# Combinatorial Reid’s recipe for consistent dimer models

Alastair Craw  Department of Mathematical Sciences, University of Bath,
Claverton Down, Bath BA2 7AY, UK<EMAIL_ADDRESS>,  Liana Heuberger
Laboratoire angevin de recherche en mathématiques, Faculté des Sciences, 2
Boulevard Lavoisier, 49045 Angers cedex 01, France.  liana.heuberger@univ-
angers.fr and  Jesus Tapia Amador  Downside School, Stratton-on-the-Fosse,
Radstock, Somerset, BA3 4RJ, UK<EMAIL_ADDRESS>
  *     * scAbstract. scReid’s recipe [Rei97, Cra05] for a finite abelian subgroup $G\subset\operatorname{SL}(3,\mathbb scC$)$isacombinatorialprocedurethatmarksthetoricfanofthe$G$-Hilbertschemewithirreduciblerepresentationsof$G$.ThegeometricMcKaycorrespondenceconjectureofCautis--Logvinenko\leavevmode\nobreak\ \cite[cite]{[\@@bibref{}{CautisLogvinenko09}{}{}]}thatdescribescertainobjectsinthederivedcategoryof$$G$-Hilb$intermsofReid^{s}c\prime$s recipe was later proved by Logvinenko et. al. [Log10, CCL17]. We generalise Reid’s recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt–Craw–Quintero–Vélez [BCQ15]. Our main tool generalises the jigsaw transformations of Nakamura [Nak01] to consistent dimer models.

scKeywords. Reid’s recipe; dimer model; quiver moduli space; jigsaw
transformations; tilting bundle

sc2020 Mathematics Subject Classification. 14E16; 14M25; 16E35; 16G20

sc[Français]

scProcd́é de Reid combinatoire pour les modèles de dimères cohérents

scRésumé. scLe procédé de Reid [Rei97, Cra05] pour un sous-groupe abélien fini
$G\subset\operatorname{SL}(3,\mathbb
scC$)$estuneproc\'{e}durecombinatoirequifournitunmarquagedel^{s}c\prime$éventail
torique du $G$-schéma de Hilbert par les composantes irréductibles de $G$. La
version géométrique de la correspondance de McKay, conjecture formulée par
Cautis–Logvinenko [CL09] qui décrit cerains objets de la catégorie dérivée de
$G\operatorname{-Hilb}$ en termes de procédé de Reid, a été démontrée ensuite
par Logvinenko et. al. [Log10, CCL17]. Nous généralisons à tout modèle de
dimère le procédé de Reid qui consiste à marquer l’éventail torique d’une
résolution crépante de l’espace des modules d’une manière compatible avec la
correspondance géométrique de Bocklandt–Craw–Quintero–Vélez [BCQ15]. Notre
outil principal généralise les transformations de puzzles de Nakamura [Nak01]
aux modèles de dimères cohérents.

  *
cDecember 10, 2020Received by the Editors on February 11, 2020\.
Accepted on January 14, 2021.

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath
BA2 7AY, UK.

sce-mail<EMAIL_ADDRESS>
Laboratoire angevin de recherche en mathématiques, Faculté des Sciences, 2
Boulevard Lavoisier, 49045 Angers cedex 01, France.

sce-mail<EMAIL_ADDRESS>
Downside School, Stratton-on-the-Fosse, Radstock, Somerset, BA3 4RJ, UK.

sce-mail<EMAIL_ADDRESS>
The second author was funded by grant EP/S004130/1 (PI Elisa Postinghel) and
the third author was funded by CONACYT.

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Nakamura’s jigsaw transformations
  4. 4 Combinatorial Reid’s recipe
  5. 5 Compatibility with Geometric Reid’s Recipe

## 1\. Introduction

For a finite subgroup $G\subset\operatorname{SL}(2,\mathbb{C})$, the
construction by McKay [McK80] of an affine Dynkin diagram of type ADE directly
from the representation theory of $G$ established a bijection between the
nontrivial irreducible representations of $G$ and the irreducible components
of the exceptional divisor on the minimal resolution $S$ of
$\mathbb{C}^{2}/G$. For a geometric interpretation, identify $S$ with the
$G$-Hilbert scheme following Ito–Nakamura [IN99] and use the induced universal
family on $S$ to obtain an equivalence of derived categories of coherent
sheaves

