(G)} \rho^{\oplus\mathrm{rank}(\mathcal{Q}_\rho|_{D_i})}$.

We set $\mathcal{E}:=[L^{-1}\otimes \omega_{D_i}]\in K_0(X)$. The cohomology sheaves of $\Phi_C(L^{-1}\otimes \omega_{D_i})$, which are $G$-equivariant $\mathcal{O}_{\mathbb{C}^3}$-modules supported at the origin, are computed as
$$\begin{aligned}
\Phi_C^j(L^{-1}\otimes \omega_{D_i})&\cong\bigoplus_{\rho\in\mathrm{Irr}(G)}H^j(X, \mathcal{R}_\rho\otimes L^{-1}\otimes \omega_{D_i})\otimes_\mathbb{C} \rho\\
&\cong\bigoplus_{\rho\in\mathrm{Irr}(G)} H^{2-j}(D_i, \mathcal{R}_\rho^\vee \otimes L)^\vee\otimes_\mathbb{C} \rho\\
&\cong\bigoplus_{\rho\in\mathrm{Irr}(G)} H^{2-i}\left(D_i, ((L^{-1})^{\oplus\mathrm{rank}(\mathcal{Q}_\rho|_{D_i})}\oplus \mathcal{S}_\rho^\vee) \otimes L \right)^\vee \otimes_\mathbb{C} \rho \\
&\cong\bigoplus_{\rho\in\mathrm{Irr}(G)} H^{2-i}\left(D_i, \mathcal{O}_{D_i}^{\oplus \mathrm{rank}(\mathcal{Q}_\rho|_{D_i})} \right)^\vee \otimes_\mathbb{C} \rho
\end{aligned}$$
where the last equality holds by Lemma \ref{cohomology}. Therefore, the class $\phi_C^* (\mathcal{E})$ is equal to $\chi(\mathcal{O}_{D_i})\cdot R_{2,i}\in R(G)$.

When $\mathcal{S}|_{D_i}$ is a trivial family, one can similarly show that $\phi_C^*([L'])$ is equal to $\chi(\mathcal{O}_{D_i})\cdot R_{1,i}\in R(G)$ for the line bundle $L'$ on $D_i$ such that $\mathcal{S}_\rho|_{D_i}\cong L'^{\bigoplus \mathrm{rank}(\mathcal{S}_\rho|_{D_i})}$.

Next we consider the case (2) in Lemma \ref{structure}. In this case each $D_i$ is the product of curves $C_1$ and $C_2$ which are regarded as moduli spaces parametrizing fibers of $\mathcal{S}|_{D_i}$ and $\mathcal{Q}|_{D_i}$ respectively. Since each $C_j$ is a union of $\mathbb{P}^1$'s, the bundle $\mathcal{S}|_{D_i}$ (resp. $\mathcal{Q}|_{D_i}$) on $D_i$ is a direct sum of line bundles such that the restrictions of these line bundles to fibers of the projection $D_i\to C_1$ (resp. $D_i\to C_2$) are mutually isomorphic. We take any line bundles $L_1$ and $L_2$ on $D_i$ appearing as direct summands of $\mathcal{S}_\rho|_{D_i}$ and $\mathcal{Q}_{\rho'}|_{D_i}$ respectively for any $\rho, \rho'$. By Lemma \ref{cohomology}, we have $H^j(D_i, L_1^{-1}\otimes L_2)=0$ for all $j$. Then either
$$H^j(C_1, (L_1^{-1}\otimes L_2)|_{C_1})=0\;(\forall j)\quad \text{or} \quad H^j(C_2, (L_1^{-1}\otimes L_2)|_{C_2})=0\;(\forall j)$$
holds where $C_1$ and $C_2$ are regarded as fibers of $D_i\to C_2$ and $D_i\to C_1$ respectively. From this we see that one has either $H^j(C_2, (L_1^{-1}\otimes L'_2)|_{C_2})=0\;(\forall j)$ for all line bundles $L'_2$ appearing as direct summands of $\mathcal{Q}|_{D_i}$ or $H^j(C_1, (L'^{-1}_1\otimes L_2)|_{C_1})=0\;(\forall j)$ for all line bundles $L'_1$ appearing as direct summands of $\mathcal{S}|_{D_i}$. In the former case one has $\phi_C([L_2^{-1}])=R_{2,i}\in R(G)$, and in the latter case $\phi_C([L_1]\otimes \omega _{D_i})=R_{1,i}$. This can be checked by noticing that the Euler characteristics $\chi(L'_2|_{C_2})$ (resp. $\chi(L'_1|_{C_1})$) are the same for all the direct summands $L'_2$ of $\mathcal{Q}|_{D_i}$ in the former case (resp. $L'_1$ of $\mathcal{S}|_{D_i}$ in the latter case). Note that the Euler characteristics of a line bundle $L$ on $C_i$ is equal to $1+\sum_k \deg(L|_{C_{i,k}})$ where $C_{i,k}$ are the irreducible components of $C_i$.

