# The McKay correspondence for dihedral groups: The Moduli Space and the
Tautological Bundles

John Ashley Capellan  Graduate School of Mathematics, Nagoya University, Furo-
cho, Chikusa-ku, Nagoya, 464-8602, Japan<EMAIL_ADDRESS>
###### Abstract.

A conjecture in [Ish20] states that for a finite subgroup $G$ of
$GL(2,\mathbb{C})$, a resolution $Y$ of $\mathbb{C}^{2}/G$ is isomorphic to a
moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for some generic
stability parameter $\theta$ if and only if $Y$ is dominated by the maximal
resolution. This paper affirms the conjecture in the case of dihedral groups
as a class of complex reflection groups, and offers an extension of McKay
correspondence (via [IN1], [IN2], and [Ish02]).

###### 2010 Mathematics Subject Classification:

Primary 14D20, 14E16, 14J17

##  1\. Introduction

The classical McKay correspondence relates representations of a finite
subgroup $G\subset GL(2,\mathbb{C})$ to the dual graph of exceptional divisors
of the minimal resolution of the quotient variety $\mathbb{C}^{2}/G$.

An algebro-geometric viewpoint of the correspondence was found by [GV] in the
case of $G\subset SL(2,\mathbb{C})$, via some locally free sheaves. In [IN1]
and [IN2], these locally free sheaves are realized as tautological bundles. An
explicit description is obtained using the $G$-Hilbert scheme
$G\operatorname{-Hilb}(\mathbb{C}^{2})$ as the minimal (crepant) resolution of
the quotient variety $\mathbb{C}^{2}/G$. The McKay correspondence of
$SL(2,\mathbb{C})$ was obtained by computing the minimal generators of the
$G$-module $I/(\mathfrak{m}I+\mathfrak{n})$ of each $G$-cluster. This
computation is related to the tops and socles of each $G$-cluster (which is
defined in Definition 5.5).

The aforementioned viewpoints of the correspondence (both [GV] and [IN1]) give
a rigorous proof of the correspondence between representations of $G\subset
SL(2)$ and the exceptional divisors of the minimal resolution of
$\mathbb{C}^{2}/G$. This correspondence can be generalized further into
equivalences of derived categories.

A natural generalization of the McKay correspondence is an equivalence between
the $G$-equivariant geometry of $\mathbb{C}^{n}$ and the geometry of a crepant
resolution $Y$ of $\mathbb{C}^{n}/G$ expressed in the language of derived
categories. When $Y\subset G\operatorname{-Hilb}(\mathbb{C}^{n})$ is the
irreducible component of $G\operatorname{-Hilb}(\mathbb{C}^{n})$ which
contains the open subset of all reduced $G$-clusters, the celebrated result of
[BKR] states some conditions that will determine that
$\tau:Y\rightarrow\mathbb{C}^{n}/G$ is a crepant resolution and that
$\Phi:D(Y)\rightarrow D^{G}(\mathbb{C}^{n})$ given by the Fourier-Mukai
transform gives a derived equivalence. More particularly and explicitly, for
$n\leq 3$ and $G\subset SL(n,\mathbb{C})$, $\Phi$ defines a derived
equivalence.

However, the story does not stop here. The moduli space of $G$-clusters
provide one candidate for a crepant resolution of the quotient variety
$\mathbb{C}^{n}/G$. With the conjecture of Reid (Conjecture 4.1 of [R]) in
mind, which states that if $\tau:Y\rightarrow\mathbb{C}^{n}/G$ is a crepant
resolution, then $\Phi:D(Y)\xrightarrow{\sim}D^{G}(\mathbb{C}^{n})$ for some
derived equivalence $\Phi$, there is an ongoing search for such crepant
resolutions. One of such candidates is the moduli space of $G$-constellations.

A generalization of Hilbert scheme of $G$-orbits is the moduli space of
$G$-constellations (on an affine space) which are introduced in [CI04]. The
moduli space depends on some stability parameter $\theta$ and the moduli space
of $\theta$-stable $G$-constellations is denoted by $\mathcal{M}_{\theta}$. If
$G$ is a subgroup of $SL(n,\mathbb{C})$ acting on $\mathbb{C}^{n}$ and $n\leq
3$, then $\mathcal{M}_{\theta}$ is a crepant resolution of $\mathbb{C}^{n}/G$
for a generic stability parameter $\theta$. The main results in [CI04], [YAb],
[YNAb] realize a (projective) crepant resolution $Y$ of $\mathbb{C}^{3}/G$
(for any finite subgroup $G\subset SL(3,\mathbb{C})$) as a moduli space
$\mathcal{M}_{\theta}$ of $G$-constellations for some generic stability
parameter $\theta$. More precisely, there is a generic stability parameter
$\theta$ such that $Y\cong\mathcal{M}_{\theta}$.

If we generalize from $SL(2,\mathbb{C})$ to $GL(2,\mathbb{C})$, which can be
either small (i.e. which does not contain pseudoreflections) or non-small, we
get a more general McKay correspondence.

The paper [W] obtained an algebraic-geometric viewpoint of the correspondence
in the case of small subgroups of $GL(2,\mathbb{C})$. Other explicit
descriptions of the correspondence, by derived functors and by the work of
[W], were obtained in [Ish02], using the $G$-Hilbert scheme
$G\operatorname{-Hilb}(\mathbb{C}^{2})$ as the minimal resolution of the
quotient variety $\mathbb{C}^{2}/G$, and determined their tops and socles of
each $G$-cluster recovering the Ito-Nakamura type of correspondence, i.e. the
socles of a $G$-cluster recover the same $G$-modules as with the module
$I/(\mathfrak{m}I+\mathfrak{n})$.

Since $\mathcal{M}_{\theta}$ is a resolution of $\mathbb{C}^{2}/G$, there is a
fully faithful functor from $D(\mathcal{M}_{\theta})\hookrightarrow
D^{G}(\mathbb{C}^{2})$. In relation to the DK hypothesis and the maximal
resolution in [Kaw20], Ishii posed the conjecture in [Ish20] that $\tilde{Y}$
is isomorphic to $\mathcal{M}_{\theta}$ for some generic stability parameter
$\theta$ if and only if $\tilde{Y}$ is between the minimal and maximal
resolution of $\mathbb{C}^{2}/G$, where the maximal resolution means the
smooth variety which has unique maximal coefficients satisfying the inequality
in Definition 2.10. So far, this conjecture is solved in the cases of abelian
subgroups and small subgroups of $GL(2,\mathbb{C})$.

It is natural to ask if it is possible to formulate the explicit descriptions
of the McKay correspondence in the case of complex reflection groups. This
case is particularly interesting because the quotient variety
$\mathbb{C}^{2}/G$ is isomorphic to $\mathbb{C}^{2}$ itself in which the
minimal resolution is the identity map which reveals no data about the
exceptional divisors. Hence, we consider the aforementioned maximal resolution
in the hopes of recovering a McKay correspondence. In this paper, we offer an
explicit description of the McKay correspondence in the case of dihedral
groups via its derived equivalence.

Notation: Let $D_{2n}=\left\langle\tau:=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\sigma:=\begin{bmatrix}\epsilon&0\\\
0&\epsilon^{-1}\end{bmatrix}(\epsilon^{n}=1)\right\rangle\subset
GL(2,\mathbb{C})$ be a dihedral group of order $2n$ embedded in the general
linear group $GL(2,\mathbb{C})$. Unless explicitly stated, $G$ is the dihedral
group $D_{2n}$. The representations of $D_{2n}$ are $\rho_{0}$ which is the
trivial representation; $\rho_{0}^{\prime},\rho_{n/2},\rho_{n/2}^{\prime}$ are
non-trivial $1$-dimensional representation; and $\rho_{j}(j\neq
0,0^{\prime},n/2,n/2^{\prime})$ are $2$-dimensional representations. The
character table are as follows:

Table 1. The Irreducible Representations of $D_{2n}$

Representation Conjugacy Class | $1$ | $\tau$ | $\sigma^{i}$
---|---|---|---
$\rho_{0}$ | $1$ | $1$ | $1$
$\rho_{0}^{\prime}$ | $1$ | $-1$ | $1$
$\rho_{j}$ | $2$ | $0$ | $\epsilon^{ij}+\epsilon^{-ij}$
$\rho_{n/2}$ ($n$ even) | $1$ | $1$ | $(-1)^{i}$
$\rho_{n/2}^{\prime}$ ($n$ even) | $1$ | $-1$ | $(-1)^{i}$

where $1\leq j\leq\frac{n-1}{2}$ and $0\leq i\leq n-1$ are integers.

The main results of this paper are the following:

######  Theorem 1.1 (Theorem 3.6).

The maximal resolution $Y_{max}$ of $(\mathbb{C}^{2}/G,\hat{B})$, defined as
the smooth variety which has unique maximal coefficients satisfying the
inequality in Definition 2.10, is isomorphic to the quotient variety
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}:=\langle\sigma\rangle\operatorname{-Hilb}(\mathbb{C}^{2})/(D_{2n}/\langle\sigma\rangle)$,
where $\hat{B}$ is a $\mathbb{Q}$-divisor defined by the equation
$K_{\mathbb{C}^{2}}=\pi^{*}(K_{\mathbb{C}^{2}/D_{2n}}+\hat{B})$, and
$\pi:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}/D_{2n}$ is the projection map. It
is also the minimal embedded resolution of $(\mathbb{C}^{2}/G,\hat{B})$.

The minimal embedded resolution of $(\mathbb{C}^{2}/G,\hat{B})$ is the smooth
surface obtained after the least number of monoidal transformations such that
the strict transform of (the support of) $\hat{B}$ is smooth.

######  Theorem 1.2 (Main Result 1, Theorem 3.12).

A resolution of singularities
$Y\rightarrow\mathbb{C}^{2}/D_{2n}\cong\mathbb{C}^{2}$ is isomorphic to
$\mathcal{M}_{\theta}$ for some generic $\theta$ if and only if $Y$ is
dominated by the maximal resolution of the pair
$(\mathbb{C}^{2}/D_{2n},\hat{B})$.

######  Remark 1.3.

We point out here that $\mathbb{C}^{2}/D_{2n}$ is non-singular. The notion of
resolution $Y\xrightarrow{f}\mathbb{C}^{2}/D_{2n}$ is a proper birational
morphism from the smooth variety $Y$.

Section 3 is devoted to the proof of the first two theorems.

We define the stack associated to the maximal resolution
$\mathcal{Y}:=[Y_{max}]$ realized as the $2$nd root stack

$\mathcal{Y}=\sqrt{(O_{Y_{max}}(\mathbb{B}),1_{\mathbb{B}})/(Y_{max})},$

where $B$ is the boundary divisor or the strict transform of $\hat{B}$,
$(f_{2})_{*}^{-1}(\hat{B})$, under the maximal resolution
$f_{2}:Y_{max}\rightarrow\mathbb{C}^{2}/G$ and $\mathbb{B}:=\left\lceil
B\right\rceil$ (for prime divisors $B_{\alpha}$ of
$B=\Sigma_{\alpha}b_{\alpha}B_{\alpha}$, $\left\lceil
B\right\rceil:=\left\lceil b_{\alpha}\right\rceil B_{\alpha}$); the global
section $1_{\mathbb{B}}$ is induced by the inclusion of divisors
$O_{Y_{max}}\hookrightarrow O_{Y_{max}}(\mathbb{B})$.

We refer to Section 2.2 of [Cad] for the detailed definition of the root
stack. Explicitly, the objects of $\mathcal{Y}$ over a scheme $S$ are
quadruples $(f,M,t,\phi)$, where $f:S\rightarrow Y_{max}$ is a morphism, $M$
is an invertible sheaf on $S$, $t\in\Gamma(S,M)$, and $\phi:M^{\otimes
2}\rightarrow f^{*}(O_{Y_{max}}(\mathbb{B}))$ is an isomorphism such that
$\phi(t^{2})=f^{*}(1_{\mathbb{B}})$.

By Theorem 3.6, we obtain the isomorphism between stacks
$\mathcal{Y}\cong[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}]$.
We have the Fourier-Mukai transforms (defined in Section 4):

$\displaystyle\Phi:D([\mathbb{C}^{2}/D_{2n}])$ $\displaystyle\rightarrow
D(\mathcal{Y})$ $\displaystyle\mathfrak{G}$
$\displaystyle\mapsto\phi_{*}(Rp_{[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}]*}(p_{[\mathbb{C}^{2}/D_{2n}]}^{*}(\mathfrak{G})\otimes
O_{[\mathcal{Z}/D_{2n}]}))$ $\displaystyle\Psi:D(\mathcal{Y})$
$\displaystyle\rightarrow D([\mathbb{C}^{2}/D_{2n}])$ $\displaystyle\epsilon$
$\displaystyle\mapsto
R(p_{[\mathbb{C}^{2}/D_{2n}]*})(p_{[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}]}^{*}(\phi^{*}(\epsilon))\otimes
det(\rho_{nat})\otimes O_{[\mathcal{Z}/D_{2n}]}^{\lor}[2])$

The functors $\Psi$ and $\Phi$ are equivalences via Theorem 4.1 of [IU15].

We define the tautological sheaf associated to a representation $\rho$ of
$D_{2n}$ as

$\hat{\mathcal{R}}_{\rho}:=\Phi(O_{\mathbb{C}^{2}}\otimes\rho^{\lor}).$

######  Theorem 1.4 (Main Result 2, Theorem 4.5).

The tautological bundles on the stack $\mathcal{Y}$ are described by the
following:

Table 2. The Tautological Sheaves for odd $n$

Tautological Bundle | Description | Chern Class
---|---|---
$\hat{\mathcal{R}}_{\rho_{0}}$ | $\mathcal{O}_{\mathcal{Y}}$ | $0$
$\hat{\mathcal{R}}_{\rho_{0}^{\prime}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{3}-\mathcal{D})$ | $\mathcal{B}_{3}-\mathcal{D}$
$\hat{\mathcal{R}}_{\rho_{i}}$ (rank 2) | $0\rightarrow\mathcal{O}_{\mathcal{Y}}\xrightarrow{i}\hat{\mathcal{R}}_{\rho_{i}}\xrightarrow{pr}\mathcal{O}_{\mathcal{Y}}(\mathcal{D}_{i}+\mathcal{B}_{3}-\mathcal{D})\rightarrow 0$ | $\mathcal{D}_{i}+\mathcal{B}_{3}-\mathcal{D}$

Table 3. The Tautological Sheaves for even $n$

Tautological Bundle | Description | Chern Class
---|---|---
$\hat{\mathcal{R}}_{\rho_{0}}$ | $\mathcal{O}_{\mathcal{Y}}$ | $0$
$\hat{\mathcal{R}}_{\rho_{0}^{\prime}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{1}-\mathcal{B}_{2})$ | $\mathcal{B}_{1}-\mathcal{B}_{2}$
$\hat{\mathcal{R}}_{\rho_{i}}$ (rank 2) | $0\rightarrow\mathcal{O}_{\mathcal{Y}}\xrightarrow{i}\hat{\mathcal{R}}_{\rho_{i}}\xrightarrow{pr}\mathcal{O}_{\mathcal{Y}}(\mathcal{D}_{i}+\mathcal{B}_{1}-\mathcal{B}_{2})\rightarrow 0$ | $\mathcal{D}_{i}+\mathcal{B}_{1}-\mathcal{B}_{2}$
$\hat{\mathcal{R}}_{\rho_{(n/2)}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{2})$ | $\mathcal{B}_{2}$
$\hat{\mathcal{R}}_{\rho_{(n/2)}^{\prime}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{1})$ | $\mathcal{B}_{1}$

where $\pi:\mathcal{Y}\rightarrow Y_{max}$ is the morphism to the coarse
moduli space and
$p:\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})\rightarrow\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}$
is the projection; $\mathcal{B}_{i}$ is a prime divisor on $\mathcal{Y}$ such
that $2\mathcal{B}_{i}=\pi^{-1}(B_{i})$ (this is the stacky locus)
$\mathcal{D}:=\pi^{-1}(D)$, where $D$ is a prime divisor of $Y_{max}$ that
satisfies all of the following properties: (1) $D$ does not intersect $B_{i}$
for all $i$, (2) $D$ is transversal to the exceptional divisor intersecting
$B_{1}$ and $B_{2}$ (or $B_{3}$), and (3) $D\cdot E_{j}=0,j\neq m$; and
$\mathcal{D}_{i}:=\pi^{-1}(D_{i})$ with
$D_{i}:=p(\tilde{D}_{i}+g\cdot\tilde{D}_{i})$, whose $\tilde{D}_{i}$ is a
transversal divisor to an exceptional divisor $\tilde{E}_{i}$ of the minimal
resolution
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})\rightarrow\mathbb{C}^{2}/\mathbb{Z}_{n}$.
(The abuse of notation is due to Theorem 3.6.)

The rank one tautological bundles on the stack are uniquely determined by
their Chern classes; and the rank two tautological bundles are determined by
an extension of two line bundles. Furthermore, there is only one possible
(non-trivial) extension class, making these descriptions unique.

The aforementioned theorems are proved by explicitly tracing the by-product of
the quotient variety $\mathbb{C}^{2}/G\cong
G\operatorname{-Hilb}(\mathbb{C}^{2})$ as a subscheme in
$G\operatorname{-Hilb}(\mathbb{C}^{3})$ under flopping operations via [NS17],
which will show that all of the two-dimensional counterparts can be realized
as a moduli space of $D_{2n}$-constellations $\mathcal{M}_{\theta}$ for some
generic stability parameter $\theta$. In particular, the maximal resolution
realized as a moduli space of $D_{2n}$-constellations can be used to construct
a similar McKay correspondence from the work of [AV].