${}_{\operatorname{KV}}\colon
D^{b}\big{(}G\text{-}\operatorname{coh}(\mathbb{C}^{2})\big{)}\longrightarrow
D^{b}\big{(}\\!\operatorname{coh}(S)\big{)}$

as in Kapranov–Vasserot [KV00]. For a nontrivial irreducible representation
$\rho$ of $G$, _ibid_. shows that the functor ${}_{\operatorname{KV}}$ sends
the simple $G$-sheaf $\mathcal{O}_{0}\otimes\rho$ on $\mathbb{C}^{2}$ to the
pure sheaf $\mathcal{O}_{C_{\rho}}(-1)[1]$ on $S$ whose support is the
irreducible component $C_{\rho}$ of the exceptional locus associated to $\rho$
by McKay’s bijection. Alternatively, the first Chern classes of the vector
bundles ${}_{\operatorname{KV}}(\mathcal{O}_{\mathbb{C}^{2}}\otimes\rho)$
introduced by Gonzalez-Sprinberg–Verdier [GSV84] provide the basis of
$H^{2}(S,\mathbb{Z})$ dual to the basis of $H_{2}(S,\mathbb{Z})$ given by the
curves classes $[C_{\rho}]$ for nontrivial $\rho$.

A similar story emerges for any finite subgroup
$G\subset\operatorname{SL}(3,\mathbb{C})$, though the final step is well-
understood only when $G$ is abelian. Indeed, building on work of Nakamura
[Nak01] in the abelian case, Bridgeland–King–Reid [BKR01] used the universal
family on $G\operatorname{-Hilb}$ to obtain an equivalence of derived
categories of coherent sheaves

${}_{\operatorname{BKR}}\colon
D^{b}\big{(}\\!\operatorname{coh}(G\operatorname{-Hilb})\big{)}\longrightarrow
D^{b}\big{(}G\text{-}\operatorname{coh}(\mathbb{C}^{3})\big{)}$

from which they deduced that $G\operatorname{-Hilb}$ is a crepant resolution
of $\mathbb{C}^{3}/G$. The conventions from [KV00] and [BKR01] are such that
the functors ${}_{\operatorname{KV}}$ and ${}_{\operatorname{BKR}}$ determined
by the universal family on $G\operatorname{-Hilb}$ point in opposite
directions, so to implement the final step in the programme extending the
McKay correspondence to dimension three, Cautis–Logvinenko [CL09] considered
the quasi-inverse functor

$\Psi:={}_{\operatorname{BKR}}^{-1}\colon
D^{b}\big{(}G\text{-}\operatorname{coh}(\mathbb{C}^{3})\big{)}\longrightarrow
D^{b}\big{(}\\!\operatorname{coh}(G\operatorname{-Hilb})\big{)}$

and studied the images under of the simple $G$-sheaves
$\mathcal{O}_{0}\otimes\rho$ on $\mathbb{C}^{3}$ when $G$ is abelian. Their
conjecture, proved by Logvinenko [Log10] when $\mathbb{C}^{3}/G$ has a single
isolated singularity and by Cautis–Craw–Logvinenko [CCL17] for any finite
abelian subgroup $G\subset\operatorname{SL}(3,\mathbb{C})$, asserted that the
objects $\Psi(\mathcal{O}_{0}\otimes\rho)$ can be computed explicitly from a
recipe of Reid [Rei97] for marking torus-invariant subvarieties of
$G\operatorname{-Hilb}$ with irreducible representations of $G$. _Reid’s
recipe_ , shown to hold for any finite abelian subgroup of
$\operatorname{SL}(3,\mathbb{C})$ by Craw [Cra05], therefore provides the
combinatorial input allowing for a complete understanding of the geometric
McKay correspondence that sends each nontrivial irreducible representation
$\rho$ of $G$ to the pure sheaf $\Psi(\mathcal{O}_{0}\otimes\rho)$ on
$G\operatorname{-Hilb}$.