We have shown that the image in $R(G)$ of a line bundle on $D_i$ is proportional to $R_{1,i}$ (and to $R_{2,i}$) modulo $\mathbb{Z} R$ in any case. Since $D_i$'s are linearly independent in $\mathrm{Pic}(X)_\mathbb{R}\cong F^1/F^2$, the classes of the line bundles on $D_i$'s in $K(X)$ are also linearly independent (cf. the proof of \cite[Proposition 5.5(ii)]{CI}). Therefore, $D_W$ must be connected. The similar argument in the proof of \cite[Proposition 5.5(iii)]{CI} also shows that we have $H^1(\mathcal{O}_{D_W})=H^2(\mathcal{O}_{D_W})=0$ in the case (1) in Lemma \ref{structure}.

To summarize, the following two cases may occur:\\
(I) we have $\chi(\mathcal{R}_\rho^\vee, \mathcal{E})=\mathrm{rank}(\mathcal{Q}_\rho)$ for the class $\mathcal{E}=[L^{-1}\otimes \omega_{D_W}]\in K_0(X)$ with an arbitrary line bundle $L$ on $D_W$ which is a direct summand of $\mathcal{Q}$, and\\
(II) we have $\chi(\mathcal{R}_\rho^\vee, \mathcal{E})=\mathrm{rank}(\mathcal{S}_\rho)$ for the class $\mathcal{E}=[L]\in K_0(X)$ with an arbitrary line bundle $L$ on $D_W$ which is a direct summand of $\mathcal{S}$.\\
In the former case (I), one has
$$\mathcal{T}_\mathcal{E}([\mathcal{R}_\rho^\vee])=[\mathcal{R}_\rho^\vee]-\mathrm{rank}(\mathcal{Q}_\rho)[L^{-1}\otimes \omega_{D_W}]=[\mathcal{R}_\rho^\vee]-[\mathcal{Q}_\rho^\vee(D)]$$
in $K(X)$. By the sequence (\ref{dual}) and the fact that $\{\mathcal{R}_\rho^\vee\}_\rho$ forms a basis of $K(X)$, we can conclude that $\mathcal{T}_\mathcal{E}$ coincides with the automorphism induced by $\Phi_{C'}^{-1}\circ \Phi_{C}$ or its composition with tensoring $\mathcal{O}_X(-D_W)$. Which of the cases (a) and (c) happens depends on whether $\mathcal{Q}_{\rho_0}=0$ or $\ne0$. The latter case (II) can be treated similarly by switching the roles of $C$ and $C'$, so that we fall into the cases (b) and (d). This completes the proof.
\qed

\vspace{3mm}

\begin{rem}
The above proposition shows that the wall-crossing gives a combination of a spherical twist and tensoring a line bundle at least at the level of the Grothendieck group. In the rigid case (1) in Lemma \ref{structure}, one can use the same argument as in the proof of \cite[Proposition 7.3]{CI} to prove the similar statement at the level of the derived category. In the case (2), however, this argument does not work since the line bundles in $\mathcal{S}$ or $\mathcal{Q}$ are not mutually isomorphic (while they have the same Euler characteristics) in contrast to the rigid case. By analogy with the abelian case \cite[Corollary 4.6]{CI}, it is expected that the case (2) does not occur.
\qed
\end{rem}

\noindent{\it Proof of Theorem \ref{main2}}

Since all projective crepant resolutions of $\mathbb{C}^3/G$ are connected with each other by a sequence of flops, we only have to show that every flop of $X=\mathfrak{M}_C$ can be realized by (not necessarily single) wall-crossings. Let $B\subset \overline{\mathrm{Amp}}(X)$ be a facet corresponding to a flop $X\to X_B\gets X'$. If we can reach a GIT-chamber $C''$ through walls of Type 0 such that $C''$ admits a wall $\tilde{B}\subset \overline{C}''$ satisfying $\psi_{C''}(\tilde{B})\subset B$, then crossing $\tilde{B}$ induces the desired flop. Indeed, otherwise $\tilde{B}$ would be contained in a facet of the orbit cone of a general point of a divisor since the homomorphism $\psi_{C''}$ must change when we cross $\tilde{B}$. In such a case, however, Lemma \ref{vertical} shows that $\tilde{B}$ would induce a divisorial contraction, which is a contradiction.