Building from the works [IN1], [IN2] and [Ish02], we formulate an analogous
description of the socles of the $G$-constellations over exceptional divisors
on the stack:

######  Theorem 1.5 (Main Result 3, Theorem 5.9).

For a given $D_{2n}$-constellation $\overline{F}_{st}$ on the exceptional
divisors over the quotient stack $\mathcal{Y}$, where the exceptional divisors
$\mathcal{E}_{i}$ satisfies $\pi^{-1}(E_{i})=\mathcal{E}_{i}$,

$\operatorname{top}(\overline{F}_{st})=\rho_{0}\oplus\rho_{0}^{\prime}$
$\operatorname{socle}(\overline{F}_{st})=\begin{cases}\rho_{i}&[\overline{F}_{st}]\in\mathcal{E}_{i},[\overline{F}_{st}]\notin\mathcal{E}_{j},i\neq
j\\\
\rho_{i}\oplus\rho_{j}&[\overline{F}_{st}]\in\mathcal{E}_{i}\cap\mathcal{E}_{j}\\\
\rho_{\frac{n}{2}-1}\oplus\rho_{n/2}\oplus\rho^{\prime}_{n/2}&[\overline{F}_{st}]\in\mathcal{E}_{\frac{n}{2}-1}\cap\mathcal{E}_{n/2}\mbox{
($n$ even)}\\\
\rho_{n/2}\oplus\rho^{\prime}_{n/2}&[\overline{F}_{st}]\in\mathcal{E}_{n/2}-(\mathcal{E}_{\frac{n}{2}-1}\cup\\{\mathcal{B}_{1},\mathcal{B}_{2}\\})\mbox{
($n$ even)}\\\ \rho_{n/2}&[\overline{F}_{st}]=\mathcal{B}_{1}\mbox{ ($n$
even)}\\\ \rho^{\prime}_{n/2}&[\overline{F}_{st}]=\mathcal{B}_{2}\mbox{ ($n$
even)}\\\ \end{cases}$

In the last two main results, the McKay correspondence was constructed over
the quotient stack. The reader may ask if such correspondence can be made over
the coarse moduli space. Unfortunately, it is not possible, and there is a
comparison between the tautological sheaves over the stack and over the moduli
space in Sections 4 and 5.

##  2\. Preliminaries

###  2.1. $G$-constellations on $\mathbb{C}^{n}$

####  2.1.1. Definitions

Let $V=\mathbb{C}^{n}$ be an affine space and $G\subset GL(V)$ be a finite
subgroup.

######  Definition 2.1.

A $G$-constellation on $V$ is a $G$-equivariant coherent sheaf $E$ on $V$ such
that $H^{0}(E)$ is isomorphic to the regular representation of $G$ as a
$\mathbb{C}[G]$-module. In symbols, $H^{0}(E)\cong\mathbb{C}[G]$.

######  Example 2.2.

When $E=O_{Z}$, the structure sheaf $O_{Z}$ of a $G$-cluster $Z$ inside $V$,
$E$ is a $G$-constellation.

Let $R(G)=\oplus_{\rho\in\operatorname{Irr}(G)}\mathbb{Z}\rho$ be the
representation ring of $G$, where $\operatorname{Irr}(G)$ denotes the set of
irreducible representations of $G$. The parameter space of stability
conditions of $G$-constellations is the $\mathbb{Q}$-vector space:

$\Theta:=\\{\theta\in\operatorname{Hom}_{\mathbb{Z}}(R(G),\mathbb{Q})|\theta(\mathbb{C}[G])=0\\},$

where $\mathbb{C}[G]$ is regarded as the regular representation of $G$.

######  Definition 2.3.

Given $\theta\in\Theta$, a $G$-constellation $E$ is $\theta$-stable (resp.
$\theta$-semistable) if every proper $G$-equivariant coherent subsheaf
$0\subsetneq F\subsetneq E$ satisfies $\theta(H^{0}(F))>0$ (resp.
$\theta(H^{0}(F))\geq 0$). We regard $H^{0}(F)$ here as an element of $R(G)$.

######  Definition 2.4.

A parameter $\theta\in\Theta$ is generic if a $\theta$-semistable
$G$-constellation is also $\theta$-stable.

By Proposition 5.3 of [Kin94], there is a fine moduli scheme
$\mathcal{M}_{\theta}=\mathcal{M}_{\theta}(V)$ of $\theta$-stable
$G$-constellations on $V$ for generic $\theta$.

There is a morphism $\tau:\mathcal{M}_{\theta}(V)\rightarrow V/G$ which sends
a $G$-constellation to its support. By Proposition 2.2 of [CI04], $\tau$ is a
projective morphism when $\theta$ is generic.

######  Definition 2.5.

The subset $\Theta^{gen}\subset\Theta$ of generic parameters is open and
dense. It is the disjoint union of finitely many convex polyhedral cones $C$
in $\Theta$ (see Lemma 3.1 of [CI04]). The convex polyhedral cone $C$ is
called a chamber in $\Theta$.

For $\theta\in\Theta^{gen}$, the moduli space $\mathcal{M}_{\theta}$ only
depends on the open Geometric Invariant Theory (GIT) chamber $C\subset\Theta$
containing $\theta\in\Theta$, so that we can write $\mathcal{M}_{C}$ instead
of $\mathcal{M}_{\theta}$ for any $\theta\subset C$. The following theorem
gives an example:

######  Theorem 2.6 (Theorem 1.1 of [CI04], Theorem 1.1 of [YNAb]).

For a finite subgroup $G\subset SL(3,\mathbb{C})$, suppose that
$Y\rightarrow\mathbb{C}^{3}/G$ is a projective crepant resolution. Then
$Y\cong\mathcal{M}_{C}$ for some GIT chamber $C\subset\Theta$.

The following theorem describes the structure of $G$-constellations for $n=2$.
The arguments on Theorems 1.1 and 1.2 in [BKR] can be adapted to guarantee not
only a resolution of singularities of $\mathbb{C}^{2}/G$, but also the
embedding of their corresponding derived categories, which tell the
relationship between canonical divisors via inequalities following the DK-
hypothesis in [Kaw20].

######  Proposition 2.7 (Theorem 3 of [Ish20]).

Let $G$ be a finite subgroup of $GL(2,\mathbb{C})$. If $\theta$ is generic,
then the moduli space $\mathcal{M}_{\theta}$ is a resolution of singularities
of $\mathbb{C}^{2}/G$. Moreover, the universal family of $G$-constellations
defines a fully faithful functor

$\Phi_{\theta}:D^{b}(coh(\mathcal{M}_{\theta}))\rightarrow
D^{b}(coh^{G}(\mathbb{C}^{2})).$

###  2.2. The Maximal Resolution

Let $G$ be a finite subgroup of $GL(2,\mathbb{C})$, not necessarily small
(i.e. the action may not be free on $\mathbb{C}^{2}-\\{0\\}$). Then the
quotient variety $X=\mathbb{C}^{2}/G$ and its projection
$\mathbb{C}^{2}\xrightarrow{\pi}X$ is equipped with a boundary divisor $B$
determined by the equality $K_{\mathbb{C}^{2}}=\pi^{*}(K_{X}+B)$ expressed as
$B=\Sigma\frac{m_{j}-1}{m_{j}}B_{j}$, where $B_{j}\subset X$ is the image of a
one-dimensional linear subspace whose pointwise stabilizer subgroup
$G_{j}\subset G$ is cyclic of order $m_{j}$. Furthermore, $G$ is small if and
only if $B=0$.

######  Example 2.8.

Consider the abelian group generated by the matrices $\begin{bmatrix}1&0\\\
0&\epsilon_{3}\end{bmatrix}$ and $\begin{bmatrix}-1&0\\\ 0&1\end{bmatrix}$,
i.e. this is the abelian group $G\cong\mathbb{Z}_{3}\times\mathbb{Z}_{2}$.

There is a relation between canonical divisors:
$K_{\mathbb{C}^{2}}=\pi^{*}(K_{\mathbb{C}^{2}/G}+\frac{1}{2}div(x^{2})+\frac{2}{3}div(y^{3})).$

######  Theorem 2.9 (Proposition 5.20 of [KM98]).

The log pair $(X,B)$ is a log terminal singularity.

From this theorem, given a resolution of singularities $\tau:Y\rightarrow X$
and write $K_{Y}+\tau_{*}^{-1}(B)=\tau^{*}(K_{X}+B)+\Sigma_{i}a_{i}E_{i}$,
where $E_{i}$ are the exceptional divisors and $a_{i}\in\mathbb{Q}$, then
$a_{i}>-1$, for all $i$. Then, among all the resolutions $Y$ which satisfy
$a_{i}\leq 0$ for all $i$, we define the maximal resolution of $(X,B)$:

######  Definition 2.10.

Let $(X,B)$ be a log terminal pair of a surface $X$ and a $\mathbb{Q}-$divisor
$B$. We can assume the surfaces $Y$ and $Z$ are smooth. A resolution of
singularities $f:Y\rightarrow X$ is a maximal resolution of $(X,B)$ if
$K_{Y}+f_{*}^{-1}(B)=f^{*}(K_{X}+B)+\Sigma_{i}a_{i}E_{i}$, where $-1<a_{i}\leq
0$, and for any proper birational morphism $g:Z\rightarrow Y$ that is not an
isomorphism, we have
$K_{Z}+h_{*}^{-1}(B)=h^{*}(K_{X}+B)+\Sigma_{j}b_{j}F_{j}$, $h=fg$ and for some
$b_{j}>0$.

######  Theorem 2.11 (expanded from [KS88], Lemma 3.13; generalized in higher-
dimension cases in Theorem 17 of [Kaw20]; and Corollary 1.4.3 of [BCHM]).

A quotient singularity $(X,B)$ of a surface has a unique maximal resolution
(which we denote by $Y_{max}$).

##  3\. Realizing Blow-ups as Moduli Spaces

In this section, we prove Theorem 1.2 (or the conjecture in the case of
dihedral groups) by embedding an affine open subset of each blow-up of
$\mathbb{C}^{2}/D_{2n}$ to a crepant resolution of $\mathbb{C}^{3}/D_{2n}$.

Throughout the rest of this paper (unless explicitly mentioned), in
$G:=D_{2n}$ represented by $\left\langle\sigma=\begin{pmatrix}e^{2\pi
i/n}&0\\\ 0&e^{-2\pi i/n}\end{pmatrix},\tau=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\right\rangle$ and
$H:=\langle\sigma\rangle\cong\mathbb{Z}_{n}$, we have the following
commutative diagram:

${\mathbb{C}^{2}}$${\mathbb{C}^{2}/H}$${X_{1}}$${Y}$${\mathbb{C}^{2}/G}$${Y_{1}}$${(Y_{1})_{max}}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\tau_{2}}$$\scriptstyle{\tau_{2}^{\prime}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{h_{max}}$

where the corresponding varieties and the morphisms are:

$\displaystyle X_{1}$
$\displaystyle:=H\operatorname{-Hilb}(\mathbb{C}^{2})=\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$
$\displaystyle Y_{1}$ $\displaystyle:=X_{1}/(G/H)=X_{1}/(\mathbb{Z}_{2})$
$\displaystyle f_{1}$ $\displaystyle:\mbox{ the minimal (crepant) resolution
of $\mathbb{C}^{2}/H$}$ $\displaystyle\tau_{1}$ $\displaystyle:\mbox{
projection morphisms to $H$-orbits}$ $\displaystyle\tau_{2},\tau_{2}^{\prime}$
$\displaystyle:\mbox{ projection morphisms to their
$G/H\cong\mathbb{Z}_{2}$-orbits}$ $\displaystyle f_{2}$ $\displaystyle:\mbox{
the induced morphism from taking $\mathbb{Z}_{2}$-orbits}$
$\displaystyle(Y_{1})_{max}$ $\displaystyle:=\mbox{ the maximal resolution
$h_{max}$ of $(Y_{1},B^{\prime})$, where $B^{\prime}$ is defined by:}$
$\displaystyle K_{X_{1}}$
$\displaystyle=\tau_{2}^{\prime*}(K_{Y_{1}}+B^{\prime})$ $\displaystyle f_{3}$
$\displaystyle:\mbox{ the minimal resolution of $Y_{1}$}$ $\displaystyle Y$
$\displaystyle:=(G/H)\operatorname{-Hilb}(X_{1})=\mathbb{Z}_{2}\operatorname{-Hilb}(X_{1})$

By the commutative diagram above, because $f_{1}$ is a birational map, $f_{2}$
is also a birational map. We can see this because the projection $\tau_{2}$
induces an inclusion between the ring of rational functions
$(k(\mathbb{C}^{2})^{H})^{G/H}\hookrightarrow k(\mathbb{C}^{2})^{H}$.

We denote the following exceptional divisors and refer to Figures 1 and 2 for
the configuration.:

(3.1) $\displaystyle\mbox{ On $X_{1}$, }\tilde{E_{i}}\mbox{ whose projective
coordinates are defined by $(x^{i}:y^{n-i}),(1\leq i\leq n-1)$}$
$\displaystyle\mbox{ On $Y_{1}$,
}E_{i}:=\tau_{2}^{\prime}(\tilde{E}_{i})\mbox{ so
$\tau_{2}^{\prime}(\tilde{E}_{n-i})=\tau_{2}^{\prime}(\tilde{E}_{i})$ for all
$1\leq i\leq m$}$ where $m=\frac{n-1}{2}$ for odd $n$ and $m=n/2$ for even
$n$.

We define the ramification divisors on $\mathbb{C}^{2}/G$ (i.e. the support of
the discriminant divisor $\hat{B}$ defined by the equation
$K_{\mathbb{C}^{2}/H}=\tau_{2}^{*}(K_{\mathbb{C}^{2}/G}+\hat{B})$) with their
corresponding explicit equations as:

(3.2) $\displaystyle\hat{B}_{1}:\langle(x^{n/2}+y^{n/2})^{2}=0\rangle$
$\displaystyle\hat{B}_{2}:\langle(x^{n/2}-y^{n/2})^{2}=0\rangle$
$\displaystyle\hat{B}_{3}:\langle(x^{n}-y^{n})^{2}=0\rangle$

so that we can define their corresponding strict transformations for $i=1,2,3$
as:

(3.3) $\displaystyle B_{i}:=(f_{2})_{*}^{-1}(\hat{B}_{i})$
$\displaystyle\tilde{B_{i}}:=(\tau_{2}^{\prime})_{*}^{-1}(B_{i}).$

•••••••• $\tilde{E}_{1}:(x:y^{n-1})$ $\tilde{E}_{2}:(x^{2}:y^{n-2})$
$\tilde{E}_{(n-1)/2}:(x^{(n-1)/2}:y^{(n+1)/2})$
$\tilde{E}_{(n+1)/2}:(x^{(n+1)/2}:y^{(n-1)/2})$ $\tilde{B}_{3}$
$\tilde{E}_{n-2}:(x^{n-2}:y^{2})$ $\tilde{E}_{n-1}:(x^{n-1}:y)$

$\downarrow{\tau_{2}^{\prime}}$

•••• $E_{1}$ $E_{2}$ $E_{(n-1)/2}$ $\tau_{2}^{\prime}(\tilde{B}_{3})$

Figure 1. Configuration of Exceptional Divisors and Boundary Divisors on
$X_{1}$ and $Y_{1}$ for Odd $n$ Case

•••••••• $\tilde{E}_{1}:(x:y^{n-1})$ $\tilde{E}_{2}:(x^{2}:y^{n-2})$
$\tilde{E}_{n/2}:(x^{n/2}:y^{n/2})$ $\tilde{B}_{1}$ $\tilde{B}_{2}$
$\tilde{E}_{n-2}:(x^{n-2}:y^{2})$ $\tilde{E}_{n-1}:(x^{n-1}:y)$

$\downarrow{\tau_{2}^{\prime}}$

•••• $E_{1}$ $E_{2}$ $E_{n/2}$ $\tau_{2}^{\prime}(\tilde{B}_{1})$
$\tau_{2}^{\prime}(\tilde{B}_{2})$

Figure 2. Configuration of Exceptional Divisors and Boundary Divisors on
$X_{1}$ and $Y_{1}$ for Even $n$ Case

Using the notations in [NS17], we define $f_{1}:=x^{2m+1}+y^{2m+1}$ and
$f_{2}:=x^{2m+1}-y^{2m+1}$ in the odd $n$ case; and $f_{1}:=x^{m}+y^{m}$ and
$f_{2}:=x^{m}-y^{m}$ in the even $n$ case. We also note here that
$\mathbb{C}[x,y]^{D_{2n}}=\mathbb{C}[xy,x^{n}+y^{n}]$.

${\circ_{E_{1}:\rho_{1}}}$${\circ_{E_{2}:\rho_{2}}}$${\cdots}$${\circ_{E_{(n-1)/2}:\rho_{\frac{n-1}{2}}}}$${\circ_{B:\rho_{\frac{n-1}{2}}}}$

${\circ_{B_{1}:\rho_{n/2}}}$${\circ_{E_{1}:\rho_{1}}}$${\circ_{E_{2}:\rho_{2}}}$${\cdots}$${\circ_{E_{n/2}:\rho_{n/2}\oplus\rho_{n/2}^{\prime}}}$${\circ_{B_{2}:\rho_{n/2}^{\prime}}}$

Figure 3. Dual graph of the exceptional divisors and boundary divisors of
$f_{2}$

We prepare some propositions.

######  Proposition 3.1.

The surface $Y_{1}$ is smooth. Hence, $f_{2}$ is a resolution of
$(\mathbb{C}^{2}/G,\hat{B})$.

###### Proof.

We compute $(X_{1})^{\mathbb{Z}_{2}}$, the fixed locus of the
$\mathbb{Z}_{2}$-action on $X_{1}$ by taking a closed subscheme $V$ to $g\cdot
V$, where $g$ is an element of $\mathbb{Z}_{2}$.

The points of $X_{1}$ are $G$-invariant $0$-dimensional subscheme of
$\mathbb{C}^{2}$ (whose space of global sections is isomorphic to the regular
representation, i.e. $H^{0}(O_{Z})\cong\mathbb{C}[G]$), and so it can be
realized as an ideal defining the aforementioned closed subscheme of
$\mathbb{C}^{2}$. Referring to Thm. 2.2 of [IN1]; and Remark 9.7, Lemma 12.2,
and Theorem 12.3 of [IN2], we can identify the points on the exceptional
divisors of $X_{1}$ as $I_{i}(a_{i}:b_{i})=\langle
a_{i}x^{i}-b_{i}y^{n-i},x^{i+1},xy,y^{n+1-i}\rangle$, where $1\leq i\leq n$
and $(a_{i}:b_{i})\in\mathbb{P}^{1}$; or equivalently, using the open affine
covers of
$X_{1}=\bigcup_{i=1}^{n}U_{i}:=\bigcup_{i=1}^{n}\operatorname{Spec}\left(\mathbb{C}\left[\frac{x^{i}}{y^{n-i}},\frac{y^{n+1-i}}{x^{i-1}}\right]\right)$,
the points
$\left(\frac{x^{i}}{y^{n-i}},\frac{y^{n+1-i}}{x^{i-1}}\right)=(0,b)$ and
$\left(\frac{x^{i}}{y^{n-i}},\frac{y^{n+1-i}}{x^{i-1}}\right)=(a,0)$ on the
exceptional divisor correspond to $I_{i-1}(b:1)$ and $I_{i}(1:a)$,
respectively.