We study a more general situation in which the singularity $\mathbb{C}^{3}/G$
is replaced by an arbitrary Gorenstein affine toric threefold
$\operatorname{Spec}R$, and where $G\operatorname{-Hilb}$ is replaced by a
moduli space of cyclic modules over a noncommutative crepant resolution (NCCR)
$A$ of $R$. Bocklandt [Boc12, Boc13] shows that every such algebra $A$ arises
as the path algebra of a quiver $Q$ modulo the ideal of relations encoded by a
consistent dimer model in a real 2-torus. To study cyclic $A$-modules we
specify a vertex of $Q$, denoted $0\in Q_{0}$, that determines a _0-generated
stability condition_ $\theta$ (see (4.1) below). The fine moduli space
$\mathcal{M}_{\theta}$ of $\theta$-stable $A$-modules of dimension vector
$(1,..,1)$ comes armed with a tautological vector bundle
$T=\bigoplusop\displaylimits_{i\in Q_{0}}L_{i}$, where each $L_{i}$ has rank
one. Results of Ishii–Ueda [IU08, IU15] and Broomhead [Bro12] imply that
$\mathcal{M}_{\theta}$ is a toric, crepant resolution of
$\operatorname{Spec}R$ and, moreover, the universal family over
$\mathcal{M}_{\theta}$ induces a derived equivalence θ with quasi-inverse

${}_{\theta}:={}_{\theta}^{-1}\colon
D^{b}(\operatorname{mod-}\\!A)\longrightarrow
D^{b}\big{(}\operatorname{coh}(\mathcal{M}_{\theta})\big{)}.$

Each $i\in Q_{0}$ defines a vertex simple $A$-module $S_{i}$ and the object
${}_{\theta}(S_{i})$ is known to be a pure sheaf for every $i\neq 0$ by work
of Bocklandt–Craw–Quintero-Vélez [BCQ15]. In light of the works of Logvinenko
et. al. [CL09, Log10, CCL17], it is natural to seek a recipe that marks cones
in the fan θ of the toric variety $\mathcal{M}_{\theta}$ with nonzero vertices
of the quiver in such a way that the objects ${}_{\theta}(S_{i})$ for $i\neq
0$ can be computed from this marking.

To state our first main result, each cone $\sigma\in{}_{\theta}$ determines a
$\theta$-stable $A$-module $M_{\sigma}$ that is isomorphic to the fibre of the
tautological bundle $T$ over a distinguished point $y\in\mathcal{M}_{\theta}$
in the torus-orbit determined by $\sigma$ (see Lemma 3.1). In what follows, it
is convenient to identify θ with the induced triangulation of the polygon
obtained as the height one slice of θ, so we refer to cones in θ of dimension
one, two and three as _lattice points, line segments_ and _lattice triangles_
respectively. The following statement combines results from Definitions 4.13
and 4.16, as well as Corollary 4.18:

######  Theorem 1.1 (Combinatorial Reid’s recipe).

Let $Q$ be the quiver dual to a consistent dimer model and choose a vertex
$0\in Q_{0}$. There is a combinatorial recipe that marks every internal
lattice point and line segment of the fan θ with one or more nonzero vertices
of $Q$; specifically, vertex $i\in Q_{0}$ marks:

  1. $(\mathrm{i})$

a lattice point $\rho$ iff $S_{i}$ lies in the socle of each torus-invariant
$\theta$-stable $A$-module $M_{\sigma}$ defined by a lattice triangle
$\sigma\in{}_{\theta}$ satisfying $\rho\subset\sigma$; and

  2. $(\mathrm{ii})$

a line segment $\tau$ iff $i$ is a source vertex of a pair of quivers
$Q(\tau)^{+}$ and $Q(\tau)^{-}$ that we obtain from the $\theta$-stable
$A$-modules $M_{\sigma_{+}}$ and $M_{\sigma_{-}}$ defined by the pair of
lattice triangles $\sigma_{\pm}\in{}_{\theta}$ satisfying
$\tau\subset\sigma_{\pm}$.

Moreover, this recipe agrees with Reid’s original recipe for marking cones in
the toric fan of $G\operatorname{-Hilb}$ in the special case when $Q$ is the
McKay quiver of a finite abelian subgroup
$G\subset\operatorname{SL}(3,\mathbb{C})$.