As shown in \cite[\S8]{CI}, each compact exceptional divisor $E\subset X$ acts on $F^1$ by
$$\xi\mapsto \xi\otimes \mathcal{O}_X(E)=\xi+[\xi \otimes \mathcal{O}_E]$$
with $[\xi \otimes \mathcal{O}_E]\in F^2$. Moreover, for the set of all irreducible compact exceptional divisors $S_1,\dots,S_d$ and for any $\xi\in \mathrm{pr}^{-1}(\mathrm{Amp}(X)\cup \mathrm{relint}(B))$, the set $[\xi\otimes\mathcal{O}_{S_1}],\dots,[\xi\otimes\mathcal{O}_{S_d}]$ forms a basis of $F^2$ where $\mathrm{Amp}(X)$ is identified with a cone in $F^1/F^2$ (see (\ref{Pic})).

The existence of the desired chamber $C''$ can be shown similarly to the proof of \cite[Proposition 8.2]{CI}. More precisely, one can show from Proposition \ref{twist} that we have
\begin{equation}\label{inclusion}
\mathrm{pr}^{-1}(\mathrm{Amp}(X)\cup \mathrm{relint}(B))\subset \bigcup_{L\in \mathrm{Pic}^c(X)}\bigcup_{\mathfrak{M}_C\cong X} L\otimes \phi_C^*(\overline{C})
\end{equation}
noticing that GIT-chambers in $\Theta$ are finite. Thus we obtain a chamber $C''$ and its wall $\tilde{B}$ such that $\mathrm{pr}(\phi_C^*(\tilde{B}))\subset B$.
\qed

\vspace{3mm}

Theorem \ref{main2} together with (\ref{inclusion}) implies that $\{\psi_C^*(\overline{C})\}_{C\subset\Theta\text{: chamber}}$ covers the movable cone of $X$. Thus, every rational contraction $X\dashrightarrow X'$ (see Subsection \ref{3.2}) is realized as the birational map $\mathfrak{M}_C\dashrightarrow\bar{\mathfrak{M}}_\theta$ induced by variation of GIT for some $\theta$. This fact has the following application:

\begin{prop}\label{prop:generate}
Let $h_1, \dots, h_\ell$ be homogeneous generators of the normalization of $\mathbb{C}[\mathcal{W}]$. Then the associated $\ell+m$ elements to $\varphi(h_1), \dots, \varphi(h_\ell)\in\mathbb{C}[V]^{[G,G]}$ (see (\ref{gens})) generate the Cox ring $\mathrm{Cox}(X)\subset \mathbb{C}[V]^{[G,G]}[t_1^{\pm1},\dots,t_m^{\pm1}]$ of any crepant resolution $X\to\mathbb{C}^3/G$.
\end{prop}

{\it Proof.} Let $f_1,\dots, f_{\ell'}\in\mathbb{C}[V]^{[G,G]}$ be homogeneous elements such that the associated elements in (\ref{gens}) generate $\mathrm{Cox}(X)$. We consider the ambient toric varieties $Y$ of $X$ and $\bar{Y}_\mathcal{W}$ of $\bar{\mathcal{W}}$ associated to $f_i$'s (see Subsection \ref{3.3}). Note that each $f_i$ corresponds to a non-exceptional torus-invariant divisor $D_i\subset Y$ and that the image of the corresponding section $f_{D_i}\in H^0(Y, \mathcal{O}(-D_i))$ (i.e. $D_i=\{f_{D_i}=0\}$) in $\mathrm{Cox}(X)$ is equal to $\tilde{f_i}$ (cf. Remark \ref{cox(V/G)}). Regarding the weight of $f_{D_i}$ as a character of $T_X$, we consider its associated GIT-quotient $Y_i$ of $\mathbb{A}^{\ell+m}$. In terms of a fan, $Y_i$ is characterized as a birational model $Y\dashrightarrow Y_i$ with $\dim\mathrm{Pic}(Y_i)_\mathbb{R}=1$ such that the toric fan of $Y_i$ contains the cone generated by the rays corresponding to the torus-invariant divisors of $Y$ except $D_i$.

For each $i$, let $X_i\subset Y_i$ be the induced GIT-quotient of $\mathrm{Spec}(\mathrm{Cox}(X))$. As observed above, there exists $\theta\in\Theta$ such that $X_i\cong \bar{\mathfrak{M}}_\theta$. Since $\bar{Y}_\mathcal{W}$ is normal,  there is a homogeneous semi-invariant $f$ in $\mathbb{C}[\bar{\mathcal{W}}]$ such that its (scheme-theoretic) zero locus in $\bar{\mathfrak{M}}_\theta\cong X_i$ is equal to the strict transform of $D_i\cap X$ under $X\dashrightarrow X_i$. If we take a GIT-chamber $C$ so that $\overline{C}$ contains $\theta$, this implies $\varphi_C(f)$ is equal to $\tilde{f_i}$ (up to constant multiplication). This particularly implies that each $\tilde{f_i}$ can be obtained as the associated element of $\varphi(h)$ for some homogeneous semi-invariant $h$. Thus, the claim follows.
\qed