Furthermore, the $\mathbb{Z}_{2}$-action sends $I_{i}(a_{i}:b_{i})=\langle
a_{i}x^{i}-b_{i}y^{n-i},x^{i+1},xy,y^{n+1-i}\rangle$ to
$I_{n-i}(b_{i}:a_{i})=\langle
a_{i}y^{i}-b_{i}x^{n-i},x^{n-i+1},xy,y^{i+1}\rangle$, so that the fixed points
on the exceptional divisors of $X_{1}$ are:

When $n$ is odd, $I_{(n-1)/2}(0:1)=I_{(n+1)/2}(1:0)$.

When $n$ is even, there are two fixed points, $I_{n/2}(1:1)$ or
$I_{n/2}(-1:1)$.

From the commutative diagram below, we can check smoothness of $Y_{1}$ by
verifying that the $\mathbb{Z}_{2}$ acts on $X_{1}$ as a pseudoreflection.
This amounts in showing that the fixed points under the
$\mathbb{Z}_{2}$-action are on
$\overline{((f_{1})_{*}^{-1}\circ(\tau_{2})_{*}^{-1})(\hat{B}_{i,0})}$.

${(\tau_{2})_{*}^{-1}(\hat{B}_{i,0})\subset\mathbb{C}^{2}/H-\\{(0,0)\\}}$${X_{1}-\Sigma(\tilde{E}_{i})}$${\hat{B}_{i}-\\{(0,0)\\}=:\hat{B}_{i,0}\subset\mathbb{C}^{2}/G-\\{(0,0)\\}}$${Y_{1}-\Sigma(E_{i})}$$\scriptstyle{\tau_{2}}$$\scriptstyle{\tau_{2}^{\prime}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$

First, we define the open affine covering of
$X_{1}=\bigcup_{i=1}^{n}U_{i}:=\bigcup_{i=1}^{n}\operatorname{Spec}\left(\mathbb{C}\left[\frac{x^{i}}{y^{n-i}},\frac{y^{n+1-i}}{x^{i-1}}\right]\right)$.

By the definition of $\hat{B}_{i}$ from (3.2), we compute the strict transform
$((f_{1})_{*}^{-1}\circ(\tau_{2})_{*}^{-1})(\hat{B}_{i})$ on each of the
affine open sets covering $X_{1}$.

On
$\operatorname{Spec}\left(\mathbb{C}\left[\frac{x^{i}}{y^{n-i}},\frac{y^{n+1-i}}{x^{i-1}}\right]\right)$,

$(x^{n}-y^{n})^{2}=\left(\frac{x^{i}}{y^{n-i}}\right)^{2(n+1-i)}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{2(n-i)}-2\left(\frac{x^{i}}{y^{n-i}}\right)^{n}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{n}+\left(\frac{x^{i}}{y^{n-i}}\right)^{2(i-1)}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{2i}$

For $\hat{B_{1}}$:

$0=x^{n}+2(xy)^{n/2}+y^{n}=\left(\frac{x^{n/2}}{y^{n/2}}\right)^{\frac{n-2}{2}}\left(\frac{y^{(n+2)/2}}{x^{(n-2)/2}}\right)^{n/2}\left[\left(\frac{x^{n/2}}{y^{n/2}}\right)+1\right]^{2}$

so that the strict transform is the line $\frac{x^{n/2}}{y^{n/2}}=-1$ on the
open set $U_{n/2}$. The coordinate
$(\frac{x^{n/2}}{y^{n/2}},\frac{y^{(n+2)/2}}{x^{(n-2)/2}})=(-1,0)$ corresponds
to the point on the $G$-Hilbert scheme $I_{n/2}(1:-1)$. This works similarly
for $U_{(n/2)+1}$ and $U_{(n+1)/2}$ (for odd $n$). The same argument works for
$\hat{B_{2}}$ in which we obtain $I_{n/2}(1:1)$; and for $\hat{B_{3}}$ in
which we obtain $I_{(n-1)/2}(0:1)=I_{(n+1)/2}(1:0)$. This completes the
description of the strict transform of the boundary divisors.

To show further that the closure
$\overline{((f_{1})_{*}^{-1}\circ(\tau_{2})_{*}^{-1})(\hat{B}_{i})}$ does not
exist on other open sets other than $U_{n/2},U_{(n/2)+1},U_{(n+1)/2}$, we
again notice that WLOG:

$(x^{n}-y^{n})^{2}=\left(\frac{x^{i}}{y^{n-i}}\right)^{2(i-1)}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{2i}\left[\left(\frac{x^{i}}{y^{n-i}}\right)^{2(n+2-2i)}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{2(n-2i)}-2\left(\frac{x^{i}}{y^{n-i}}\right)^{n+2-2i}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{n-2i}+1\right]$

This implies that plugging any of the coordinates of $U_{i}$ to zero does not
lie on the strict transform defined by:
$0=\left(\frac{x^{i}}{y^{n-i}}\right)^{2(n+2-2i)}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{2(n-2i)}-2\left(\frac{x^{i}}{y^{n-i}}\right)^{n+2-2i}\left(\frac{y^{n+1-i}}{x^{i-1}}\right)^{n-2i}+1$.
This completes the description of the strict transform of the boundary
divisors.

These all imply that $dim((X_{1})^{\mathbb{Z}_{2}})=1$, or equivalently,
$\mathbb{Z}_{2}$ acts as a pseudo-reflection on $X_{1}$, which implies that
the boundary divisor $B^{\prime}$ on $Y_{1}$ determined by the equation
$K_{X_{1}}=\tau_{2}^{\prime*}(K_{Y_{1}}+B^{\prime})$ is smooth and so $Y_{1}$
is smooth. ∎

######  Remark 3.2.

A more general statement for the smoothness of $\tau_{2}^{\prime}$ is as
follows:

######  Corollary 3.3.

For $H\subset SL(2),Y_{1}:=H\operatorname{-Hilb}(\mathbb{C}^{2})$, so that
$G/H$ is cyclic, then $Y_{1}/(G/H)$ is smooth iff $G/H$ is a (cyclic) complex
reflection group (or equivalently, if $G/H$ has a local linear action on
$Y_{1}$ by pesudoreflections).

The dihedral group is a special case of this.

######  Proposition 3.4.

The morphism $f_{2}:Y_{1}\rightarrow(\mathbb{C}^{2}/G,\hat{B})$ is a crepant
resolution.

###### Proof.

From the fact that $f_{1}$ is a crepant resolution:

$\displaystyle K_{X_{1}}$ $\displaystyle=f_{1}^{*}(K_{\mathbb{C}^{2}/H})$
$\displaystyle K_{Y_{1}}+(f_{2})_{*}^{-1}(\hat{B})$
$\displaystyle=f_{2}^{*}(K_{\mathbb{C}^{2}/G}+\hat{B})+\Sigma_{j}a_{j}F_{j}$
$\displaystyle K_{X_{1}}$
$\displaystyle=(\tau_{2}^{\prime})^{*}(K_{Y_{1}}+B^{\prime})$

We need to show that $(f_{2})_{*}^{-1}(\hat{B})=B^{\prime}=B_{Y_{1}}$ by
showing that $a_{j}=0$ for all $j$.

$\displaystyle K_{\mathbb{C}^{2}/H}$
$\displaystyle=\tau_{2}^{*}(K_{\mathbb{C}^{2}/G}+\hat{B})$ $\displaystyle
K_{X_{1}}=f_{1}^{*}(K_{\mathbb{C}^{2}/H})$
$\displaystyle=f_{1}^{*}(\tau_{2}^{*}(K_{\mathbb{C}^{2}/G}+\hat{B}))$
$\displaystyle=(\tau_{2}\circ f_{1})^{*}(K_{\mathbb{C}^{2}/G}+\hat{B})$
$\displaystyle=(f_{2}\circ\tau_{2}^{\prime})^{*}(K_{\mathbb{C}^{2}/G}+\hat{B})$
$\displaystyle=\tau_{2}^{\prime*}(K_{Y_{1}}+(f_{2})_{*}^{-1}(\hat{B})-\Sigma_{j}a_{j}F_{j}),$
(where $F_{j}$ are exceptional divisors of $f_{2}$)
$\displaystyle=\tau_{2}^{\prime*}(K_{Y_{1}}+(f_{2})_{*}^{-1}(\hat{B}))-\Sigma_{j}a_{j}\tau_{2}^{\prime*}(F_{j})$

This implies that $\tau_{2}^{\prime*}(F_{j})$ are exceptional divisors for
$f_{1}$ which forces the discrepancies to be zero. ∎

This proposition implies that $Y_{1}$ is also a crepant resolution of
$(\mathbb{C}^{2}/G,\hat{B})$. Also, we recall the notion of the minimal
embedded resolution of $(\mathbb{C}^{2}/G,\hat{B})$.

######  Proposition 3.5 (Proposition 3.8 (Ch. V) of [Hart]).

Let $C_{0}$ be an irreducible curve in the surface $X_{0}$. Then there exists
a finite sequence of monoidal transformations (with suitable centers)
$X_{n}\rightarrow X_{n-1}\rightarrow...\rightarrow X_{1}\rightarrow X_{0}$
such that the strict transform $C_{n}$ of $C_{0}$ on $X_{n}$ is nonsingular.

The minimum $n$ that satisfies this proposition is the minimal embedded
resolution of $(X_{0},C_{0})$. This is minimal in the sense that if $Y$ is a
smooth surface and dominates $X_{0}$, then it dominates $X_{n}$ as well. For
instance, a smooth surface $Y$ with normal crossings dominates the minimal
embedded resolution. A more detailed description is given in Theorem 3.9 (Ch.
V) of [Hart].

######  Theorem 3.6.

The maximal resolution $Y_{max}$ of $(\mathbb{C}^{2}/G,\hat{B})$ is isomorphic
to the quotient variety $Y_{1}$. It is also the minimal embedded resolution of
$(\mathbb{C}^{2}/G,\hat{B})$. Furthermore, the iterated Hilbert scheme $Y$ is
also isomorphic to the maximal resolution $Y_{max}$.

###### Proof.

First, from the proof of Theorem 1 in [Kaw20], the maximal resolution of
$(Y_{1},B^{\prime})$ defined by $h_{max}$ is also the maximal resolution of
$(\mathbb{C}^{2}/G,\hat{B})$ defined by $h_{max}\circ f_{2}$.

Because $Y_{1}$ has at worst cyclic quotient singularities, and
$(Y_{1},B_{Y_{1}})$ has the smoothness property for both the variety and the
boundary divisor, the minimal resolution of
$(Y_{1},(f_{2})_{*}^{-1}(\hat{B}))$, i.e.
$f_{3}:Y\rightarrow(Y_{1},(f_{2})_{*}^{-1}(\hat{B}))$ is crepant, and more
strongly, $f_{3}=id$, which implies that the maximal resolution of
$(\mathbb{C}^{2}/G,\hat{B})$ is $Y_{1}$. ∎

######  Remark 3.7.

An explicit way to do this is to consider the affine open covers of $Y_{1}$
via the open affine covers of $X_{1}$. We show this for the odd $n$ case,
since the argument for the even case is similar.

Because $f_{2}$ is a crepant resolution, it remains to compute the discrepancy
of the blow-up $h:Z:=Blp(Y_{1})\rightarrow Y_{1}$, and we divide it into two
cases depending on where the center of the blow-up is. The more interesting
case is where the center of $h$ is on $f_{2*}^{-1}(\hat{B})$ (this can be
realized also as the boundary divisor for the morphism $\tau_{2}^{\prime}$):

In the odd $n$ case, in $\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$,
the open set
$Spec\left(\mathbb{C}\left[\frac{x^{(n+1)/2}}{y^{(n-1)/2}},\frac{y^{(n+1)/2}}{x^{(n-1)/2}}\right]\right)$
which covers the invariant locus under the $\mathbb{Z}_{2}$-action is
$\mathbb{Z}_{2}$-invariant. Thus, we consider the open set
$Spec\left(\mathbb{C}\left[\frac{x^{(n+1)/2}}{y^{(n-1)/2}},\frac{y^{(n+1)/2}}{x^{(n-1)/2}}\right]\right)^{\mathbb{Z}_{2}}=Spec\left(\mathbb{C}\left[xy,\frac{f_{1}}{(xy)^{m}}\right]\right)$.
The boundary locus in $\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$ is
$\left(\frac{x^{(n+1)/2}}{y^{(n-1)/2}}-\frac{y^{(n+1)/2}}{x^{(n-1)/2}}\right)^{2}=0$,
which translates to $\left(\frac{f_{1}}{(xy)^{m}}\right)^{2}-4xy=0$ on the
invariant open set.

For any point
$\left(xy,\frac{f_{1}}{(xy)^{m}}\right)=\left(\frac{1}{4}a^{2},a\right)$ on
$\tilde{B}_{m}$, performing the coordinate change, we obtain the new equation:
$\left(\frac{f_{1}}{(xy)^{m}}\right)^{2}+2a\frac{f_{1}}{(xy)^{m}}+a^{2}=\left(\frac{f_{1}}{(xy)^{m}}+a\right)^{2}=4(xy+\frac{1}{4}a^{2})=4xy+a^{2}$.

On
$Spec\left(\mathbb{C}\left[\frac{(xy)^{m+1}}{f_{1}},\frac{f_{1}}{(xy)^{m}}\right]\right)$,
the equation transforms to
$\frac{f_{1}}{(xy)^{m}}-2a=4\frac{(xy)^{m+1}}{f_{1}}$. The exceptional divisor
defines the equation $\frac{f_{1}}{(xy)^{m}}=0$.

Thus, the intersection number of $h_{*}^{-1}(B^{\prime})$ with the exceptional
divisor is $1/2$. Using the relation between canonical divisors, the
discrepancy $a_{m+1}=1/2$. For the even $n$ case, this reduces to a blow-up
along lines which is treated similarly.

Thus, $Y_{1}$ is the maximal resolution of $(\mathbb{C}^{2}/D_{2n},\hat{B})$.
Furthermore, because $f_{3}$ is a crepant resolution of $Y_{1}$, $f_{3}$ must
be an isomorphism. Hence, $Y\cong(\mathbb{C}^{2}/G,\hat{B})_{max}\cong Y_{1}$.

This particular assertion tells us that the maximal resolution can be realized
as a moduli space of $G$-constellations which will help in our computations
later.

Once again, we refer to (3.1) for the definition of the exceptional divisors
for the next lemma:

######  Lemma 3.8.

For an exceptional divisor $\tilde{E}$ (resp. $E$) of $X_{1}$ (resp. $Y_{1}$),
we know that the normal bundles $\mathcal{N}_{\tilde{E}/X_{1}}$ are of degree
$-2$, or equivalently, $\mathcal{N}_{\tilde{E}/X_{1}}\cong O_{\tilde{E}}(-2)$.
Then:

$\mathcal{N}_{E/Y_{1}}\cong\begin{cases}O_{E}(-1)&\mbox{ if $E=E_{m}$}\\\
O_{E}(-2)&\mbox{ if $E\neq E_{m}$}\\\ \end{cases}$

###### Proof.

The first statement for $X_{1}$ is well-known since it is the (minimal)
crepant resolution of the quotient singularity
$\mathbb{C}^{2}/\mathbb{Z}_{n}$.

We can compute the self-intersection number $E^{2}$ via the adjunction formula
and given our computations in Proposition 3.1 regarding the fixed points of
$X_{1}$ under the $\mathbb{Z}_{2}$-action.

Because $f_{2}$ is crepant, we have
$K_{Y_{1}}+(f_{2})_{*}^{-1}(\hat{B})=(f_{2})^{*}(K_{\mathbb{C}^{2}/G}+\hat{B})$,
so that $K_{Y_{1}}\cdot E_{m}=-1$ and $K_{Y_{1}}\cdot E=0$ for $E\neq E_{m}$.
∎

######  Corollary 3.9.

The only resolutions dominated by the maximal resolution of
$(\mathbb{C}^{2}/G,\hat{B})$ are essentially the blow-ups from
$(\mathbb{C}^{2}/G,\hat{B})$ with center the singular point of the (strict
transforms of the) boundary divisor $\hat{B}$.

###### Proof.

Using the same argument as in Lemma 3.8, after the blow-down of the
$(-1)$-curve on $Y_{1}$ and so on, we obtain the result. ∎

The next lemma provides an isomorphism between the minimal resolution of the
variety and the quotient variety.

######  Lemma 3.10.

For a complex reflection group $G\subset GL(n,\mathbb{C})$, there is an
isomorphism between the $G$-Hilbert scheme and the quotient variety. In
symbols:

$G\operatorname{-Hilb}(\mathbb{C}^{n})\cong\mathbb{C}^{n}/G.$

###### Proof.

We consider the moduli functor for $G$-clusters:

$h:S\mapsto\\{\mbox{flat families of $G$-clusters parametrized by
$S$}\\}/\equiv$

for a locally Noetherian scheme $S$ over $\mathbb{C}$ where $E_{S}\equiv
F_{S}$ if and only if there is an $L$ in $\operatorname{Pic}(S)$ such that
$E_{S}\cong F_{S}\otimes L$.

The $G$-Hilbert scheme $G\operatorname{-Hilb}(\mathbb{C}^{n})$ represents the
functor $h$. Thus:
$h(S)\cong\operatorname{Hom}_{Sch}(S,G\operatorname{-Hilb}(\mathbb{C}^{n}))$.
We wish to show that $h(S)\cong\operatorname{Hom}_{Sch}(S,\mathbb{C}^{n}/G)$.

We construct the map first from $h(S)$ to
$\operatorname{Hom}_{Sch}(S,\mathbb{C}^{n}/G)$.

(1) From $\upsilon:\operatorname{Hom}_{Sch}(S,\mathbb{C}^{n}/G)$ to $h(S)$.

Given $\gamma_{S}\in\operatorname{Hom}_{Sch}(S,\mathbb{C}^{n}/G)$, consider
the fiber product diagram:

${S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}}$${\mathbb{C}^{n}}$${S}$${\mathbb{C}^{n}/G}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{p}$$\scriptstyle{\gamma_{S}}$

It remains to show that every fiber of $p$ is a $G$-cluster. This implies that
the fiber product $S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}$ is a flat family
of $G$-clusters.

By the Chevalley-Shephard-Todd theorem, the morphism $p$ is flat. Then by the
decomposition of
$p_{*}(O_{\mathbb{C}^{n}/G})=\bigoplus_{\rho\in\operatorname{Irr}(G)}M_{\rho}\otimes\rho$
over representations of a finite group $G$ with characteristic $0$, where
$M_{\rho}=(p_{*}(O_{\mathbb{C}^{n}/G})\otimes\rho^{\lor})^{G}$ is a finitely
generated $O_{\mathbb{C}^{n}/G}$-module, each of the modules $M_{\rho}$ is
locally free.