The torus-invariant $\theta$-stable $A$-modules $M_{\sigma}$ that provide the
input for Theorem 1.1 are each encoded combinatorially by a fundamental domain
in $\mathbb{R}^{2}$ for the action of the group $\mathbb{Z}^{2}$ of deck
transformations of the real 2-torus containing the dimer model . These
_fundamental hexagons_ , introduced originally by Ishii–Ueda [IU08, Section
4], are easy to calculate by hand in small examples, while a computer program
written by Raf Bocklandt can perform the calculation for larger examples. As a
result, our recipe is easy to implement in practice, and we illustrate this by
presenting two examples in Section 4.4. These examples exhibit many new
features that were not present in Reid’s original recipe for the toric fan of
$G\operatorname{-Hilb}$ (see Remark 4.21).

The statement in Theorem 1.1 that is by no means obvious, however, is that the
quivers $Q(\tau)^{\pm}$ associated to the interior line segment $\tau$ share
the same source vertices. To define these quivers we first reconstruct the
$A$-module $M_{\sigma_{-}}$ directly from $M_{\sigma_{+}}$, and vice versa,
using a procedure that we call a _jigsaw transformation_. This procedure (see
Theorem 3.12) establishes that the choice of the facet $\tau\subset\sigma_{+}$
determines a way to cut up the fundamental hexagon
$\operatorname{Hex}(\sigma_{+})$ into ‘jigsaw pieces’, each of which is a
union of (lifts to $\mathbb{R}^{2}$ of) tiles in , and moreover, each piece
can be translated by a uniquely determined element in the group
$\mathbb{Z}^{2}$ of deck transformations such that the newly arranged jigsaw
pieces reconstruct $\operatorname{Hex}(\sigma_{-})$; see Figure 9 for an
illustration. Now, a unique jigsaw piece contains the tile dual to the vertex
$0\in Q_{0}$, and once we remove this jigsaw piece from each of
$\operatorname{Hex}(\sigma_{\pm})$, we are left with the combinatorial data
encoding the quivers $Q(\tau)^{\pm}$. These two quivers do not coincide in
general (see Remark 4.8), but the striking fact is that the source vertices of
these two quivers are the same (see Corollary 4.11). This key ingredient
allows us to mark vertices on line segments in θ as in Theorem
1.1$(\mathrm{ii})$. That our recipe generalises that of Reid [Rei97] when $Q$
is the McKay quiver of a finite abelian subgroup
$G\subset\operatorname{SL}(3,\mathbb{C})$ follows from the fact that our
jigsaw transformations recover the so-called ‘$G$-igsaw transformations’ of
Nakamura [Nak01] in that case (see Corollary 3.13).

By combining Theorem 1.1 with results of [BCQ15], we provide the following
simple combinatorial description of the objects ${}_{\theta}(S_{i})$ on
$\mathcal{M}_{\theta}$ associated to some of the nonzero vertices $i\in Q_{0}$
(see Proposition 5.2).

######  Corollary 1.2 (Compatibility with Geometric Reid’s recipe).

The recipe from Theorem 1.1 is such that:

  1. $(\mathrm{i})$

if a vertex $i$ marks at least one interior lattice point in θ, then
${}_{\theta}(S_{i})\cong L_{i}^{-1}\otimes\mathcal{O}_{Z_{i}}$ where $Z_{i}$
is the (connected) union of all torus-invariant divisors $D_{\rho}$ indexed by
lattice points $\rho$ marked with $i$;

  2. $(\mathrm{ii})$

if a vertex $i$ marks a unique interior line segment $\tau$ in θ, then
${}_{\theta}(S_{i})\cong L_{i}^{-1}\otimes\mathcal{O}_{C_{\tau}}$ where
$C_{\tau}$ is the torus-invariant curve defined by $\tau$;

Corollary 1.2 makes no mention of the vertices $i$ marking two or more line
segments; instead, we conjecture in such cases that the support of
${}_{\theta}(S_{i})$ is the union of all torus-invariant divisors $D_{\rho}$
defined by lattice points $\rho$ in θ attached to two or more line segments
marked with vertex $i$ (see Conjecture 5.5 for a stronger statement). In fact,
we anticipate in such cases that the object ${}_{\theta}(S_{i})$ can be
described explicitly in terms of the lattice points at the endpoints of the
line segments singled out by Theorem 1.1$(\mathrm{ii})$ in a manner similar to
that given by Cautis–Craw–Logvinenko [CCL17, Theorem 1.2] for the toric fan of
$G\operatorname{-Hilb}$. The key technical tool in that work, namely the
construction of CT-subdivisions in the fan, required as input the original
combinatorial recipe from [Rei97, Cra05]. Thus, Theorem 1.1$(\mathrm{ii})$
paves the way for a dimer model generalisation.