\section{The symplectic resolutions for the binary tetrahedral group}\label{5}

In this section we treat crepant resolutions of a symplectic quotient singularity defined by the binary tetrahedral group $T\subset Sp_4(\mathbb{C})$ (cf. \cite{LS}). It is generated by two elements
$${\footnotesize
g_1=\begin{pmatrix}
i&0&0&0\\
0&-i&0&0\\
0&0&i&0\\
0&0&0&-i
\end{pmatrix}, \quad
g_2=-\frac{1}{2}\begin{pmatrix}
(1+i)\omega&(-1+i)\omega&0&0\\
(1+i)\omega&(1-i)\omega&0&0\\
0&0&(1+i)\omega^2&(-1+i)\omega^2\\
0&0&(1+i)\omega^2&(1-i)\omega
\end{pmatrix}}$$
where $\omega$ is the third root of unity. $T$ is isomorphic to a subgroup of $SL_2(\mathbb{C})$ corresponding to $E_6$-singularity via
$$g_1\mapsto
\begin{pmatrix}
i&0\\
0&-i
\end{pmatrix},\quad
g_2\mapsto
-\frac{1}{2}\begin{pmatrix}
1+i&-1+i\\
1+i&1-i
\end{pmatrix}.$$
We denote this representation by $V_0$. Also, $\mathrm{Irr}(G)$ has three 1-dimensional representations $\rho_k:G\to\mathbb{C}^*\,(k=0,1,2)$ defined by $\rho_k(g_1)=1,\, \rho_k(g_2)
=\omega^k$. Note that the inclusion $T\subset Sp_4(\mathbb{C})$ corresponds to the direct sum $V_1\oplus V_2$ where $V_k=V_0\otimes \rho_k$. Moreover, $\mathrm{Irr}(G)$ contains a unique 3-dimensional  representation $W$ given by
$$g_1\mapsto\begin{pmatrix}
1&0&0\\
0&-1&0\\
0&0&-1
\end{pmatrix},\quad
g_2\mapsto\begin{pmatrix}
0&0&1\\
1&0&0\\
0&1&0
\end{pmatrix},$$
so that $\mathrm{Irr}(G)=\{\rho_0,\rho_1,\rho_2,V_0,V_1,V_2,W\}$. In the subsequent computations, we use the basis of the regular representation of $G$ coming from the above presentations similarly as before.

The McKay quiver $Q_T$ is as follows:

\[\xymatrix@=48pt{
&\rho_0 \ar@<-0.5ex>[ld]_-{a_{01}} \ar@<0.5ex>[rd]^-{a_{02}}
&\\
V_1 \ar@<0.5ex>[d]^-{b_{12}} \ar@<-0.5ex>[r]_-{B_{1i}} \ar@<-0.5ex>[ru]_-{b_{10}}
& W \ar@<-0.5ex>[d]_{A_{0i}} \ar@<-0.5ex>[l]_{A_{1i}} \ar@<0.5ex>[r]^{A_{2i}}
&V_2 \ar@<-0.5ex>[d]_{b_{21}} \ar@<0.5ex>[l]^-{B_{2i}} \ar@<0.5ex>[lu]^-{b_{20}}\\
\rho_2 \ar@<-0.5ex>[r]_-{a_{20}} \ar@<0.5ex>[u]^-{a_{21}}
&V_0 \ar@<-0.5ex>[l]_-{b_{02}} \ar@<0.5ex>[r]^-{b_{01}} \ar@<-0.5ex>[u]_-{B_{0i}}
&\rho_1 \ar@<-0.5ex>[u]_-{a_{12}} \ar@<0.5ex>[l]^{a_{10}}
}\]
where the arrow $A_{ki}$ (resp. $B_{ki}$) stands for two arrows $A_{k1},A_{k2}$ (resp. $B_{k1},B_{k2}$).