Over the free locus on $\mathbb{C}^{n}/G$, the fiber consists of a
$G$-cluster. Thus, considering a family of representations of a finite group
$G$, the fiber over the non-free locus is also a $G$-cluster.

(2) From $\Lambda:h(S)$ to $\operatorname{Hom}_{Sch}(S,\mathbb{C}^{n}/G)$.

Given a flat family $\mathcal{Z}$ of $G$-clusters over a scheme $S$, which is
a subscheme of $S\times\mathbb{C}^{n}$, we wish to construct a scheme morphism
$\delta_{\mathcal{Z}}:S\rightarrow\mathbb{C}^{n}/G$. We consider first the
following diagram:

${\mathcal{Z}}$${\mathbb{C}^{n}}$${S}$${\mathbb{C}^{n}/G}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{p}$$\scriptstyle{\delta_{\mathcal{Z}}}$

Taking note that the action of $G$ on $S$ is trivial, so that again, there is
a decomposition of
$(p_{1})_{*}(O_{\mathcal{Z}})=\bigoplus_{\rho\in\operatorname{Irr}(G)}S_{\rho}\otimes\rho$.

Taking the $G$-invariant sections gives
$[(p_{1})_{*}(O_{\mathcal{Z}})]^{G}=S_{\rho_{0}}$, which is of rank $1$, which
is generated by the non-vanishing global section $1$. This implies that
$S_{\rho_{0}}=O_{S}$ and $\mathcal{Z}/G=S$.

Thus, the morphism
$\delta_{\mathcal{Z}}:\mathcal{Z}/G=S\rightarrow\mathbb{C}^{n}/G$ is induced
by the morphism $p_{2}:\mathcal{Z}\rightarrow\mathbb{C}^{n}$.

(3) Now that we have constructed the maps, we want to show now that the maps
$\upsilon$ and $\Lambda$ induce a bijection between sets. First, we show that
$\Lambda\circ\upsilon=1_{\operatorname{Hom}_{Sch}(S,\mathbb{C}^{n}/G)}$.

By the construction of the map $\Lambda$, $p_{2}$ induces the map
$(p_{2})/G:S=(S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}/G)=(S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n})/G\rightarrow\mathbb{C}^{n}/G$.
From the construction, $S$ is a categorical quotient $p_{1}$. Thus, by the
universality property of the categorical quotient applied to the morphism
$p\circ p_{2}$, $\gamma_{S}=(p_{2})/G$.

(4) We now show that $\upsilon\circ\Lambda=1_{h(S)}$. This amounts to show
that $\mathcal{Z}=S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}$.

Consider the inclusion morphism $i:\mathcal{Z}\hookrightarrow
S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}$, which is a closed immersion (via
the closed immersion $S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}\hookrightarrow
S\times_{\mathbb{C}}\mathbb{C}^{n}$), induced by the universal property of the
fiber product diagram. Then we have the exact sequence:

$0\rightarrow\mathcal{I}\rightarrow
O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}}\rightarrow
i_{*}(O_{\mathcal{Z}})\rightarrow 0$

where $\mathcal{I}$ is the kernel of
$O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}}\rightarrow
i_{*}(O_{\mathcal{Z}})$.

Because $p_{1}$ is finite, the pushforward functor $(p_{1})_{*}$ is exact:

$0\rightarrow(p_{1})_{*}\mathcal{I}\rightarrow(p_{1})_{*}(O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}})\rightarrow(p_{1})_{*}(i_{*}(O_{\mathcal{Z}}))=(p_{1})_{*}(O_{\mathcal{Z}})\rightarrow
0$

Because both $S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}$ and $\mathcal{Z}$ are
flat families of $G$-clusters over $S$, the sheaf
$(p_{1})_{*}(O_{\mathcal{Z}})$ is flat and every fiber of both
$(p_{1})_{*}(O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}})$ and
$(p_{1})_{*}(O_{\mathcal{Z}})$ are $G$-clusters. Taking the fibers over $s\in
S$ in the exact sequence, both
$(p_{1})_{*}(O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}})_{s}\otimes(O_{S,s}/\mathfrak{m}_{S,s})=:[(p_{1})_{*}(O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}})](s)$
and $[(p_{1})_{*}(i_{*}(O_{\mathcal{Z}}))=(p_{1})_{*}(O_{\mathcal{Z}})](s)$
have the same dimension as vector spaces over $\mathbb{C}$. Thus, they are
isomorphic as vector spaces. This leaves the fiber
$[(p_{1})_{*}\mathcal{I}](s)=0$ for all $s\in S$. By Nakayama Lemma for local
rings applied to the coherent sheaf $(p_{1})_{*}\mathcal{I}$, for
$\mathcal{I}$ is coherent and $p_{1}$ is finite, this implies that the stalk
$(p_{1})_{*}(\mathcal{I})_{s}=0$, which implies that
$(p_{1})_{*}(\mathcal{I})=0$.

Again, because $p_{1}$ is finite, then the natural map
$0=(p_{1})^{*}(p_{1})_{*}(\mathcal{I})\rightarrow\mathcal{I}$ is surjective,
which implies that $\mathcal{I}=0$. This implies now that
$O_{S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}}\cong i_{*}(O_{\mathcal{Z}})$,
and so $\mathcal{Z}=S\times_{\mathbb{C}^{n}/G}\mathbb{C}^{n}$. ∎

######  Remark 3.11.

The $G$-cluster that corresponds to the points on the boundary divisor of
$\mathbb{C}^{n}/G$ is a non-reduced closed subscheme.

######  Theorem 3.12.

Given a log pair of the quotient variety $(\mathbb{C}^{2}/D_{2n},\hat{B})$
determined by the projection morphism
$\pi:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}/D_{2n}$ via the relation
$K_{\mathbb{C}^{2}}=\pi^{*}(K_{\mathbb{C}^{2}/D_{2n}}+\hat{B})$ from the
previous section, then the following hold for the blow-ups of the quotient
variety $\mathbb{C}^{2}/D_{2n}$:

  1. (1)

The maximal resolution for $(\mathbb{C}^{2}/D_{2n},\hat{B})$ is obtained after
$m=\frac{n-1}{2}$ (for odd case) and $m=\frac{n}{2}$ (for even case) blowing
ups with singular points of the boundary divisors as the center which
satisfies the inequality in Definition 2.10.

  2. (2)

For each resolution $\tilde{Y}\rightarrow\mathbb{C}^{2}/D_{2n}$ dominated by
the maximal resolution $Y_{max}$, there is a generic $\theta$ in the parameter
space of $G$-constellations such that $\tilde{Y}\cong\mathcal{M}_{\theta}$.

###### Proof.

(1) Because $X_{1}$ has $n-1$ exceptional divisors, $Y_{1}$ has
$\frac{n-1}{2}$ (for $n$ odd) or $\frac{n}{2}$ (for even $n$) exceptional
divisors. The surface $Y_{1}$ being the maximal resolution follows from
Theorem 3.6. And the contraction of exceptional divisors yielding a smooth
resolution follows from Corollary 3.9.

(2) Because $(Y_{1})_{max}\cong Y_{1}\cong Y$, we can look instead at the
iterated Hilbert scheme and naturally embed $Y$ into
$\mathbb{Z}_{2}\operatorname{-Hilb}(\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{3}))$.

We embed the group $D_{2n}\subset GL(2)\hookrightarrow SL(3)$ via taking the
determinant so that $SL(3)\supset D_{2n}=\left\langle\begin{bmatrix}0&1&0\\\
1&0&0\\\ 0&0&-1\end{bmatrix},\sigma:=\begin{bmatrix}\epsilon&0&0\\\
0&\epsilon^{-1}&0\\\ 0&0&1\end{bmatrix}\right\rangle$, so that it induces the
group action of $D_{2n}$ on $\mathbb{C}^{3}$ by matrix multiplication as
well). This iterated Hilbert scheme
$\mathbb{Z}_{2}\operatorname{-Hilb}(\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{3}))$
is identified with $X_{0...(m-1)}$ via Theorems 5.1 and 5.2 of [NS17] and
Example 6.1 of [IIN]. The quotient variety $\mathbb{C}^{2}/D_{2n}$ can be
realized as a subscheme of $D_{2n}\operatorname{-Hilb}(\mathbb{C}^{3})$.

We refer to the table of the normal bundles of the exceptional divisors (and
their flops) $\mathcal{N}_{X/E}$ with their corresponding open covers in the
same Theorems 5.1 and 5.2 of [NS17] to see which open sets cover the floppable
$(-1,-1)$ curve.

From the notation in [NS17], under a suitable generic parameter $\theta^{i}$
satisfying the inequalities of Theorem 6.4 of [NS17], we define
$Y_{i}^{\prime}:=\mathcal{M}_{\theta^{i}}(\mathbb{C}^{2})$ realized as moduli
space of $G$-constellations of $\mathbb{C}^{2}$ which can be embedded in
$X_{0...i}:=\mathcal{M}_{\theta^{i}}(\mathbb{C}^{3})$ realized as moduli space
of $G$-constellations of $\mathbb{C}^{3}$.

${Y_{m-1}^{\prime}=Y_{max}=\mathbb{Z}_{2}\operatorname{-Hilb}(\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2}))}$${Y_{m-2}^{\prime}}$${\cdots}$${Y_{0}^{\prime}}$${\mathbb{C}^{2}/G\cong
G\operatorname{-Hilb}(\mathbb{C}^{2})}$${X_{0...(m-1)}=\mathbb{Z}_{2}\operatorname{-Hilb}(\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{3}))}$${X_{0...(m-2)}}$${\cdots}$${X_{0...0}}$${G\operatorname{-Hilb}(\mathbb{C}^{3})}$$\scriptstyle{T_{m-1}}$$\scriptstyle{T_{m-2}}$$\scriptstyle{T_{1}}$$\scriptstyle{T_{0}}$

The number of flops from the iterated Hilbert scheme $X_{0...(m-1)}$ to the
$G$-Hilbert scheme $G\operatorname{-Hilb}(\mathbb{C}^{3})$ is the same as the
number of exceptional divisors of the morphism
$f_{2}:Y_{max}\rightarrow\mathbb{C}^{2}/G$.

After each flop of $(-1,-1)$ curve, the number of exceptional divisors over
the surface must either: (a) increase by one, (b) decrease by one, or (c)
stays the same.

Furthermore, after $m$ flops from $X_{0...(m-1)}$, we eventually reach the
$G\operatorname{-Hilb}(\mathbb{C}^{3})$, whose image of $Y_{max}$ must be
$G\operatorname{-Hilb}(\mathbb{C}^{2})\cong\mathbb{C}^{2}/G$. Thus, all of the
birational transformations over the surfaces must decrease the number of
exceptional curves by one.

We examine the flop restricted to the surface, most especially the open cover
containing the exceptional divisor $E_{i}$, in Figure 4.

Figure 4. The process of flop in relation to the surface containing the
exceptional curve

Because the surface contains the $(-1,-1)$ flopping curve over the threefold,
blowing up with the flopping curve as the center in the threefold, over the
surface, the exceptional curve remains the same. Over the threefold, this
produces the $\mathbb{P}^{1}\times\mathbb{P}^{1}$ exceptional surface.
Contracting in the other direction, this contracts the exceptional divisor on
the surface.

Because each of the surfaces $Y_{i}^{\prime}$ are smooth, this now implies
that each of the broken arrows over the two-dimensional variants are blowdown
morphisms of a $(-1)$-curve, and comparing with Corollary 3.9, every
resolution dominated by $Y_{max}$ can be realized as a moduli space of
$G$-constellations, which proves the main theorem. ∎

######  Remark 3.13.

We fix the following notations for the open sets for odd $n$:

$U_{1}^{\prime}\cong Spec(\mathbb{C}[\frac{z}{f_{2}},f_{1},xy])$ | $U_{i}^{\prime}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{i-1}z}{f_{2}},\frac{f_{1}}{(xy)^{i-1}},xy\right]\right)\mbox{ for }i\leq m+1$
---|---
$U_{i}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{i-1}z}{f_{2}},\frac{f_{2}}{(xy)^{i-2}z},\frac{zf_{1}}{f_{2}}\right]\right),\mbox{ for }i\leq m+1$ | $U_{i}^{\prime\prime}\cong Spec\left(\mathbb{C}\left[\frac{zf_{1}}{f_{2}},\frac{(xy)^{i}}{f_{1}},\frac{f_{1}}{(xy)^{i-1}}\right]\right)\mbox{ for }i\leq m$
$U_{m+2}\cong Spec\left(\mathbb{C}\left[z^{2},\frac{f_{1}}{(xy)^{m}},\frac{f_{2}}{(xy)^{m}z}\right]\right)$ | $X_{0...i}=\bigcup_{k=1}^{i+1}U_{k}^{\prime\prime}\cup U_{i+2}^{\prime}\cup\bigcup_{k=i+3}^{m+2}U_{k}$

We fix the notations also for the open sets for even $n$:

$U_{i}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{i-1}z}{f_{1}f_{2}},\frac{f_{1}f_{2}}{(xy)^{i-2}z},\frac{zf_{1}}{f_{2}}\right]\right),\mbox{ for }i\leq m$ | $V_{m+1}^{\prime}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{m-1}z^{2}}{f_{2}^{2}},\frac{f_{2}^{2}}{(xy)^{m-1}},\frac{f_{1}f_{2}}{(xy)^{m-1}z}\right]\right)$
---|---
$U_{m+1}\cong Spec\left(\mathbb{C}\left[\frac{zf_{2}}{f_{1}},\frac{f_{1}f_{2}}{(xy)^{m-1}z},\frac{zf_{1}}{f_{2}}\right]\right)$ | $V_{m+2}^{\prime}\cong Spec\left(\mathbb{C}\left[z^{2},\frac{f_{2}^{2}}{(xy)^{m-1}},\frac{f_{1}}{zf_{2}}\right]\right)$
$U_{i}^{\prime}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{i-1}z}{f_{1}f_{2}},\frac{f_{1}^{2}}{(xy)^{i-1}},xy\right]\right)\mbox{ for }i\leq m$ | $V_{i}^{\prime\prime}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{i-2}z^{2}}{f_{2}^{2}},\frac{f_{2}^{2}}{(xy)^{i-3}z^{2}},\frac{zf_{1}}{f_{2}}\right]\right)\mbox{ for }i\leq m+1$
$U_{m+1}^{\prime}\cong Spec\left(\mathbb{C}\left[\frac{zf_{2}}{f_{1}},\frac{f_{1}^{2}}{f_{2}^{2}},\frac{f_{2}^{2}}{(xy)^{m-1}}\right]\right)$ | $V_{m+2}^{\prime\prime}\cong Spec\left(\mathbb{C}\left[z^{2},\frac{f_{2}^{2}}{(xy)^{m-1}z^{2}},\frac{zf_{1}}{f_{2}}\right]\right)$
$U_{i}^{\prime\prime}\cong Spec\left(\mathbb{C}\left[\frac{zf_{1}}{f_{2}},\frac{(xy)^{i}}{f_{1}^{2}},\frac{f_{1}^{2}}{(xy)^{i-1}}\right]\right)\mbox{ for }i\leq m-1$ | $V_{m+3}^{\prime\prime}\cong Spec\left(\mathbb{C}\left[z^{2},\frac{f_{1}^{2}}{(xy)^{m-1}},\frac{f_{2}}{zf_{1}}\right]\right)$
$U_{m}^{\prime\prime}\cong Spec\left(\mathbb{C}\left[\frac{zf_{1}}{f_{2}},\frac{f_{2}^{2}}{f_{1}^{2}},\frac{f_{1}^{2}}{(xy)^{m-1}}\right]\right)$ | $X_{0...i}^{m...(m-j)}=\bigcup_{k=1}^{i+1}U_{k}^{\prime\prime}\cup U_{i+2}^{\prime}\cup\cdots$
$V_{i}^{\prime}\cong Spec\left(\mathbb{C}\left[\frac{(xy)^{i-2}z^{2}}{f_{2}^{2}},xy,\frac{f_{1}f_{2}}{(xy)^{i-2}z}\right]\right)\mbox{ for }i\leq m$ | $\bigcup_{k=i+3}^{m-j}U_{k}\cup V_{m-j+1}^{\prime}\cup\bigcup_{k=m-j+2}^{m+3}V_{k}^{\prime\prime}$

An explicit way to approach the theorem is to consider also the open sets so
that $U_{m}^{\prime\prime}\cup U_{m+1}^{\prime}$ cover the exceptional divisor
with coordinates

$E_{m}:\begin{cases}(f_{1}:(xy)^{m})&\mbox{for odd $n$}\\\
(f_{1}^{2}:(xy)^{m})&\mbox{for even $n$}\end{cases}$

From the proof of Theorems 5.1 and 5.2 of [NS17], under the flopping
transformation of the exceptional divisor $E_{m}$, the open sets
$U_{m}^{\prime\prime}\cup U_{m+1}^{\prime}$ maps to the open sets
$U_{m}^{\prime}\cup U_{m+1}$ covering the flop of $E_{m}$. The coordinate of
the flop of $E_{m}$ is

$\hat{E}_{m}:\begin{cases}((xy)^{m-1}z:f_{2})&\mbox{for odd $n$}\\\
((xy)^{m-1}z:f_{1}f_{2})&\mbox{for even $n$}\end{cases}$

so that on the new variety $X_{0...(m-2)}$, the exceptional divisor vanishes
(or is contracted in the two dimensional case).