We have focused on the geometric interpretation of Reid’s recipe that
describes the objects ${}_{\theta}(S_{i})$, but the application that Reid
originally had in mind was to encode a minimal set of relations in
$\operatorname{Pic}(G\operatorname{-Hilb})$ between the tautological line
bundles $L_{i}$ for $i\neq 0$ (see [Cra05, Theorem 6.1]). Using Theorem 1.1,
we formulate a conjecture presenting a similar set of minimal relations in
$\operatorname{Pic}(\mathcal{M}_{\theta})$ for any consistent dimer model with
a chosen vertex $0\in Q_{0}$ (see Conjecture 5.7). A proof of this conjecture
should lead to the following:

  * •

the construction of a $\mathbb{Z}$-basis of the cohomology
$H^{*}(\mathcal{M}_{\theta},\mathbb{Z})$ following the approach from [Cra05,
Section 7]. This would provide the appropriate analogue of the original
geometric construction of the McKay correspondence by Gonzalez-
Sprinberg–Verdier [GSV84].

  * •

a description of the GIT chamber containing the 0-generated stability
parameter $\theta$ defining $\mathcal{M}_{\theta}$, in a manner similar to
that given recently by Wormleighton [Wor20] for the GIT chamber of
$G\operatorname{-Hilb}$.

As a word of warning on terminology, we use three different combinatorial
objects determined by points and lines: the dimer model ; the quiver $Q$; and
the triangulation of a polygon that determines the fan θ. To avoid confusion,
we use different terminology and notation in each case that we summarise here:

| 0-dimensional | 1-dimensional | 2-dimensional
---|---|---|---
Dimer model | nodes ($\mathfrak{n}$) | edges ($\mathfrak{e}$) | tiles ($\mathfrak{t}$)
Quiver $Q$ | vertices ($i$) | arrows ($a$) |
Triangulation θ | lattice points ($\rho$) | line segments ($\tau$) | lattice triangles ($\sigma$)

### Acknowledgements

We thank Raf Bocklandt for sharing with us the computer-generated examples
upon which the results of this paper are based. Thanks to Timothy Logvinenko
and Alastair King for examining the PhD thesis of the third author that
contains many of our results. Thanks also to the anonymous referee for several
helpful comments.

## 2\. Background

### 2.1. Dimer models

Let be a dimer model in the real two-torus $\mathbb{T}$, _i. e._ is a
polygonal cell decomposition of $\mathbb{T}$, where elements of the set of
$i$-cells i are called _nodes_ , _edge_ and _tiles_ when $i=0,1,2$
respectively, that satisfies the following properties: first, the set 0
decomposes as the disjoint union of a set of _black_ nodes and a set of
_white_ nodes, such that every edge $\mathfrak{e}\in{}_{1}$ joins a black node
to a white node; and secondly, every tile is a simply-connected convex
polygon. We may assume that contains no bivalent nodes.

Dually, we obtain a quiver $Q=(Q_{0},Q_{1})$ embedded in $\mathbb{T}$, where
each vertex $i\in Q_{0}$ lies in the interior of a tile of , and each arrow
$a\in Q_{1}$ crosses a unique edge $\mathfrak{e}_{a}\in{}_{1}$ in such a way
that the white endnode of $\mathfrak{e}_{a}$ lies to the right of $a$. Let
$\operatorname{\mathsf{h}}(a),\operatorname{\mathsf{t}}(a)\in Q_{0}$ denote
the head and tail of an arrow $a\in Q_{1}$. Let $Q_{2}$ denote the set of
connected components of the complement of this quiver in $\mathbb{T}$, and we
refer to elements of $Q_{2}$ as _faces_ of $Q$. Each face $f\in Q_{2}$
contains a unique node of , and the arrows in the boundary of $f$ form a cycle
in $Q$ that is clockwise (resp. anticlockwise) when the dual node is white
(resp. black). Every arrow $a\in Q_{1}$ therefore appears in the cycle
traversing the boundary of precisely one white face and one black face, and we
may write these cycles in the form $ap_{a}^{+}$ and $ap_{a}^{-}$ respectively,
where $p_{a}^{\pm}$ are paths in $Q$ with tail at
$\operatorname{\mathsf{h}}(a)$ and head at $\operatorname{\mathsf{t}}(a)$. The
_Jacobian algebra_ of the dimer model is defined to be the quotient