We denote the standard coordinates of $V=\mathbb{C}^4$ by $x,y,z,w$. Then the generic matrices are given as
{\footnotesize
$$\begin{aligned}
&a_{01}=\begin{pmatrix}x\\y\end{pmatrix},b_{10}=\begin{pmatrix}-w&z\end{pmatrix},
a_{12}=\begin{pmatrix}x\\y\end{pmatrix},b_{21}=\begin{pmatrix}-w&z\end{pmatrix}, a_{20}=\begin{pmatrix}x\\y\end{pmatrix},b_{02}=\begin{pmatrix}-w&z\end{pmatrix},\\
&a_{02}=\begin{pmatrix}z\\w\end{pmatrix},b_{20}=\begin{pmatrix}-y&x\end{pmatrix},
a_{10}=\begin{pmatrix}z\\w\end{pmatrix},b_{01}=\begin{pmatrix}-y&x\end{pmatrix}, a_{21}=\begin{pmatrix}z\\w\end{pmatrix},b_{12}=\begin{pmatrix}-y&x\end{pmatrix},\\
&A_{01}=\begin{pmatrix}\omega^2 x&y&-i\omega y\\ -\omega^2 y&x&i\omega x\end{pmatrix}, A_{02}=\begin{pmatrix}i\omega^2 z&i\omega w&w\\ -i\omega^2 w&i\omega z&-z\end{pmatrix}, B_{01}=\begin{pmatrix}y&x\\ -\omega x&\omega y\\ -i\omega^2 x& -i\omega^2 y\end{pmatrix}, B_{02}=\begin{pmatrix}w&z\\ -\omega^2 z&\omega^2 w\\ -i\omega z& -i\omega w\end{pmatrix}, \\
&A_{11}=\begin{pmatrix}x&y&-i y\\ -y&x&i x\end{pmatrix}, A_{12}=\begin{pmatrix}\omega^2 z&w&-i\omega w\\ -\omega^2 w&z&i\omega z\end{pmatrix}, B_{11}=\begin{pmatrix}y&x\\ -\omega^2 x&\omega^2 y\\ -i\omega x& -i\omega y\end{pmatrix}, B_{12}=\begin{pmatrix}w&z\\ -z&w\\ -i z& -i w\end{pmatrix}, \\
&A_{21}=\begin{pmatrix}i\omega^2 x&i\omega y& y\\ -i\omega^2 y&i\omega x&-x\end{pmatrix}, A_{22}=\begin{pmatrix} z&w&-i w\\ -w&z&i z\end{pmatrix}, B_{21}=\begin{pmatrix}y&x\\ -x&y\\ -ix& -i y\end{pmatrix}, B_{22}=\begin{pmatrix}w&z\\ -\omega z&\omega w\\ -i\omega^2 z& -i\omega^2 w\end{pmatrix}\end{aligned}$$}
for some choices of basis of the vector spaces $\mathrm{Hom}_{\mathbb{C} G}(V^*\otimes_\mathbb{C} \rho,\rho')$.

The commutator group $[T,T]=\langle g_1,g_2 g_1 g_2^{-1}\rangle$ is isomorphic to the quaternion group of order eight, and we can choose 18 generators
$$\bar{w}_{01},\dots,\bar{w}_{07},\bar{w}_{11},\dots,\bar{w}_{15},\bar{w}_{21},\dots,\bar{w}_{25},\bar{w}_3$$
of $\mathbb{C}[V]^{[T,T]}$ such that the associated 20 elements $w_{01},\dots,w_3,t_1^{-3},t_2^{-3}$ in $\mathbb{C}[V]^{[T,T]}[t_1^{\pm1},t_2^{\pm1}]$ generate the Cox ring of a crepant (or equivalently symplectic) resolution $X$ of $V/T$ \cite[Example 3]{Y2}, \cite{G}, where the variables $t_1, t_2$ correspond to junior elements $g_2,g_2^2$ respectively.

To state the main result for this section, let us recall that $V/T$ admits two projective symplectic resolutions which are isomorphic as varieties (but not as varieties over $V/G$). The nef cones in $\mathrm{Pic}(X)_\mathbb{R}=\mathbb{R} E_1\oplus \mathbb{R} E_2$ of these two resolutions are given as $\mathrm{Cone}(-E_1-E_2, -2E_1-E_2)$ and $\mathrm{Cone}(-E_1-E_2, -E_1-2E_2)$, and we denote the corresponding resolutions by $X_1$ and $X_2$ respectively.

\begin{thm}\label{main3}
The symplectic resolution $X_1$ (resp. $X_2$) of $V/T$ is isomorphic to the coherent component of the moduli space $\mathfrak{M}_{\theta_{X_1}}$ of $T$-constellations for certain generic $\theta_{X_1}$ (resp. $\theta_{X_2}$). Moreover, the coherent component of the $T$-Hilbert scheme is isomorphic to the common blow-up of $X_1$ and $X_2$ along closed subschemes which are isomorphic to $\mathbb{P}^2$.
\end{thm}

In the rest of this section we give a proof of this theorem. We will first show that $X_1$ is isomorphic to $\mathfrak{M}_{\theta_{X_1}}$ for some $\theta_{X_1}$. The claim for $X_2$ can be shown similarly using the symmetry of $T$ which we will see below.