We also note the gluing between the open sets $U_{m}^{\prime\prime}$ and
$U_{m+1}^{\prime}$ via

$\begin{cases}(\frac{zf_{1}}{f_{2}},(xy)^{m}/f_{1},f_{1}/(xy)^{m-1})\mapsto(\frac{zf_{1}}{f_{2}}\cdot\frac{(xy)^{m}}{f_{1}},(\frac{(xy)^{m}}{f_{1}})^{-1},\frac{(xy)^{m}}{f_{1}}\cdot\frac{f_{1}}{(xy)^{m-1}})&\mbox{for
odd $n$}\\\
(\frac{zf_{1}}{f_{2}},\frac{f_{1}^{2}}{(xy)^{m-1}},\frac{f_{2}^{2}}{f_{1}^{2}})\mapsto(\frac{zf_{1}}{f_{2}}\cdot\frac{f_{2}^{2}}{f_{1}^{2}},(\frac{f_{2}^{2}}{f_{1}^{2}})^{-1},\frac{f_{1}^{2}}{(xy)^{m-1}}\cdot\frac{f_{2}^{2}}{f_{1}^{2}})&\mbox{for
even $n$}\end{cases}$

Thus, with the realization of the iterated Hilbert scheme over the surface as
the locus from earlier, the loci $zf_{1}/f_{2}=0$ and $(xy)^{m}z/f_{2}=0$
collapses the open set $U_{m+1}$. The open cover implies that this is
identical to the blow-down of the exceptional curve $\hat{E}_{m}$. In symbols,
we have the commutative diagram:

${X_{0...(m-2)}\supset U_{m}^{\prime}\cup
U_{m+1}\supset\hat{E}_{m}}$${E_{m}\subset U_{m}^{\prime\prime}\cup
U_{m+1}\subset X_{0...(m-1)}}$${Y_{m-2}^{\prime}\supset
T_{m-1}(E_{m})=pt}$${E_{m}\subset
Y_{max}}$$\scriptstyle{\Psi_{m-1}}$$\scriptstyle{i}$$\scriptstyle{T_{m-1}}$$\scriptstyle{i}$

From the flop of $E_{m}$, we obtain another crepant resolution. This time, we
do this similarly for the open sets $U_{i}^{\prime\prime}\cup
U_{i+1}^{\prime}$ for $1\leq i\leq m-1$, which gives another crepant
resolution. This works for both cases.

##  4\. The Tautological Bundles and the McKay Correspondence

In the next sections, we explicitly construct a McKay correspondence by
investigating the exceptional divisors of the maximal resolution
$Y_{max}\rightarrow\mathbb{C}^{2}/D_{2n}$ using the realization of the maximal
resolution as a moduli space of $D_{2n}$-constellations.

In this section, in the spirit of [GV], we consider the tautological bundles
and consider the stacky descriptions (or equivalently their
$\mathbb{Z}_{2}$-equivariant sheaves) to construct the McKay Correspondence
for the dihedral groups. The main result in this section is the description
parallel to the proof of Theorem 1.11 in [AV], in particular the description
of a rank $n$ indecomposable reflexive module as an extension of two vector
bundles (one of which is a line bundle) of the form $0\rightarrow
O_{\tilde{X}}^{\oplus(n-1)}\rightarrow\tilde{M}\rightarrow
O_{\tilde{X}}(\tilde{D})\rightarrow 0$, whose Chern class $c_{1}(\tilde{M})$
of the locally free sheaf $\tilde{M}$ corresponds to a vertex of the Dynkin
diagram associated to $\tilde{X}$.

We also insert a comparison between the tautological sheaves on the stack and
the tautological sheaves on the coarse moduli space to show how they behave
differently in those spaces. Together with the results on the final section,
we show evidences why constructing a McKay correspondence is more plausible
over the stack than on the coarse moduli space.

We consider the representations of $\mathbb{Z}_{n}$ and $D_{2n}$ as described
by the following character table. There are four one-dimensional
representations $\rho_{0},\rho_{0}^{\prime},\rho_{n/2},\mbox{and
}\rho_{n/2}^{\prime}$. For $1\leq j\leq n/2$ (for even $n$) or $1\leq
j\leq(n-1)/2$ (for odd $n$), the two-dimensional representations $\rho_{j}$
are defined by their corresponding matrix representations:
$\rho_{i}=\left\langle\tau:=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\sigma^{i}:=\begin{bmatrix}\epsilon^{i}&0\\\
0&\epsilon^{-i}\end{bmatrix}(\epsilon^{n}=1)\right\rangle$.

Table 4. The Irreducible Representations of $\mathbb{Z}_{n}$

Representation Conjugacy Class | $1$ | $\sigma^{i}$
---|---|---
$\epsilon_{0}$ | $1$ | $1$
$\epsilon_{j}$ | $1$ | $\epsilon^{ij}$

where $1\leq j\leq n-1$ and $0\leq i\leq n-1$ are integers.

Table 5. The Irreducible Representations of $D_{2n}$

Representation Conjugacy Class | $1$ | $\tau$ | $\sigma^{i}$
---|---|---|---
$\rho_{0}$ | $1$ | $1$ | $1$
$\rho_{0}^{\prime}$ | $1$ | $-1$ | $1$
$\rho_{j}$ | $2$ | $0$ | $\epsilon^{ij}+\epsilon^{-ij}$
$\rho_{n/2}$ ($n$ even) | $1$ | $1$ | $(-1)^{i}$
$\rho_{n/2}^{\prime}$ ($n$ even) | $1$ | $-1$ | $(-1)^{i}$

where $1\leq j\leq\frac{n-1}{2}$ and $0\leq i\leq n-1$ are integers.

Consider the diagram:

${\mathcal{Z}\subset
X_{1}\times\mathbb{C}^{2}}$${X_{1}}$${\mathbb{C}^{2}}$$\scriptstyle{p_{X_{1}}}$$\scriptstyle{p_{\mathbb{C}^{2}}}$

where $\mathcal{Z}\subset X_{1}\times\mathbb{C}^{2}$ is the universal
subscheme and $p_{X_{1}}$ and $p_{\mathbb{C}^{2}}$ are natural projections.

Since the group $D_{2n}$ acts on each of the schemes $X_{1}$ and
$\mathbb{C}^{2}$, by defining $[X/G]$ as the (quotient) stack associated to
the scheme $X/G$, we can construct the diagram:

${\left[\mathcal{Z}/D_{2n}\right]\subset\left[(X_{1}\times\mathbb{C}^{2})/D_{2n}\right]}$${\left[X_{1}/D_{2n}\right]}$${\left[\mathbb{C}^{2}/D_{2n}\right]}$$\scriptstyle{p_{\left[X_{1}/D_{2n}\right]}}$$\scriptstyle{p_{\left[\mathbb{C}^{2}/D_{2n}\right]}}$

Because $\mathbb{Z}_{n}$ acts trivially on $X_{1}$, there is a natural
morphism
$\phi:\left[X_{1}/D_{2n}\right]\rightarrow\left[X_{1}/\mathbb{Z}_{2}\right]$,
so that the pushforward morphism $\phi_{*}$ sends a $D_{2n}$-equivariant
coherent sheaf $\mathcal{F}$ on $X_{1}$ to the $\mathbb{Z}_{2}$-equivariant
coherent sheaf $\mathcal{F}^{\mathbb{Z}_{n}}$.

We also use the fact that the category of (quasi-)coherent sheaves on the
quotient stack $[X/G]$ is equivalent to the category of $G$-equivariant
(quasi-)coherent sheaves on the scheme $X$, i.e.
$(Q)Coh([X/G])\cong(Q)Coh^{G}(X)$.

By Theorem 3.6, the maximal resolution and the quotient variety
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}$ are
identical. This means that the corresponding boundary divisors are identical
as well. This means that by considering the 2nd root stack
$\sqrt{(O_{\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}}(\left\lceil
B^{\prime}\right\rceil),1_{\left\lceil
B^{\prime}\right\rceil})/(\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2})}$,
we obtain the isomorphism between stacks
$\mathcal{Y}\cong[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}]$.

Defining
$\mathcal{Z}\subset\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})\times\mathbb{C}^{2}$
as the universal family of $\mathbb{Z}_{n}$-constellations/clusters and
consider the diagram whose
$p_{[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}]}$ and
$p_{[\mathbb{C}^{2}/D_{2n}]}$ are natural projections:

${\left[\mathcal{Z}/D_{2n}\right]\subset\left[(\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})\times\mathbb{C}^{2})/D_{2n}\right]}$${\left[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}\right]}$${\left[\mathbb{C}^{2}/D_{2n}\right]}$$\scriptstyle{p_{\left[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}\right]}}$$\scriptstyle{p_{\left[\mathbb{C}^{2}/D_{2n}\right]}}$

We have the Fourier-Mukai transforms:

$\displaystyle\Phi:D([\mathbb{C}^{2}/D_{2n}])$ $\displaystyle\rightarrow
D(\mathcal{Y})$ $\displaystyle\mathfrak{G}$
$\displaystyle\mapsto\phi_{*}(Rp_{[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}]*}(p_{[\mathbb{C}^{2}/D_{2n}]}^{*}(\mathfrak{G})\otimes
O_{[\mathcal{Z}/D_{2n}]}))$ $\displaystyle\Psi:D(\mathcal{Y})$
$\displaystyle\rightarrow D([\mathbb{C}^{2}/D_{2n}])$ $\displaystyle\epsilon$
$\displaystyle\mapsto
R(p_{[\mathbb{C}^{2}/D_{2n}]*})(p_{[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/D_{2n}]}^{*}(\phi^{*}(\epsilon))\otimes
det(\rho_{nat})\otimes O_{[\mathcal{Z}/D_{2n}]}^{\lor}[2])$

The functors $\Psi$ and $\Phi$ are equivalences via Theorem 4.1 of [IU15].

We define the tautological sheaf associated to a representation $\rho$ of
$D_{2n}$ as

$\hat{\mathcal{R}}_{\rho}:=\Phi(O_{\mathbb{C}^{2}}\otimes\rho^{\lor})=(p_{\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})*}(O_{\mathcal{Z}})\otimes\rho^{\lor})^{\mathbb{Z}_{n}}=\left([\oplus_{\epsilon\in\operatorname{Irr}(\mathbb{Z}_{n})}\mathcal{R}_{\epsilon}\otimes\epsilon]\otimes\rho^{\lor}\right)^{\mathbb{Z}_{n}},$

which are $\mathbb{Z}_{2}$-equivariant locally free sheaves since
$\mathcal{R}_{\epsilon}$ are locally free sheaves, and the
$\mathbb{Z}_{2}$-sheaf structure came from the induced representation from
$\mathbb{Z}_{n}$ to $D_{2n}$.

We consider also the diagram:

${\mathcal{Y}}$${\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})}$${\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\scriptstyle{p}$

to define the corresponding tautological sheaves on
$\mathcal{M}_{\theta}=\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}$,
for some $\theta\in\Theta$.

The tautological bundle $\mathcal{R}:=p_{X_{1}*}(O_{\mathcal{Z}})$ on
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$ decomposes as:

$\mathcal{R}=\bigoplus_{\epsilon\in\operatorname{Irr}(\mathbb{Z}_{n})}\mathcal{R}_{\epsilon}^{\circ}\otimes\epsilon.$

By considering
$\mathcal{R}_{\epsilon}^{\circ}:=(\mathcal{R}\otimes\epsilon^{\lor})^{\mathbb{Z}_{n}}$
and
$\mathcal{R}_{\rho}^{\circ}:=(p_{*}(\mathcal{R})\otimes\rho^{\lor})^{D_{2n}}$
as subsheaves of $K(\mathbb{C}^{2})\otimes\epsilon_{i}^{\lor}$ and
$K(\mathbb{C}^{2})\otimes\rho_{i}^{\lor}$, respectively, we have the following
images of the tautological bundles on
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$ under the pushforward
$p_{*}$:

$\displaystyle
p_{*}(\mathcal{R}_{\epsilon_{i}}\otimes\epsilon_{i}\oplus\mathcal{R}_{\epsilon_{n-i}}\otimes\epsilon_{n-i})$
$\displaystyle=(\mathcal{R}_{\rho_{i}}\otimes\rho_{i})^{\oplus 2}(i\neq n/2)$
$\displaystyle p_{*}(\mathcal{R}_{\epsilon_{0}}\otimes\epsilon_{0})$
$\displaystyle=\mathcal{R}_{\rho_{0}}\otimes\rho_{0}\oplus\mathcal{R}_{\rho_{0}^{\prime}}\otimes\rho_{0}^{\prime}$
$\displaystyle p_{*}(\mathcal{R}_{\epsilon_{n/2}}\otimes\epsilon_{n/2})$
$\displaystyle=\mathcal{R}_{\rho_{n/2}}\otimes\rho_{n/2}\oplus\mathcal{R}_{\rho_{n/2}^{\prime}}\otimes\rho_{n/2}^{\prime}$

We collect some lemmas in order to establish a parallel statement of the
correspondence:

######  Lemma 4.1.

For a (Cartier) divisor $D$ on $X_{1}$, the followings hold:

  1. (1)

$\operatorname{det}(p_{*}(O_{X_{1}}(D))=\operatorname{det}(p_{*}(O_{X_{1}}))\otimes
O_{Y_{1}}(p_{*}(D))$

  2. (2)

$\operatorname{det}(p_{*}(O_{X_{1}}))=O_{Y_{1}}(L)$, where $L$ is a
$\mathbb{Q}$-divisor satisfying $2L=-(B_{1}+B_{2})$ (for even $n$) and
$2L=-B_{3}$ (for odd $n$). Please refer again to (3.2) and (3.3) for the
definition of the ramification divisors. This shows that the sheaf
$O_{Y_{1}}(L)$ is a line bundle.

  3. (3)

$p_{*}(O_{X_{1}})=O_{Y_{1}}\oplus O_{Y_{1}}(L)$

###### Proof.

The statements (1) and (2) are essentially (Ch. IV.2, Ex. 2.6) of [Hart], but
we provide the complete proof. In the following, we distinguish $K_{X}$ as the
canonical divisor, and $\omega_{X}=O_{X}(K_{X})$ is the canonical sheaf.

For an effective divisor $D_{1}$, consider the exact sequence:

$0\rightarrow O_{X_{1}}\rightarrow O_{X_{1}}(D_{1})\rightarrow
O_{D_{1}}(D_{1})\rightarrow 0.$

Applying the push-forward $p_{*}$ gives the exact sequence:

$0\rightarrow p_{*}(O_{X_{1}})\rightarrow p_{*}(O_{X_{1}}(D_{1}))\rightarrow
p_{*}(O_{D_{1}}(D_{1}))\rightarrow 0.$

Taking the Chern classes over the exact sequence gives statement (1) for the
effective divisor $D_{1}$.

If $D$ is not effective, we can write $D$ as $D=D_{1}-D_{2}$, where both
$D_{1}$ and $D_{2}$ are effective. We have shown the statement for the
effective divisor $D_{1}$. For the divisor $-D_{2}$, we apply the similar
sequence:

$0\rightarrow O_{X_{1}}(-D_{2})\rightarrow O_{X_{1}}\rightarrow
O_{D_{2}}\rightarrow 0.$

Applying the push-forward $p_{*}$ and taking the Chern classes gives the
statement (1) for any divisor $D=D_{1}-D_{2}$.

By the duality for a finite flat morphism (Ch. III.6, Ex. 6.19 of [Hart]),
with $\omega_{Y_{1}}$ being the dualizing sheaf of $Y_{1}$, so that
$\omega_{X_{1}}\cong
p^{!}(\omega_{Y_{1}}):=\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\mathcal{H}om(p_{*}(O_{X_{1}}),\omega_{Y_{1}})$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}$ (where for a $p_{*}(O_{X_{1}})$-module
$\mathcal{M}$, the sheaf
$\ThisStyle{\stackengine{-.1\LMpt}{$\SavedStyle\mathcal{M}$}{\stretchto{\scaleto{\SavedStyle\mkern
0.2mu\AC}{.5150}}{.6}}{O}{c}{F}{T}{S}}$ is the associated $O_{X_{1}}$-module,
as seen in Ch. II.5, Ex. 5.17 of [Hart]), is also a dualizing sheaf for
$X_{1}$, and we have the isomorphism:

$p_{*}(O_{X_{1}})\cong
p_{*}\mathcal{H}om_{X_{1}}(\omega_{X_{1}},\omega_{X_{1}})\cong\mathcal{H}om_{Y_{1}}(p_{*}(\omega_{X_{1}}),\omega_{Y_{1}})\cong(p_{*}(\omega_{X_{1}}))^{\lor}\otimes\omega_{Y_{1}}.$

Taking the Chern classes gives the isomorphism:

$\operatorname{det}(p_{*}(\omega_{X_{1}}))\cong\operatorname{det}(p_{*}(O_{X_{1}}))^{-1}\otimes\omega_{Y_{1}}^{\otimes
2}.$

By plugging $D=K_{X_{1}}$ on statement (1), we compare different expressions
for $\operatorname{det}(p_{*}(\omega_{X_{1}}))$:

$\displaystyle\operatorname{det}(p_{*}(O_{X_{1}}))^{-1}\otimes\omega_{Y_{1}}^{\otimes
2}$ $\displaystyle\cong\operatorname{det}(p_{*}(O_{X_{1}}))\otimes
O_{Y_{1}}(p_{*}(K_{X_{1}}))$
$\displaystyle\operatorname{det}(p_{*}(O_{X_{1}}))^{\otimes 2}$
$\displaystyle\cong O_{Y_{1}}(2K_{Y_{1}}-p_{*}(K_{X_{1}})).$

From the relation $K_{X_{1}}=p^{*}(K_{Y_{1}}-L)$, where $-2L=B_{1}+B_{2}$ (in
the even $n$ case) and $-2L=B_{3}$ (in the odd $n$ case), taking the
pushforward $p_{*}$, we get statement (2) thereafter:

$\displaystyle p_{*}(K_{X_{1}})$ $\displaystyle=2K_{Y_{1}}-2L$
$\displaystyle\operatorname{det}(p_{*}(O_{X_{1}}))^{\otimes 2}$
$\displaystyle\cong O_{Y_{1}}(2L).$

Because $\mathbb{Z}_{2}$ acts trivially on $Y_{1}$ and $p$ is a finite flat
morphism, $p_{*}(O_{X_{1}})$ decomposes into a $\mathbb{Z}_{2}$-invariant
locally free sheaf and $\mathbb{Z}_{2}$-anti-invariant locally free sheaf; or
more precisely:

$p_{*}(O_{X_{1}})=\mathcal{F}_{0}\oplus\mathcal{F}_{1}\otimes\delta,$

where $\delta$ is the nontrivial representation of $\mathbb{Z}_{2}$.
Certainly, $\mathcal{F}_{0}=O_{Y_{1}}$, and it follows that
$\mathcal{F}_{1}=\operatorname{det}(p_{*}(O_{X_{1}}))=O_{Y_{1}}(L)$ showing
(3). ∎

######  Lemma 4.2.

There exists a (Weil) divisor $D$ on the maximal resolution $Y_{max}$ such
that:

  1. (1)

$D$ is transversal to the exceptional divisor $E_{k}$; $D\cdot E_{k}=1$, for
some $1\leq k\leq m$.

  2. (2)

$D\cdot E_{j}=0,j\neq k$.

  3. (3)

For even $n$, $D$ does not coincide with either of $B_{1}$ or $B_{2}$; $D\neq
B_{1},B_{2}$.

###### Proof.

We prove this in odd $n$ case because the same argument will work in the even
$n$ case.

We refer to the open sets of the maximal resolution from Remark 3.13.

Consider the locus in $\mathbb{C}^{2}/G$ defined by the equation
$W_{k}:f_{1}-(xy)^{k}=0;(1\leq k\leq m=\frac{n-1}{2})$.

We shall illustrate for $k=1$ as the rest of the cases can be performed very
similarly.

On $U_{1}^{\prime\prime}$, the locus $W_{1}$ is the line $xy/f_{1}=1$, which
only intersects the coordinate axis at $(xy/f_{1},f_{1})=(1,0)$, which
corresponds to a point on the exceptional divisor $E_{1}$.

By the gluing between the open sets $U_{1}^{\prime\prime}$ and
$U_{2}^{\prime\prime}$, this identifies the same point
$(xy/f_{1},f_{1})=(1,0)=(f_{1}/xy,(xy)^{2}/f_{1})$.

Once again, by the gluing between the open sets $U_{2}^{\prime\prime}$ and
$U_{3}^{\prime\prime}$ (and so on), the following equation defines the locus,
which has no intersection with the coordinate axes of
$U_{i}^{\prime\prime},i\neq 1,2;$ which implies that there is no intersection
with the exceptional divisors other than $E_{1}$:

$0=\frac{f_{1}}{xy}-1=\left(\frac{(xy)^{k+1}}{(f_{1})}\right)^{k-1}\cdot\left(\frac{(f_{1})}{(xy)^{k}}\right)^{k-1}\cdot\frac{f_{1}}{(xy)^{k}}-1.$

∎

######  Remark 4.3.

For odd $n$, the boundary divisor $B_{3}$ is defined by a quadratic equation
via Remark 3.7, hence, any divisor transversal to $E_{m}$ must intersect the
boundary divisor.

######  Lemma 4.4.

From the equivalence of categories $Coh^{\mathbb{Z}_{2}}(X_{1})\cong
Coh(\mathcal{Y})$:

$O_{X_{1}}\otimes\delta\mapsto
O_{\mathcal{Y}}(\mathcal{C}):=\begin{cases}O_{\mathcal{Y}}(\mathcal{B}_{3}-\mathcal{D})&\text{if
}$n$\text{ is odd}\\\
O_{\mathcal{Y}}(\mathcal{B}_{1}-\mathcal{B}_{2})&\text{if }$n$\text{ is
even}\end{cases}$

where $\mathcal{B}_{i}$ is a prime divisor on $\mathcal{Y}$ such that
$2\mathcal{B}_{i}=\pi^{-1}(B_{i})$ (this is the stacky locus); and
$\mathcal{D}:=\pi^{-1}(D)$, where $D$ satisfies the conditions in Lemma 4.2.
(Refer to (3.2) and (3.3) for the definition of $B$.)

###### Proof.

We comment first on the derivation for this description. The
$\mathbb{Z}_{2}$-equivariant sheaf $O_{X_{1}}\otimes\delta$ is a torsion
element of $Pic^{\mathbb{Z}_{2}}(X_{1})$ of order $2$, so that the
corresponding sheaf on the stack is also a torsion element of
$Pic(\mathcal{Y})$ of order $2$. Because of the following relations:

$\displaystyle O_{\mathcal{Y}}(2\mathcal{B}_{3}-2\mathcal{D})$
$\displaystyle=\pi^{*}(O_{Y_{1}}(B_{3}-2D))\cong\pi^{*}(O_{Y_{1}})=O_{\mathcal{Y}}\mbox{
for odd $n$ }$ $\displaystyle
O_{\mathcal{Y}}(2\mathcal{B}_{1}-2\mathcal{B}_{2})$
$\displaystyle=\pi^{*}(O_{Y_{1}}(B_{1}-B_{2}))\cong\pi^{*}(O_{Y_{1}})=O_{\mathcal{Y}}\mbox{
for even $n$ }$

so that $O_{\mathcal{Y}}(\mathcal{B}_{3}-\mathcal{D})$ and
$O_{\mathcal{Y}}(\mathcal{B}_{1}-\mathcal{B}_{2})$ are torsion elements of
$Pic(\mathcal{Y})$. It is important to point out that these are distinct
divisors by considering $\mathcal{B}_{i}=[B_{i}/\mathbb{Z}_{2}]$. Another way
is to consider the Fourier-Mukai images of $\mathcal{B}$ under $\Phi$ via
Theorem 5.3. The stabilizer groups of $\mathcal{B}_{i}$ and $\mathcal{D}$ are
$\mathbb{Z}_{2}$ and $\\{e\\}$ (except for the intersection point with
$\mathcal{B}_{i}$), respectively.

We further note that such $D$ exists by the previous Lemma 4.2.

We consider the commutative diagram. This particular diagram defines the
equivalence between the two categories. Our main argument here is to trace via
the diagram.

${X_{1}\times\mathbb{Z}_{2}}$${X_{1}}$${X_{1}}$${\mathcal{Y}}$$\scriptstyle{pr_{X_{1}}}$$\scriptstyle{a}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

which gives the isomorphism
$\alpha:a^{*}\pi^{*}(O_{\mathcal{Y}}(\mathcal{C}))\xrightarrow{\sim}pr_{X_{1}}^{*}\pi^{*}(O_{\mathcal{Y}}(\mathcal{C}))$.

By considering the fiber at each point, the stabilizer group of $\mathcal{C}$
is $\mathbb{Z}_{2}$ and for the other points being $\\{e\\}$. This induces the
non-trivial action of $\mathbb{Z}_{2}$, and thus $\alpha$ is not identity. ∎

######  Theorem 4.5.

The tautological bundles on the stack
$\mathcal{Y}=[\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}]$
are described by the following:

Table 6. The Tautological Sheaves for odd $n$

Tautological Bundle | Description | Chern Class
---|---|---
$\hat{\mathcal{R}}_{\rho_{0}}$ | $\mathcal{O}_{\mathcal{Y}}$ | $0$
$\hat{\mathcal{R}}_{\rho_{0}^{\prime}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{3}-\mathcal{D})$ | $\mathcal{B}_{3}-\mathcal{D}$
$\hat{\mathcal{R}}_{\rho_{i}}$ (rank 2) | $0\rightarrow\mathcal{O}_{\mathcal{Y}}\xrightarrow{i}\hat{\mathcal{R}}_{\rho_{i}}\xrightarrow{pr}\mathcal{O}_{\mathcal{Y}}(\mathcal{D}_{i}+\mathcal{B}_{3}-\mathcal{D})\rightarrow 0$ | $\mathcal{D}_{i}+\mathcal{B}_{3}-\mathcal{D}$

Table 7. The Tautological Sheaves for even $n$

Tautological Bundle | Description | Chern Class
---|---|---
$\hat{\mathcal{R}}_{\rho_{0}}$ | $\mathcal{O}_{\mathcal{Y}}$ | $0$
$\hat{\mathcal{R}}_{\rho_{0}^{\prime}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{1}-\mathcal{B}_{2})$ | $\mathcal{B}_{1}-\mathcal{B}_{2}$
$\hat{\mathcal{R}}_{\rho_{i}}$ (rank 2) | $0\rightarrow\mathcal{O}_{\mathcal{Y}}\xrightarrow{i}\hat{\mathcal{R}}_{\rho_{i}}\xrightarrow{pr}\mathcal{O}_{\mathcal{Y}}(\mathcal{D}_{i}+\mathcal{B}_{1}-\mathcal{B}_{2})\rightarrow 0$ | $\mathcal{D}_{i}+\mathcal{B}_{1}-\mathcal{B}_{2}$
$\hat{\mathcal{R}}_{\rho_{(n/2)}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{2})$ | $\mathcal{B}_{2}$
$\hat{\mathcal{R}}_{\rho_{(n/2)}^{\prime}}$ | $\mathcal{O}_{\mathcal{Y}}(\mathcal{B}_{1})$ | $\mathcal{B}_{1}$

The rank one tautological bundles on the stack are uniquely determined by
their Chern classes; and the rank two tautological bundles are determined by
an extension of two line bundles. Furthermore, there is only one possible non-
trivial extension class, making these descriptions unique.

###### Proof.

We write an exact sequence involving (rank two) tautological sheaves
$\tilde{\mathcal{R}}_{\rho_{i}}$ (as $\mathbb{Z}_{2}$-equivariant sheaves)
similar to the proof of Theorem 1.11 of [AV]:

$0\rightarrow\mathcal{O}_{X_{1}}\xrightarrow{i}\mathcal{O}_{X_{1}}(\tilde{D}_{i})\oplus\mathcal{O}_{X_{1}}(g\cdot\tilde{D}_{i})=\tilde{\mathcal{R}}_{\rho_{i}}\xrightarrow{pr}\mathcal{O}_{X_{1}}(\tilde{D}_{i}+g\cdot\tilde{D}_{i})\otimes\delta\rightarrow
0,$

where $i$ is the inclusion, $\delta$ is the nontrivial representation of
$\mathbb{Z}_{2}$ (and $g$ correspond to the non-trivial element of
$\mathbb{Z}_{2}$), and $pr(h_{1},h_{2})=h_{1}-h_{2}$.

We show that this extension over $X_{1}$ is unique, which implies on the
tautological sheaves on the stack. We let $F=\Sigma\tilde{E}_{i}$ be the
fundamental cycle. In the following,
$\mathbb{Z}_{2}\operatorname{-Ext}^{1}_{X}(-,-)$ is defined as the
$\mathbb{Z}_{2}$-invariant part of $Ext_{X}^{1}(-,-)$.

$\displaystyle\mathbb{Z}_{2}\operatorname{-Ext}_{O_{X_{1}}}^{1}(\mathcal{O}_{X_{1}}(\tilde{D}_{i}+g\cdot\tilde{D}_{i})\otimes\delta,\mathcal{O}_{X_{1}})$
$\displaystyle=\mathbb{Z}_{2}\operatorname{-Ext}_{O_{X_{1}}}^{1}(\mathcal{O}_{X_{1}},\mathcal{O}_{X_{1}}(-\tilde{D}_{i}-g\cdot\tilde{D}_{i})\otimes\delta)$
$\displaystyle=H^{1}(X_{1},\mathcal{O}_{X_{1}}(-\tilde{D}_{i}-g\cdot\tilde{D}_{i})\otimes\delta)^{\mathbb{Z}_{2}}$
$\displaystyle=H^{1}(F,\mathcal{O}_{X_{1}}(-\tilde{D}_{i}-g\cdot\tilde{D}_{i})|_{F})^{\mathbb{Z}_{2}}$
$\displaystyle=H^{1}(F,\mathcal{O}_{F}(-\tilde{D}_{i}-g\cdot\tilde{D}_{i}))^{\mathbb{Z}_{2}}$
$\displaystyle\cong\mathbb{C}$

which is one dimensional over $\mathbb{C}$. Thus, there is only one possible
non-trivial extension (up to scalar).

From the Lemma 4.4, this corresponds over the global quotient stack
$\mathcal{Y}$:

$0\rightarrow\mathcal{O}_{\mathcal{Y}}\xrightarrow{i}\hat{\mathcal{R}}_{\rho_{i}}\xrightarrow{pr}\mathcal{O}_{\mathcal{Y}}(\mathcal{D}_{i}+\mathcal{C})\rightarrow
0,$

where $\hat{\mathcal{R}}_{\rho_{i}}$ and
$\mathcal{O}_{\mathcal{Y}}(\mathcal{D}_{i})$ is the corresponding image in the
isomorphism of the tautological sheaf $\tilde{\mathcal{R}}_{\rho_{i}}$ and the
invertible sheaf $\mathcal{O}_{X_{1}}(\tilde{D_{i}}+g\cdot\tilde{D_{i}})$,
respectively.

The rank one tautological sheaves
$\hat{\mathcal{R}}_{\rho_{0}^{\prime}},\hat{\mathcal{R}}_{\rho_{n/2}},\hat{\mathcal{R}}_{\rho_{n/2}^{\prime}}$
are obtained via the commutative diagram, where $\Phi^{\mathbb{Z}_{n}}$ is the
equivalence defined in [BKR]:

${D(coh^{D_{2n}}(\mathbb{C}^{2}))}$${D(coh^{\mathbb{Z}_{2}}(X_{1}))}$${D(coh^{\mathbb{Z}_{n}}(\mathbb{C}^{2}))}$${D(coh(X_{1}))}$$\scriptstyle{\operatorname{for}}$$\scriptstyle{\Phi}$$\scriptstyle{\operatorname{for}}$$\scriptstyle{\Phi^{\mathbb{Z}_{n}}}$

We demonstrate this for $\rho_{n/2}$ and the rest of the cases are similar.
From the definition of $\hat{\mathcal{R}}_{\rho_{n/2}}$, this must come from a
$\mathbb{Z}_{2}$-lift of
$(\Phi_{\mathbb{Z}_{n}}\circ\operatorname{for})(O_{\mathbb{C}^{2}}\otimes\rho_{n/2})=\tilde{\mathcal{R}}_{\epsilon_{n/2}}$,
which must be a $\mathbb{Z}_{2}$-line bundle with degree $1$. But the only
divisors that satisfy such property are $\tilde{B}_{1}$ and $\tilde{B}_{2}$.
Since $\tilde{B}_{2}$ is invariant under the action of $\rho_{n/2}$, this
describes the tautological sheaf
$\tilde{\mathcal{R}}_{\rho_{n/2}}=O_{X_{1}}(\tilde{B}_{2})$. In addition,
$\tilde{\mathcal{R}}_{\rho_{n/2}^{\prime}}=\tilde{\mathcal{R}}_{\rho_{n/2}}\otimes\delta$,
which gives the complete description for the tautological sheaves on $X_{1}$.

∎

######  Remark 4.6.

The theory of Chern classes also work for Deligne-Mumford stacks via the
theory of Chow groups with rational coefficients. This is shown in Section 3
of [V].

We now compare the tautological sheaves over the stack, and the bundles over
the coarse moduli space. We will see that the rank two bundles split over the
coarse moduli space.

Taking the pushforward and the $\mathbb{Z}_{2}$-invariants of the exact
sequence above, we get the exact sequence on $Y_{1}$:

$0\rightarrow O_{Y_{1}}\rightarrow\mathcal{R}_{\rho_{i}}\rightarrow
O_{Y_{1}}(D_{i}+L)\rightarrow 0,$

where $D_{i}$ is the image $p(\tilde{D_{i}})$.

######  Lemma 4.7.

The exact sequence over $Y_{1}$ is split. Equivalently,
$H^{1}(Y_{1},\mathcal{O}_{Y_{1}}(L+D_{i}))=0$.

###### Proof.

Let $Z$ be the fundamental cycle of the maximal resolution
$Y_{max}\rightarrow\mathbb{C}^{2}/D_{2n}$. The sheaf in question is
$\mathcal{F}:=\mathcal{O}_{Y_{1}}(L+D_{i})$. Computing the degrees of
$\mathcal{F}|_{E_{j}}$, $\mathcal{F}|_{Z}$, and
$\mathcal{F}\otimes\mathcal{O}(-nZ)|_{Z}$:

$\displaystyle deg(\mathcal{F}|_{E_{j}})$ $\displaystyle=(L+D_{i})\cdot E_{j}$
$\displaystyle deg(\mathcal{O}(Z)|_{Z})$ $\displaystyle=-1$ $\displaystyle
deg(\mathcal{F}\otimes\mathcal{O}(-nZ)|_{Z})$ $\displaystyle=n$

Thus, $H^{1}(\mathcal{F}|_{Z})=0$, and
$H^{1}(\mathcal{F}\otimes\mathcal{O}(-nZ)|_{Z})=0$. Using induction and the
exact sequence:

$0\rightarrow\mathcal{F}\otimes\mathcal{O}(-nZ)|_{Z}\rightarrow\mathcal{F}|_{(n+1)Z}\rightarrow\mathcal{F}|_{nZ}\rightarrow
0,$

we have $H^{1}(\mathcal{F}|_{nZ})=0$, for all $n>0$. By the theorem of formal
functions (see Ch.III, Sec.11 of [Hart]), since
$R^{1}(f_{2})_{*}(\mathcal{F})$ is supported on the origin of
$\mathbb{C}^{2}/D_{2n}$ with $f_{2}^{-1}(0)=Z$, this implies the lemma. ∎

######  Remark 4.8.

The lemma can also be applied if $D_{i}$ is replaced by any of the boundary
divisors $B_{1}$ or $B_{2}$ since it will use the very same argument. This is
needed for the following description of tautological bundles on the coarse
moduli space.

######  Theorem 4.9.

The tautological bundles on the coarse moduli space
$Y_{1}=\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}$ are
described by the following:

Table 8. The Tautological Sheaves for odd $n$

Tautological Bundle | Description | Chern Class
---|---|---
$\mathcal{R}_{\rho_{0}}$ | $\mathcal{O}_{Y_{1}}$ | $0$
$\mathcal{R}_{\rho_{0}^{\prime}}$ | $\mathcal{O}_{Y_{1}}(L)$ | $L$
$\mathcal{R}_{\rho_{i}}$ (rank 2) | $\mathcal{R}_{\rho_{i}}=\mathcal{O}_{Y_{1}}\oplus\mathcal{O}_{Y_{1}}(D_{i}+L)$ | $D_{i}+L$

Table 9. The Tautological Sheaves for even $n$

Tautological Bundle | Description | Chern Class
---|---|---
$\mathcal{R}_{\rho_{0}}$ | $\mathcal{O}_{Y_{1}}$ | $0$
$\mathcal{R}_{\rho_{0}^{\prime}}$ | $\mathcal{O}_{Y_{1}}(L)$ | $L$
$\mathcal{R}_{\rho_{i}}$ (rank 2) | $\mathcal{R}_{\rho_{i}}=\mathcal{O}_{Y_{1}}\oplus\mathcal{O}_{Y_{1}}(D_{i}+L)$ | $D_{i}+L$
$\mathcal{R}_{\rho_{(n/2)}}$ | $\mathcal{O}_{Y_{1}}(B_{1}+L)$ | $B_{1}+L$
$\mathcal{R}_{\rho_{(n/2)}^{\prime}}$ | $\mathcal{O}_{Y_{1}}(B_{2}+L)$ | $B_{2}+L$

###### Proof.

Statement (3) of Lemma 4.1 describes the tautological sheaves
$\mathcal{R}_{\rho_{0}}$ and $\mathcal{R}_{\rho_{0}^{\prime}}$. Lemma 4.7
implies the description of the rank $2$ tautological bundles
$\mathcal{R}_{\rho_{i}}$.