Let us consider the action of $\mathbb{C}^*\times \mathbb{C}^*$ on $V=V_1\oplus V_2$ given by the component-wise dilation. Then $\mathbb{C}^*\times \mathbb{C}^*$ naturally acts on the set of ideals of $\mathbb{C}[V]$ and we can find 13 $T$-clusters $\mathbb{C}[V]/I$ which are $\mathbb{C}^*\times \mathbb{C}^*$-invariant as mentioned in \cite{LS}. We describe some of them as $T$-invariant $\mathbb{C}$-vector subspaces of $\mathbb{C}[V]$ using the formal variables $z_1,z_2$ corresponding to the weight for the $\mathbb{C}^*\times \mathbb{C}^*$-actions as in \cite{LS}:
{\footnotesize
$$\begin{aligned}
&p_1:& &\rho_0\oplus V_2 z_2\oplus W z_2^2\oplus (V_1\oplus V_2)z_2^3\oplus (\rho_1\oplus W)z_2^4 \oplus (V_0\oplus V_1)z_2^5 \oplus W z_2^6 \oplus V_0 z_2^7 \oplus \rho_2 z_2^8\\
&p_2:& &\rho_0 \oplus V_1 z_1 \oplus V_2 z_2 \oplus W z_1z_2 \oplus W z_2^2 \oplus V_0 z_1 z_2^2 \oplus (V_1\oplus V_2) z_2^3 \oplus \rho_2 z_1 z_2^3\oplus (\rho_1\oplus W) z_2^4 \oplus V_0 z_2^5
\end{aligned}$$
$$\begin{alignedat}{8}
&p_{3,1}: F \oplus (V_2 & &z_2^3 & &\oplus \rho_1 & &z_2^4) \oplus (V_1 z_1 & &z_2^2 & &\oplus \rho_2 z_1 ^2 z_2^2& & )\\
&p_{3,3}:F \oplus (V_2 & &z_2^3 & &\oplus \rho_1 & &z_2^4) \oplus (V_1 & &z_2^3 & &\oplus \rho_2 z_1  z_2^3& & )\\
&p_{4,2}:F \oplus (V_2 z_1 & &z_2^2 & &\oplus \rho_1 z_1^2 & &z_2^2) \oplus (V_1 z_1^3 & & & &\oplus \rho_2 z_1^4& & )\\
&p_{4,3}: F \oplus (V_2 z_1 & &z_2^2 & &\oplus \rho_1 z_1^2 & &z_2^2) \oplus (V_1 & &z_2^3 & &\oplus \rho_2 z_1  z_2^3& & )\\
&p_{5,1}: F \oplus (V_2 z_1^3 & & & &\oplus \rho_1 z_1^3 & &z_2) \oplus (V_1 z_1 & & z_2^2 & &\oplus \rho_2 z_1 ^2 z_2^2& & )\\
&p_{5,2}: F \oplus (V_2 z_1^3 & & & &\oplus \rho_1 z_1^3 & &z_2) \oplus (V_1 z_1^3 & & & &\oplus \rho_2 z_1^4& & )
\end{alignedat}$$
$$\begin{aligned}
&p_6:& &\rho_0 \oplus V_1 z_1 \oplus V_2 z_2 \oplus W z_1^2 \oplus W z_1z_2 \oplus (V_1\oplus V_2) z_1^3 \oplus V_0 z_1^2 z_2 \oplus (\rho_2\oplus W) z_1^4 \oplus \rho_1 z_1^3 z_2 \oplus V_0 z_1^5\\
&p_7:& &\rho_0\oplus V_1 z_1\oplus W z_1^2\oplus (V_1\oplus V_2)z_1^3\oplus (\rho_2\oplus W)z_1^4 \oplus (V_0\oplus V_2)z_1^5 \oplus W z_1^6 \oplus V_0 z_1^7 \oplus \rho_1 z_1^8
\end{aligned}$$}
where
$$F=\rho_0 \oplus V_1 z_1 \oplus V_2 z_2 \oplus W z_1^2 \oplus W z_1z_2 \oplus W z_2^2\oplus V_0 z_1^2 z_2 \oplus V_0 z_1 z_2^2.$$
The corresponding ideals are also presented similarly. For example, the ideal for $p_1$ is generated by $V_1 z_1, \rho_0 z_2^4, \rho_0 z_2^6, V_2 z_2^9$.

One can give an explicit basis of each direct summand of $p_i$ and $p_{i,j}$ as polynomials using the generic matrices above. For example, a basis of $Wz_1z_2$ in $F$ is given as the three entries of $B_{12}a_{01}$ (or $B_{21}a_{02}$). Note that $p_1$ (resp. $p_2$) plays a dual role to $p_7$ (resp. $p_6$) with respect to the involutions $\rho_1\leftrightarrow\rho_2,\,V_1\leftrightarrow V_2$ and $z_1\leftrightarrow z_2$. We have three other $\mathbb{C}^*\times \mathbb{C}^*$-invariant $T$-clusters but in fact they do not lie in the coherent component of the $T$-Hilbert scheme.