Using the similar exact sequence as applied in Lemma 4.7:

$0\rightarrow O_{X_{1}}\rightarrow O_{X_{1}}(\tilde{B}_{1})\oplus
O_{X_{1}}(\tilde{B}_{2})\rightarrow
O_{X_{1}}(\tilde{B}_{1}+\tilde{B}_{2})\otimes\delta\rightarrow 0.$

Taking the push-forward $p_{*}$ and $\mathbb{Z}_{2}$-invariants gives the
exact sequence:

$0\rightarrow O_{Y_{1}}\rightarrow[p_{*}(O_{X_{1}}(\tilde{B}_{1})\oplus
O_{X_{1}}(\tilde{B}_{2}))]^{\mathbb{Z}_{2}}\rightarrow O_{Y_{1}}\rightarrow
0.$

Because
$\operatorname{Ext}^{1}(O_{Y_{1}},O_{Y_{1}})=H^{1}(Y_{1},O_{Y_{1}})=H^{1}(\mathbb{C}^{2}/G,O_{\mathbb{C}^{2}/G})=0$,
this implies that $[p_{*}(O_{X_{1}}(\tilde{B}_{1})\oplus
O_{X_{1}}(\tilde{B}_{2}))]^{\mathbb{Z}_{2}}=(O_{Y_{1}})^{\oplus 2}$.
Furthermore, by Statement (1) of Lemma 4.1, we have the corresponding
descriptions for $\mathcal{R}_{\rho_{3}}$ and
$\mathcal{R}_{\rho_{3}^{\prime}}$. ∎

######  Remark 4.10.

  1. (1)

A consequence of Theorem 4.5 is that the rank $2$ tautological sheaves can be
expressed as an extension of two line bundles on the stack. Because of the
structure of the line bundles which are torsion elements of the Picard group
of the stack as given in the Lemma 4.4, certainly, it is not generated by
global sections, which is an obstruction in writing an exact sequence similar
to Theorem 1.1 of [AV].

  2. (2)

Lemma 4.7 implies that over the coarse moduli space $Y_{1}$, the rank two
tautological bundles split, whereas over the stack $[X_{1}/\mathbb{Z}_{2}]$,
the tautological bundle does not split. Furthermore, there is a parallel
description of [AV] can be applied to stacks (especially when identified as
$\mathbb{Z}_{2}$-equivariant sheaves on $X_{1}$). This makes the stack a more
appropriate venue to establish the McKay Correspondence for the dihedral group
than the coarse moduli space.

  3. (3)

The rank two tautological bundles $\tilde{\mathcal{R}}_{\rho_{i}}$ ($i\neq
0,0^{\prime},n/2,(n/2)^{\prime}$) can be realized as an extension via the
exact sequences:

$\displaystyle 0\rightarrow
p^{*}(\mathcal{R}_{\rho_{i}})\rightarrow\tilde{\mathcal{R}}_{\rho_{i}}\rightarrow
O_{\tilde{B}}\rightarrow 0\mbox{ and }0\rightarrow O_{Y_{max}}\rightarrow
p_{*}(O_{\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})})\rightarrow
O_{Y_{max}}(L)\rightarrow 0$

where $\tilde{\mathcal{R}}_{\rho_{i}}=O_{X_{1}}(\tilde{D_{i}})\oplus
O_{X_{1}}(g\cdot\tilde{D_{i}})$ and
$p_{*}(\tilde{\mathcal{R}}_{\rho_{i}})=O_{Y_{max}}\oplus
O_{Y_{max}}(D_{i})\otimes O_{Y_{max}}(L)\otimes\delta$, where $D_{i}$ is any
transversal to the exceptional divisor $E_{i}$ not intersecting $E_{j},j\neq
i$, and not intersecting the boundary divisors; and correspondingly for
$\tilde{D_{i}}$, which is a transversal to $\tilde{E_{i}}$ not intersecting on
other exceptional divisors and not intersecting the boundary divisors. The
exact sequence is obtained by evaluating Chern classes of each sheaves with
the fact that the boundary divisors $\tilde{B}$ being affine line/s.

The extension is classified by:

$\displaystyle Ext^{1}(O_{\tilde{B}},p^{*}(\mathcal{R}_{\rho_{i}}))$
$\displaystyle\cong
Ext^{1}(O_{\tilde{B}},O(\tilde{D_{i}}+\tilde{D}_{n-i}-\tilde{B}))$
$\displaystyle\cong
H^{0}(\tilde{B},O(\tilde{D_{i}}+\tilde{D}_{n-i})|_{\tilde{B}})$
$\displaystyle\cong H^{0}(\tilde{B},O_{\tilde{B}}^{\oplus 2})$
$\displaystyle\cong(\mathbb{C}[\tilde{B}])^{\oplus 2}$

##  5\. The McKay Correspondence via the Top and the Socles

In this section, we reveal that the top and socles (defined in Definition 5.5)
can be described over the stacks, but not enough to construct such a
correspondence over the coarse moduli space.

We refer to the previous section for the functors $\Phi$ and $\Psi$.

We wish to evaluate $\Phi(O_{0}\otimes\rho_{i}^{*})$ for
$\rho_{i}\in\operatorname{Irr}(D_{2n})$.

By referring to the results in [Ish02] regarding the functor
$\Phi^{\mathbb{Z}_{n}}$, the computation of $\Phi(O_{0}\otimes\rho_{i}^{*})$
reduces to the task of determining $\mathbb{Z}_{2}$-equivariant structures on
$\Phi^{\mathbb{Z}_{n}}(\operatorname{for}(O_{0}\otimes\rho_{i}^{*}))$.

We define the representations $(P_{i},\rho_{i})$ as a representation of
$D_{2n}$ and $(D_{i},\epsilon_{i}:=(\rho_{i})|_{\mathbb{Z}_{n}})$ as its
corresponding restriction to the cyclic group $\mathbb{Z}_{n}$. Furthermore,
we consider the exact sequence $0\rightarrow D_{i}\hookrightarrow
P_{i}\rightarrow P_{i}/D_{i}\rightarrow 0$, so that $(P_{i}/D_{i},\delta_{i})$
is a representation of $D_{2n}/\mathbb{Z}_{n}\cong\mathbb{Z}_{2}$. We have the
following:

######  Proposition 5.1.

Using the Fourier-Mukai transform

$\displaystyle\Phi:D^{D_{2n}}(\mathbb{C}^{2})$ $\displaystyle\rightarrow
D^{\mathbb{Z}_{2}}(X_{1})$ $\displaystyle\mathfrak{G}$ $\displaystyle\mapsto
Rp_{X_{1}*}(p_{\mathbb{C}^{2}}^{*}(\mathfrak{G})\otimes O_{\mathcal{Z}}),$

the following images of structure sheaves at the origin has the following
images:

$\displaystyle\Phi(O_{0}\otimes\rho_{0}^{*})$
$\displaystyle=O_{F}\otimes\delta_{0}\mbox{ (for any $n$)}$
$\displaystyle\Phi(O_{0}\otimes\rho_{0}^{\prime*})$
$\displaystyle=O_{F}\otimes\delta_{0}^{\prime}\mbox{ (for any $n$)}$
$\displaystyle\Phi(O_{0}\otimes\rho_{j}^{*})$
$\displaystyle=(O_{\tilde{E}_{j}}(-1)\oplus O_{\tilde{E}_{n-j}}(-1))[1]\mbox{
(for any $n$, and $j\neq n/2,(n-1)/2$)}$
$\displaystyle\Phi(O_{0}\otimes\rho_{n/2}^{*})$
$\displaystyle=O_{\tilde{E}_{n/2}}(-\tilde{B}_{2})[1]\mbox{ (for even $n$)}$
$\displaystyle\Phi(O_{0}\otimes\rho_{n/2}^{\prime*})$
$\displaystyle=O_{\tilde{E}_{n/2}}(-\tilde{B}_{1})[1]\mbox{ (for even $n$)}$
$\displaystyle\Phi(O_{0}\otimes\rho_{(n-1)/2}^{*})$
$\displaystyle=O_{\tilde{E}_{(n-1)/2}}\left(-\tilde{B}_{3}\right)[1]\mbox{
(for odd $n$)}$

where we refer to (3.1), (3.2), and (3.3) for the definition of $\tilde{E}$
and $\tilde{B}$; and $F$ is the fundamental cycle $\Sigma\tilde{E_{i}}$. Also,
the group $D_{2n}/\mathbb{Z}_{n}\cong\mathbb{Z}_{2}$ fixes the subschemes $F$,
and $\tilde{E_{i}}\cup\tilde{E}_{n-i}$; thus, the $\mathbb{Z}_{2}$ acts on the
line bundles $O_{F}$, $O_{\tilde{E}_{i}}(-1)\oplus O_{\tilde{E}_{n-i}}(-1)$,
and $O_{\tilde{E}_{n/2}}(-1)$.

###### Proof.

From the commutative diagram above,

$\displaystyle\Phi^{\mathbb{Z}_{n}}(\operatorname{for}(O_{0}\otimes\rho_{0}))$
$\displaystyle=\Phi^{\mathbb{Z}_{n}}(O_{0}\otimes\epsilon_{0})=O_{F}$
$\displaystyle\Phi^{\mathbb{Z}_{n}}(\operatorname{for}(O_{0}\otimes\rho^{\prime}_{0}))$
$\displaystyle=\Phi^{\mathbb{Z}_{n}}(O_{0}\otimes\epsilon_{0})=O_{F}$
$\displaystyle\Phi^{\mathbb{Z}_{n}}(\operatorname{for}(O_{0}\otimes\rho_{j}))$
$\displaystyle=\Phi^{\mathbb{Z}_{n}}(O_{0}\otimes\epsilon_{j}\oplus
O_{0}\otimes\epsilon_{n-j})=O_{\tilde{E_{j}}}(-1)[1]\oplus
O_{\tilde{E}_{n-j}}(-1)[1]$
$\displaystyle\Phi^{\mathbb{Z}_{n}}(\operatorname{for}(O_{0}\otimes\rho_{n/2}))$
$\displaystyle=\Phi^{\mathbb{Z}_{n}}(O_{0}\otimes\epsilon_{n/2})=O_{\tilde{E}_{n/2}}(-1)[1]$
$\displaystyle\Phi^{\mathbb{Z}_{n}}(\operatorname{for}(O_{0}\otimes\rho^{\prime}_{n/2}))$
$\displaystyle=\Phi^{\mathbb{Z}_{n}}(O_{0}\otimes\epsilon_{n/2})=O_{\tilde{E}_{n/2}}(-1)[1]$

The sheaf $O_{\tilde{E_{j}}}(-1)[1]\oplus O_{\tilde{E}_{n-j}}(-1)[1]$ is
certainly a $\mathbb{Z}_{2}$-equivariant sheaf. Thus, it remains to determine
$\mathbb{Z}_{2}$ equivariant structures on $O_{F}$ and
$O_{\tilde{E}_{n/2}}(-1)$.

By considering the canonical isomorphism
$\mu_{g}:g^{*}O_{F}\xrightarrow{\sim}O_{g^{-1}(F)}=O_{F}$, and given the
$G$-sheaf $\lambda_{g}^{O_{F}}:O_{F}\rightarrow g^{*}(O_{F})$, the composition
$\mu_{g}\circ\lambda_{g}^{O_{F}}\in\operatorname{Hom}(O_{F},O_{F})=\mathbb{C}$.
Thus, $\mu_{g}\circ\lambda_{g}=c$, so that $\lambda_{g}=c\mu_{g}^{-1}$. Using
the condition for $G$-sheaves, then $c=\pm 1$ which in turn determines the
$\mathbb{Z}_{2}$-equivariant sheaves.

Similarly, this gives the $\mathbb{Z}_{2}$ equivariant structures on
$O_{\tilde{E}_{n/2}}(-1)$, as
$\operatorname{Hom}(O_{\tilde{E}_{n/2}}(-1),g^{*}(O_{\tilde{E}_{n/2}}(-1)))\cong\operatorname{Hom}(O_{\tilde{E}_{n/2}},g^{*}(O_{\tilde{E}_{n/2}}))\cong\operatorname{Hom}(O_{\tilde{E}_{n/2}},O_{\tilde{E}_{n/2}})\cong\mathbb{C}$.
Unfortunately, as a $\mathbb{Z}_{2}$-equivariant sheaf,
$O_{\tilde{E}_{n/2}}(-1)$ must be a fixed locus, which can only be any of the
$\tilde{B}_{i}$. Since $O_{\tilde{E}_{n/2}}(-\tilde{B}_{1})$ (resp.
$-\tilde{B}_{2}$) is invariant under the action of $\rho_{n/2}^{\prime}$
(resp. $\rho_{n/2}$), this completes the description of the Fourier-Mukai
images of the skyscraper sheaves. ∎

Because there is a stacky structure on the fixed points of the
$\mathbb{Z}_{2}$, we restate the proposition above in terms of coherent
sheaves on the global quotient stack $[X_{1}/\mathbb{Z}_{2}]$.

For the next proposition, we define some notations: We consider the morphism
of schemes $p:X_{1}\rightarrow Y_{1}$ and stacks
$\pi:[X_{1}/\mathbb{Z}_{2}]\rightarrow Y_{1}$. We define the following closed
substacks on $[X_{1}/\mathbb{Z}_{2}]$:

(5.1) $\mathcal{E}_{i}:=[p(\tilde{E}_{i}\cup\tilde{E}_{n-i})/\mathbb{Z}_{2}]\cong p(\tilde{E_{i}}\cup\tilde{E}_{n-i})/\mathbb{Z}_{2}\left(i\neq\frac{n-1}{2},\frac{n}{2},\frac{n+1}{2}\right)$ | $\mathcal{F}:=[p(F)/\mathbb{Z}_{2}]$
---|---
$\mathcal{E}_{(n-1)/2}:=[p(\tilde{E}_{(n-1)/2}\cup\tilde{E}_{(n+1)/2})/\mathbb{Z}_{2}]=:\mathcal{E}_{(n+1)/2}$ | $\mathcal{B}_{1}:=[p(\tilde{B}_{1})/\mathbb{Z}_{2}]$
$\mathcal{E}_{n/2}:=[p(\tilde{E}_{n/2})/\mathbb{Z}_{2}]$ | $\mathcal{B}_{2}:=[p(\tilde{B}_{2})/\mathbb{Z}_{2}]$
| $\mathcal{B}_{3}:=[p(\tilde{B}_{3})/\mathbb{Z}_{2}]$

######  Remark 5.2.

The exceptional divisors on the stack $\mathcal{E}$ are smooth except for
$\mathcal{E}_{(n-1)/2}$. The exceptional divisor $\mathcal{E}_{(n-1)/2}$ is
not smooth because the fixed point of
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$ lies on the intersection
of two distinct exceptional divisors.

######  Theorem 5.3.

Using the Fourier-Mukai transform

$\displaystyle\Phi:D^{D_{2n}}(\mathbb{C}^{2})$ $\displaystyle\rightarrow
D^{\mathbb{Z}_{2}}(X_{1})\cong D([X_{1}/\mathbb{Z}_{2}])$
$\displaystyle\mathfrak{G}$ $\displaystyle\mapsto
Rp_{X_{1}*}(p_{\mathbb{C}^{2}}^{*}(\mathfrak{G})\otimes O_{\mathcal{Z}}),$

the following images of structure sheaves at the origin has the images on the
quotient stack $[X_{1}/\mathbb{Z}_{2}]$:

$\displaystyle\Phi(O_{0}\otimes\rho_{0}^{*})$
$\displaystyle=O_{\mathcal{F}}\mbox{ (for any $n$)}$
$\displaystyle\Phi(O_{0}\otimes\rho_{0}^{\prime*})$
$\displaystyle=O_{\mathcal{F}}(\mathcal{B}_{1}-\mathcal{B}_{2})\mbox{ (for any
$n$)}$ $\displaystyle\Phi(O_{0}\otimes\rho_{j}^{*})$
$\displaystyle=O_{\mathcal{E}_{j}}(-1)[1]\mbox{ (for any $n$, and $j\neq
n/2,(n-1)/2$)}$ $\displaystyle\Phi(O_{0}\otimes\rho_{n/2}^{*})$
$\displaystyle=O_{\mathcal{E}_{n/2}}\left(-\mathcal{B}_{2}\right)[1]\mbox{
(for even $n$)}$ $\displaystyle\Phi(O_{0}\otimes\rho_{n/2}^{\prime*})$
$\displaystyle=O_{\mathcal{E}_{n/2}}\left(-\mathcal{B}_{1}\right)[1]\mbox{
(for even $n$)}$ $\displaystyle\Phi(O_{0}\otimes\rho_{(n-1)/2}^{*})$
$\displaystyle=O_{\mathcal{E}_{(n-1)/2}}\left(-\mathcal{B}_{3}\right)[1]\mbox{
(for odd $n$)}$

###### Proof.

This a restatement in terms of coherent sheaves on the global quotient stack.
Furthermore, the correspondence for $O_{F}\otimes\delta_{i}$ can be found in
Lemma 4.4. ∎

######  Remark 5.4.

Take the even $n$ case and the divisor $\mathcal{E}_{n/2}$ for instance. The
following justification uses the root stack construction whose introduction
and details to the said concepts can be found in [Cad], [AGV], and [Ols].

Since $p(\tilde{E}_{n/2})/\mathbb{Z}_{2}$ is smooth, we can realize
$\mathcal{E}_{n/2}$ as the $2$nd root stack

$\mathcal{E}_{n/2}:=\sqrt{(O_{p(\tilde{E}_{n/2})}(p(\tilde{B}_{1})+p(\tilde{B}_{2})),1)/(p(\tilde{E}_{n/2}))},$

where we perform the appropriate modification of the definition of the 2nd
root stack seen after Remark 1.3. Refer to Section 2.2 of [Cad] for further
details.

The setup propositions and theorems earlier in this section will be used to
compute the top and socles in the hopes of obtaining a similar description as
in the McKay correspondence of Ito-Nakamura [IN2] for the $SL(2)$ case and
Ishii [Ish02] for the small $GL(2)$ case. The top and socles can be
(explicitly) computed via their ideals that define a closed subscheme, i.e.
$I_{y}$, to obtain the corresponding quotients $I_{y}/mI_{y}$ and
$(I_{y}:m)/I_{y}$ needed to compute the top and the socle respectively.