We also consider $\mathbb{C}^*\times \mathbb{C}^*$-invariant $T$-constellations $p_3,p_4,p_5$ which are not $T$-clusters:
{\footnotesize
$$\begin{alignedat}{8}
&p_3: F \oplus (V_2 & &z_2^3 & &\oplus \rho_1 & &z_2^4) \oplus (V_1 z_1^2 & & z_2^{-1} & &\oplus \rho_2 z_1^3& & z_2^{-1})\\
&p_4:F \oplus (V_2 z_1 & &z_2^2 & &\oplus \rho_1 z_1^2 & &z_2^2) \oplus (V_1 & & z_2 & &\oplus \rho_2 z_1& & z_2)\\
&p_5: F \oplus (V_2 z_1^3 & & & &\oplus \rho_1 z_1^3 & &z_2) \oplus (V_1 z_1^{-1}& & z_2^2  & &\oplus \rho_2  & & z_2^2)\\
\end{alignedat}$$
}where each $p_i$ is indecomposable as a $G$-equivariant $\mathbb{C}[V]$-module, admits a quotient module which is isomorphic to $V_1\oplus \rho_2$, and the complement of this quotient is generated by $\rho_0$. Note that $p_{i,j}$'s $(i=3,4,5)$ have submodules which are isomorphic to $V_1\oplus \rho_2$, in contrast to these $p_i$'s.

For each $i$, we consider the orbit cone $C_{p_i}=\{\theta\in\Theta \mid p_i\text{ is }\chi_\theta\text{-semistable}\}$, which is equal to
\begin{equation}\label{orbitcone}
\{\theta=(\theta_\rho)_\rho\in\Theta \mid \sum_\rho \theta_\rho\cdot c_\rho\ge0\text{ for any submodule }p_i\supset M \cong \bigoplus_\rho \rho^{c_\rho}\}.
\end{equation}
by King's result (Proposition \ref{King}). Each $C_{p_i}$ defines a neighborhood $\tilde{U}_i\subset \mathcal{W}=\mathcal{V}/\!/SL_R$ of $p_i$ consisting of points $p\in\mathcal{W}$ such that $C_p=C_{p_i}$. From the structure of $p_i$ presented above, one can compute $C_{p_i}$ and check that the intersection $\bigcap_{i=1}^7 C_{p_i}$ is full-dimensional. In particular we can take a stability condition $\theta_{X_1}$ from its interior.

By \cite[Theorem 4.18]{G}, the $\chi$-(semi)stable locus of $\mathfrak{X}$ is a union of seven affine open subsets $U_1,\dots,U_7$:
$$\begin{aligned}
&U_1=\{w_{12}t_1^{-3}\ne0\},\,U_2=\{w_{12}w_{23}\ne0\},\,U_3=\{w_{12}w_3\ne0\},\,U_4=\{w_{11}w_3\ne0\},\\
&U_5=\{w_{13}w_3\ne0\},\,U_6=\{w_{13}w_{22}\ne0\},\,U_7=\{w_{22}t_2^{-3}\ne0\}.
\end{aligned}$$
We show that the quotient of $\tilde{U}_i$ by the torus $T_\Theta$ is isomorphic to $U_i$ and that $p_i$ corresponds to the origin of $U_i\cong\mathbb{C}^2$ via this isomorphism.
For this, we define a ring map $\varphi_{X_1}:\mathbb{C}[\mathcal{W}]\to \mathbb{C}[V]^{[T,T]}[t_1^{\pm1},t_2^{\pm1}]$ by sending a semi-invariant $h$ of weight $(c_0,\dots,c_6)$ to $\varphi(h)t_1^{c_1+c_4+c_5+c_6} t_2^{c_2+c_3+c_5+c_6}$ where the ordering for $\mathrm{Irr}(G)$ is given by $\rho_0,\rho_1,\rho_2,V_0,V_1,V_2,W$.

We will construct an isomorphism $U_i\cong \tilde{U}_i/\!/T_\Theta$ for $i=1$. The other cases can be done in a similar way. We choose determinantal semi-invariants $\{h_i\}_i$ such that\\
(1) their weights in $\Theta$ generate a full-dimensional cone (contained in $C_{p_1}$),\\
(2) they do not vanish at $p_1$, and\\
(3) each $\varphi(h_i)$ is equal to $\bar{w}_{12}=z^4 + (-4\omega-2)z^2 w^2+w^4$ up to scalar multiplication.\\
For instance, $\{h_i\}_i$ can be chosen as the determinants of the following six matrices:
{\footnotesize$$A_{02}B_{12}, \begin{pmatrix}b_{02}&b_{02}A_{02}B_{02}\end{pmatrix}, \begin{pmatrix}b_{21}&b_{21}A_{22}B_{22}\end{pmatrix}, \begin{pmatrix}a_{10}\\mathbb{A}_{02}B_{02}a_{10}\end{pmatrix}, \begin{pmatrix}a_{02}\\mathbb{A}_{22}B_{22}a_{02}\end{pmatrix}, \begin{pmatrix}b_{21}A_{22}&A_{02}\end{pmatrix}.$$}

\begin{rem}
One can find $h_i$'s as semi-invariants whose weights are part of generators of the cone $C_{p_1}$.
\qed
\end{rem}