The data for the McKay Correspondence is given by the following:

######  Definition 5.5.

For a given $G$-constellation $\overline{F}$ on the moduli space of
$\theta$-stable $G$-constellations $\mathcal{M}_{\theta}$:

$\displaystyle\operatorname{top}(\overline{F})$
$\displaystyle:=\overline{F}/\langle x,y\rangle\overline{F}$
$\displaystyle\operatorname{socle}(\overline{F})$
$\displaystyle:=\\{a\in\overline{F}|\langle x,y\rangle a=0\\}$

Before we state the proposition, the maximal resolution $Y_{max}$ can be
realized as a moduli space of $\theta$-stable $G$-constellations
$\mathcal{M}_{\theta}$ for some generic stability parameter $\theta$ via the
isomorphism $\mathcal{M}_{\theta}\cong Y\xrightarrow{\sim}Y_{1}\cong Y_{max}$.
The isomorphism of iterated Hilbert schemes to a moduli space of
$G$-constellations is justified in Theorem 1.5 of [IIN]. The specific
stability parameter $\theta$ is computed in Table 5 of [IIN].

######  Proposition 5.6.

For a given $D_{2n}$-constellation $\overline{F}$ on the maximal resolution
$Y_{max}$,

$\operatorname{top}(\overline{F})=\begin{cases}\rho_{0}\oplus\rho_{0}^{\prime}&[\overline{F}]\in
Exc(Y_{max}\rightarrow\mathbb{C}^{2}/G)\\\ 0&otherwise\end{cases}$

$\operatorname{socle}(\overline{F})=\begin{cases}\rho_{i}&[\overline{F}]\in
E_{i},[\overline{F}]\notin E_{j},i\neq j\\\
\rho_{i}\oplus\rho_{j}&[\overline{F}]\in E_{i}\cap E_{j}\\\
\rho_{\frac{n}{2}-1}\oplus\rho_{n/2}\oplus\rho^{\prime}_{n/2}&[\overline{F}]\in
E_{\frac{n}{2}-1}\cap E_{n/2}\mbox{ ($n$ even)}\\\
\rho_{n/2}\oplus\rho^{\prime}_{n/2}&[\overline{F}]\in
E_{n/2}-E_{\frac{n}{2}-1}\mbox{ ($n$ even)}\\\ \end{cases}$

where $E_{i}$ is an exceptional divisor on $Y_{max}$ such that given the
projection morphism $p:X_{1}\xrightarrow{\pi}Y_{1}\cong Y_{max}$,
$p^{-1}(E_{i})=\tilde{E_{i}}\cup\tilde{E}_{n-i}$ (refer again to (3.1) for the
definition of $E$).

###### Proof.

We consider the diagram:

${\mathcal{Y}\times\mathbb{C}^{2}}$${\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})\times\mathbb{C}^{2}}$${\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})/\mathbb{Z}_{2}\times\mathbb{C}^{2}}$$\scriptstyle{\pi\times
id_{\mathbb{C}^{2}}}$$\scriptstyle{f\times
id_{\mathbb{C}^{2}}}$$\scriptstyle{p\times id_{\mathbb{C}^{2}}}$

The universal flat family of $\mathbb{Z}_{n}$-clusters on
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})\times\mathbb{C}^{2}$ is
denoted by $\mathcal{U}_{0}$.

The universal flat family over the stacks $\tilde{\mathcal{U}}$ and the coarse
moduli space $\mathcal{U}$ are defined as follows:

$\tilde{\mathcal{U}}=(f\times id_{\mathbb{C}^{2}})_{*}(\mathcal{U}_{0});\mbox{
}\mathcal{U}=(\pi\times id_{\mathbb{C}^{2}})_{*}(\tilde{\mathcal{U}})$

Our main interest here is to know the socles over the fixed point. The non-
stacky points follow from the Fourier-Mukai image of the skyscraper sheaf
$\mathcal{O}_{\tilde{y}}\oplus\mathcal{O}_{g\cdot\tilde{y}}$ on
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$.

For a fixed point $\tilde{y}$ in
$\mathbb{Z}_{n}\operatorname{-Hilb}(\mathbb{C}^{2})$ under the
$\mathbb{Z}_{2}$-action and $y=p(\tilde{y})$, we consider the exact sequence
of skyscraper sheaves over the stack $\mathcal{Y}$:

$O\rightarrow\mathcal{O}_{\tilde{y}}\otimes\delta_{0}^{\prime}\rightarrow\mathcal{O}_{\pi^{-1}y}\rightarrow\mathcal{O}_{\tilde{y}}\otimes\delta_{0}\rightarrow
0$

which realizes the skyscraper sheaf $\mathcal{O}_{\pi^{-1}y}$ as a nontrivial
extension of two $\mathbb{Z}_{2}$-equivariant sheaves.

Using the Fourier-Mukai transform $\Psi$, the exact sequence translates to:

$O\rightarrow\mathcal{U}_{0,\tilde{y}}\otimes\delta_{0}^{\prime}\rightarrow\tilde{\mathcal{U}}_{\pi^{-1}y}\rightarrow\mathcal{U}_{0,\tilde{y}}\otimes\delta_{0}\rightarrow
0$

By the definition of the universal families, the fiber is realized as:

$\mathcal{U}_{x}=((\pi\times
id_{\mathbb{C}^{2}})_{*}\tilde{\mathcal{U}})_{x}=\tilde{\mathcal{U}}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{O}_{\pi^{-1}(x)}.$

Realizing $\mathcal{U}_{\tilde{y}}$ as a $\mathbb{Z}_{2}$-invariant cluster
yields that the socle of $\mathcal{U}_{\tilde{y}}\otimes\delta_{0}$ is
$\rho_{n/2}$ and similarly, the socle of
$\mathcal{U}_{\tilde{y}}\otimes\delta_{1}$ is $\rho_{n/2}^{\prime}$. ∎

######  Remark 5.7.

We first enumerate the $D_{2n}$-constellations on each open subset of the
maximal resolution $Y$. We are mainly interested in the open sets
$U_{m+1}^{\prime}$ and $U_{m}^{\prime\prime}$ which cover the exceptional
divisor $E_{n/2}$:

Open set:
$U_{m}^{\prime\prime}=Spec\left(\mathbb{C}\left[\frac{(x^{m}+y^{m})^{2}}{(xy)^{m-1}},\frac{(x^{m}-y^{m})^{2}}{(x^{m}+y^{m})^{2}}\right]\right)$

$\mathbb{Z}_{n}$-constellation:

$\displaystyle Z_{m}$
$\displaystyle=\\{1,y,y^{2},\cdots,y^{m-1},x,x^{2},\cdots,x^{m-1},x^{m}-y^{m}\\}$

$D_{2n}$-constellation:

$1$ |  $(x,y)$ |  $(x^{2},y^{2})$ |  $\cdots$ |  $x^{m}-y^{m}$
---|---|---|---|---
$\alpha:=\frac{x^{m}-y^{m}}{x^{m}+y^{m}}$ |  $\alpha(y,-x)$ |  $\alpha(y^{2},-x^{2})$ |  $\cdots$ |  $\alpha(x^{m}-y^{m})$

Open set:
$U_{m+1}^{\prime}=Spec\left(\mathbb{C}\left[\frac{(x^{m}+y^{m})^{2}}{(x^{m}-y^{m})^{2}},\frac{(x^{m}-y^{m})^{2}}{(xy)^{m-1}}\right]\right)$

$\mathbb{Z}_{n}$-constellation:

$\displaystyle Z_{m}$
$\displaystyle=\\{1,y,y^{2},\cdots,y^{m-1},x,x^{2},\cdots,x^{m-1},x^{m}+y^{m}\\}$

$D_{2n}$-constellation:

$1$ |  $(x,y)$ |  $(x^{2},y^{2})$ |  $\cdots$ |  $x^{m}+y^{m}$
---|---|---|---|---
$\beta:=\frac{x^{m}+y^{m}}{x^{m}-y^{m}}$ |  $\beta(y,-x)$ |  $\beta(y^{2},-x^{2})$ |  $\cdots$ |  $\beta(x^{m}+y^{m})$

We compute the top and the socle by the following: for example in
$U_{m}^{\prime\prime}$ (and similarly for $U_{m+1}^{\prime}$), the
corresponding open set in $X_{1}$ is given by
$A=Spec(\mathbb{C}\left[\frac{(x^{m}+y^{m})^{2}}{(xy)^{m-1}},\frac{x^{m}-y^{m}}{x^{m}+y^{m}}\right])$,
so that every $\mathbb{Z}_{n}$-cluster is given by the ideal
$I_{a,b}=\langle(x^{m}+y^{m})^{2}-a(xy)^{m-1},(x^{m}-y^{m})-b(x^{m}+y^{m}),(x^{2m}-y^{2m})-ab(xy)^{m-1}\rangle$.
Thus, $x\cdot(x^{m}-y^{m})=x^{m+1}-xy^{m}=0$ (and also
$x\cdot\alpha(x^{m}-y^{m})=x\cdot b(x^{m}-y^{m})=x\cdot b^{2}(x^{m}+y^{m})=0$,
and similarly for the multiplication by $y$) by interpreting the ideals as a
$\mathbb{Z}_{n}$-cluster defined by the quotient $\mathbb{C}[x,y]/\langle
p_{i}x^{m}-q_{i}y^{m},x^{m+1},y^{m+1},xy\rangle$.

Based on the character table, given that the natural presentation $\rho_{nat}$
is isomorphic to its dual, i.e. $\rho_{nat}\cong\rho_{nat}^{\lor}$, the McKay
quiver for $D_{2n}$ is given by the following (with the first quiver for odd
$n$ and second quiver for even $n$ respectively), which is identical to the
McKay quiver of the binary dihedral group in the $SL(2)$ case:

${\circ_{\rho_{0}}}$${\circ_{\rho_{1}}}$${\circ_{\rho_{2}}}$${\cdots}$${\circ_{\rho_{\frac{n-1}{2}}}}$${\circ_{\rho_{0}^{\prime}}}$${\circ_{\rho_{0}}}$${\circ_{\rho_{n/2}}}$${\circ_{\rho_{1}}}$${\circ_{\rho_{2}}}$${\cdots}$${\circ_{\rho_{\frac{n}{2}-1}}}$${\circ_{\rho_{0}^{\prime}}}$${\circ_{\rho_{n/2}^{\prime}}}$

Consider the following bijection:

$\displaystyle E_{i}$ $\displaystyle\mapsto\rho_{i}$ $\displaystyle E_{n/2}$
$\displaystyle\mapsto\rho_{n/2}\oplus\rho_{n/2}^{\prime}\mbox{ ($n$ even)}$

It is imperative to comment on the possible McKay correspondence via top and
socles. Compared to the representation of a binary dihedral group as a small
finite subgroup of $GL(2,\mathbb{C})$, particularly in the even case, the
exceptional divisor $E_{n/2}$ corresponds to the two-dimensional
representation $\rho_{n/2}\oplus\rho_{n/2}^{\prime}$. This is because the
socle failed to separate the two 1-dimensional representations $\rho_{n/2}$
and $\rho_{n/2}^{\prime}$. This tells us that such a correspondence is not the
’ideal’ correspondence on the coarse moduli space.

######  Remark 5.8.

Considering the dual graph of exceptional divisors on $Y_{max}$, possibly
referring to the computation of socles over the coarse moduli space, that
there is a bijection between exceptional divisors and representations of the
group $G$. More precisely:

For odd $n$, we consider such bijection: $E_{i}\mapsto\rho_{i}$.

However, for even $n$, the mapping given by

$\displaystyle E_{i}$ $\displaystyle\mapsto\rho_{i}$ $\displaystyle E_{n/2}$
$\displaystyle\mapsto\rho_{n/2}\oplus\rho_{n/2}^{\prime}$

gives a bijection between two-dimensional irreducible representations of $G$
and the exceptional divisors whose self-intersection number is $-2$; and the
exceptional divisor corresponding to the two-dimensional decomposable
representation has self-intersection number $-1$.

######  Theorem 5.9.

For a given $D_{2n}$-constellation $\overline{F}_{st}$ on the exceptional
divisors over the quotient stack $\mathcal{Y}$ (refer to (5.1) for the
definitions),

$\operatorname{top}(\overline{F}_{st})=\rho_{0}\oplus\rho_{0}^{\prime}$
$\operatorname{socle}(\overline{F}_{st})=\begin{cases}\rho_{i}&[\overline{F}_{st}]\in\mathcal{E}_{i},[\overline{F}_{st}]\notin\mathcal{E}_{j},i\neq
j\\\
\rho_{i}\oplus\rho_{j}&[\overline{F}_{st}]\in\mathcal{E}_{i}\cap\mathcal{E}_{j}\\\
\rho_{\frac{n}{2}-1}\oplus\rho_{n/2}\oplus\rho^{\prime}_{n/2}&[\overline{F}_{st}]\in\mathcal{E}_{\frac{n}{2}-1}\cap\mathcal{E}_{n/2}\mbox{
($n$ even)}\\\
\rho_{n/2}\oplus\rho^{\prime}_{n/2}&[\overline{F}_{st}]\in\mathcal{E}_{n/2}-(\mathcal{E}_{\frac{n}{2}-1}\cup\\{\mathcal{B}_{1},\mathcal{B}_{2}\\})\mbox{
($n$ even)}\\\ \rho_{n/2}&[\overline{F}_{st}]=\mathcal{B}_{1}\mbox{ ($n$
even)}\\\ \rho^{\prime}_{n/2}&[\overline{F}_{st}]=\mathcal{B}_{2}\mbox{ ($n$
even)}\\\ \end{cases}$

###### Proof.

Now, we consider the derived equivalences $\Psi$ and $\Phi$ and compute in the
level of the global quotient stacks. We use the derived equivalence in (7.1)
of [Ish02] and let
$\phi:D(\mathcal{Y})\xrightarrow{\sim}D^{\mathbb{Z}_{2}}(X_{1})$ be the
derived equivalence between the coherent sheaves on $\mathcal{Y}$ and
$\mathbb{Z}_{2}$-equivariant sheaves on $X_{1}$:

$\operatorname{Hom}_{D(\mathcal{Y})}^{k}(\Phi(O_{0}\otimes\rho_{i}^{*}),O_{\mathbb{y}})\cong\operatorname{Hom}_{D^{\mathbb{Z}_{2}}(X_{1})}^{k}(\phi(\Phi(O_{0}\otimes\rho_{i}^{*})),\phi(O_{\mathbb{y}}))\cong\operatorname{Hom}_{D^{G}(\mathbb{C}^{2})}^{k}(O_{0}\otimes\rho_{i}^{*},\Psi(O_{\mathbb{y}})).$

It is imperative to notice here that $\phi(O_{\mathbb{y}})$ depends on whether
the point is stacky or not.

$\phi(O_{\mathbb{y}})=\begin{cases}O_{y}&\mbox{ if $\mathbb{y}$ is a stacky
point}\\\ O_{y}\oplus O_{g\cdot y}&\mbox{ if $\mathbb{y}$ is not a stacky
point}\end{cases}$

Once again, from (7.2) of [Ish02], where $Z_{y}$ is the subscheme of
$\mathbb{C}^{2}$ corresponding to $\mathbb{y}$ and
$\mathcal{F}^{\lor}:=\operatorname{RHom}_{O_{\mathbb{C}^{2}}}(\mathcal{F},O_{\mathbb{C}^{2}})$
is the derived dual:

$\Psi(O_{\mathbb{y}})=\begin{cases}O_{Z_{y}}^{\lor}\otimes
K_{\mathbb{C}^{2}}[2]&\mbox{ if $\mathbb{y}$ is a stacky point}\\\
(O_{Z_{y}}^{\lor}\oplus O_{Z_{g\cdot y}}^{\lor})\otimes
K_{\mathbb{C}^{2}}[2]&\mbox{ if $\mathbb{y}$ is not a stacky
point}\end{cases}$

By Serre duality:

$\displaystyle\operatorname{Hom}_{D^{G}(\mathbb{C}^{2})}^{k}(O_{0}\otimes\rho_{i}^{*},\Psi(O_{\mathbb{y}}))$
$\displaystyle=G\operatorname{-Hom}_{\mathbb{C}^{2}}^{k}(O_{0}\otimes\rho_{i}^{*},\Psi(O_{\mathbb{y}}))$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{i}\otimes
det(\rho_{nat}),O_{Z_{y}}(\oplus O_{Z_{g\cdot y}}))$ $\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{i}\otimes\rho_{0}^{\prime},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))$

We are now in the position to compile each of the equivalences:

$\displaystyle\operatorname{Hom}_{D(\mathcal{Y})}^{k}(O_{\mathcal{F}},O_{\mathbb{y}})$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{0}^{\prime},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))\mbox{ for ($i=0$)}$
$\displaystyle\operatorname{Hom}_{D(\mathcal{Y})}^{k}(O_{\mathcal{F}}(\mathcal{B}_{1}-\mathcal{B}_{2}),O_{\mathbb{y}})$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{0},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))\mbox{ for ($i=0^{\prime}$)}$
$\displaystyle\operatorname{Hom}_{D(\mathcal{Y})}^{k}(O_{\mathcal{E}_{i}}(-1),O_{\mathbb{y}})$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{i},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))\mbox{ for ($1\leq i\leq m-1$)}$
$\displaystyle\operatorname{Hom}_{D(\mathcal{Y})}^{k}(O_{\mathcal{E}_{m}}(-\mathcal{B}_{3}),O_{\mathbb{y}})$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{m},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))\mbox{ for ($i=m$; $n$ odd)}$
$\displaystyle\operatorname{Hom}_{D(\mathcal{Y})}^{k}(O_{\mathcal{E}_{m}}(-\mathcal{B}_{2}),O_{\mathbb{y}})$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{n/2}^{\prime},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))\mbox{ for ($i=n/2$; $n$ even)}$
$\displaystyle\operatorname{Hom}_{D(\mathcal{Y})}^{k}(O_{\mathcal{E}_{m}}(-\mathcal{B}_{1}),O_{\mathbb{y}})$
$\displaystyle\cong
G\operatorname{-Hom}_{\mathbb{C}^{2}}^{2-k}(O_{0}\otimes\rho_{n/2},O_{Z_{y}}(\oplus
O_{Z_{g\cdot y}}))\mbox{ for ($i=n/2^{\prime}$; $n$ even)}$

Setting $k=2$ in the above equivalences, we obtain a more refined structure of
the socles over the quotient stack. ∎

######  Remark 5.10.

We can define the top and the socle of a $D_{2n}$-constellation on the stacky
locus on the quotient stack as the $1/2$ of the $D_{2n}$-constellation on the
coarse moduli space. In simpler terms, this is simply the
$\mathbb{Z}_{2}$-invariant $\mathbb{Z}_{n}$-cluster corresponding to the fixed
point of $\mathbb{Z}_{n}\operatorname{-Hilb}$ under the action of
$\mathbb{Z}_{2}$. This is reflected in the proof of the proposition.

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