Since each $\varphi(h_i)$ is of the form $c\bar{w}_{12}\,(c\in\mathbb{C}^*)$, $\varphi_{X_1}$ induces a homomorphism
$$(\mathbb{C}[\mathcal{W}]_{h_1\cdots h_6})^{T_\Theta}\to (\mathbb{C}[V]^{[T,T]}[t_1^{\pm1},t_2^{\pm1}]_{w_{12}})^{T_X}$$
between the torus-invariant parts of the localizations. The latter ring contains $H^0(\mathcal{O}_{U_1})=\mathbb{C}[w_{02},w_{05},\frac{w_{23}}{w_{12}^2 t_1^{-3}}, \frac{w_{24}}{w_{12}^2 t_1^{-3}}]\subset \mathbb{C}(V)^T$ \cite[Theorem 4.18]{G}, and we can show that the above homomorphism is an isomorphism onto $H^0(\mathcal{O}_{U_1})$ by constructing its inverse by
{\small $$\begin{aligned}
&w_{02}\mapsto \det(b_{10}A_{12}B_{12}A_{12}B_{22}a_{02}),\, w_{05}\mapsto \det(b_{10}A_{12}B_{22}a_{02}),\\
&\frac{w_{23}}{w_{12}^2 t_1^{-3}}\mapsto \frac{\det\begin{pmatrix}b_{12}&b_{02}A_{02}B_{12}\end{pmatrix}}{h_2 h_4},\; \frac{w_{24}}{w_{12}^2 t_1^{-3}}\mapsto \frac{\det\begin{pmatrix}b_{12}A_{02}B_{11}&b_{02}A_{02}B_{12}\end{pmatrix}}{h_2 h_4}
\end{aligned}$$}
where
{\small $$\begin{aligned}
&\bar{w}_{02}=z^5 w-zw^5,\,\bar{w}_{05}=z^4+(4\omega+2)z^2 w^2+w^4,\\
&\bar{w}_{23}=xz^3+(2\omega +1)yz^2 w+(2\omega+1)xzw^2+yw^3,\;\bar{w}_{24}= yz^5+5xz^4 w-5yzw^4-xw^5.\end{aligned}$$}
One can check that the elements of the both sides of this assignment indeed give the same rational functions on $V/T$. This implies that the open subset $\{h_1\cdots h_6\ne0\}\subset \mathcal{W}$ descends to an open subset of the moduli space $\mathfrak{M}_{\theta_{X_1}}$ which is isomorphic to $U_1$. Note that here we use the assumption that the weights of $h_i$'s generate a full-dimensional cone.

In this way we obtain open subsets of $\mathfrak{M}_{\theta_{X_1}}$ which are isomorphic to $U_i$ for all $i$. Thus we can regard $X_1$ as an open subset of $\mathfrak{M}_{\theta_{X_1}}$. Since $X_1$ is already proper over $V/T$, we obtain $X_1\cong \mathfrak{M}_{\theta_{X_1}}$.

Finally, we show that the coherent component of the $T$-Hilbert scheme is isomorphic to the blowup $\tilde{X}$ of $X_1$ along a closed subscheme $P\cong\mathbb{P}^2$. The center $P\subset X_1$ is defined as the zero locus of all the coordinates $w_{01},\dots,w_{25}$ but $w_{11},w_{12},w_{13},w_3$ \cite[Proposition 4.11]{G}, and is covered with $U_3, U_4$ and $U_5$. Inside
$$U_3=\mathrm{Spec}\,\mathbb{C}\!\left[\frac{w_{11}}{w_{12}},\frac{w_{13}}{w_{12}},\frac{w_{12}w_{23}}{w_3}, \frac{w_{12}w_{21}}{w_3}\right],$$
for instance, $P$ is the zero locus of the last two coordinates, and the blowup along $P\cap U_3$ has the following affine open covering
$$U_{3,1}=\mathrm{Spec}\,\mathbb{C}\!\left[\frac{w_{11}}{w_{12}},\frac{w_{13}}{w_{12}},\frac{w_{23}}{w_{21}},\frac{w_{12}w_{21}}{w_3}\right],\,
U_{3,3}=\mathrm{Spec}\,\mathbb{C}\!\left[\frac{w_{11}}{w_{12}},\frac{w_{13}}{w_{12}},\frac{w_{12}w_{23}}{w_3}, \frac{w_{21}}{w_{23}}\right].$$
Similar to the above argument, we can construct an isomorphism between a neighborhood of $p_{3,1}$ (resp. $p_{3,3}$) and $U_{3,1}$ (resp. $U_{3,3}$). Applying the same construction to $U_4$ and $U_5$ as well, we finally obtain the claim.
\qed

redpajama_set_name: RedPajamaArXiv