(2.1) $A:=\mathbb{C}Q/\langle p_{a}^{+}-p_{a}^{-}\mid a\in Q_{1}\rangle$

of the path algebra $\mathbb{C}Q$ of the quiver $Q$ by the ideal of relations
$\langle p_{a}^{+}-p_{a}^{-}\mid a\in Q_{1}\rangle$.

A _perfect matching_ of is a set of edges $\Pi\subset{}_{1}$ such that for
each node $\mathfrak{n}\in{}_{0}$, there is a unique edge $\mathfrak{e}\in\Pi$
for which $\mathfrak{n}$ is an endnode of $\mathfrak{e}$. Given a pair of
perfect matchings $\Pi,{}^{\prime}$, the locus
$\mathbb{R}^{2}\setminus(\Pi\cup{}^{\prime})$ is a union of connected
components. The _height function_
$h_{\Pi,{}^{\prime}}\colon\mathbb{R}^{2}\to\mathbb{R}$, which is only well-
defined up to the choice of an additive constant, is locally-constant on
$\mathbb{R}^{2}\setminus(\Pi\cup{}^{\prime})$ and is determined as follows:
the value of $h_{\Pi,{}^{\prime}}$ increases (resp. decreases) by 1 when a
path passing between connected components crosses either an edge
$\mathfrak{e}\in\Pi$ with a black (resp. white) node to the right, or an edge
$\mathfrak{e}\in{}^{\prime}$ with the white (resp. black) node to the right.
The ambiguity in the choice of additive constant is removed by defining
functions

$h_{x}(\Pi,{}^{\prime}):=h_{\Pi,{}^{\prime}}\big{(}p+(1,0)\big{)}-h_{\Pi,{}^{\prime}}(p)\quad\text{and}\quad
h_{y}(\Pi,{}^{\prime}):=h_{\Pi,{}^{\prime}}\big{(}p+(0,1)\big{)}-h_{\Pi,{}^{\prime}}(p)$

for any point $p\in\mathbb{R}^{2}\setminus(\Pi\cup{}^{\prime})$, and the
_height change_ of with respect to ′ is defined to be

(2.2)
$h(\Pi,{}^{\prime})=\big{(}h_{x}(\Pi,{}^{\prime}),h_{y}(\Pi,{}^{\prime})\big{)}\in
H^{1}(\mathbb{T},\mathbb{Z}).$

Fix a reference perfect matching 0, and define the _characteristic polygon_ of
to be the convex polygon

$\Delta(\Gamma):=\operatorname{conv}\big{\\{}h(\Pi,{}_{0})\in
H^{1}(\mathbb{T},\mathbb{R})\mid\Pi\text{ is a perfect matching of
}\Gamma\big{\\}}$

obtained as the convex hull of the lattice points in
$H^{1}(\mathbb{T},\mathbb{R})\cong\mathbb{R}^{2}$. Given perfect matchings i
and j, the line segment joining the lattice points is
$h({}_{i},{}_{j}):=h({}_{i},{}_{0})-h({}_{j},{}_{0})$. If we choose a
different reference perfect matching then $\Delta(\Gamma)$ undergoes a
translation. For the rank three lattice
$N:=H^{1}(\mathbb{T},\mathbb{Z})\oplus\mathbb{Z}$, let $\sigma_{0}\subset
N\otimes{Z}\mathbb{Q}$ denote the cone over the lattice polygon
$\Delta(\Gamma)\times\\{1\\}$. Then for the dual lattice
$M:=\operatorname{Hom}(N,\mathbb{Z})$, the toric threefold
$X:=\operatorname{Spec}\mathbb{C}[\sigma_{0}^{\vee}\cap M]$ determined by the
characteristic polygon $\Delta(\Gamma)$ is Gorenstein.

######  Example 2.1.

Consider the dimer model shown in black in Figure 1. The dual quiver $Q$ is
drawn in grey in the same picture. This dimer model admits 60 perfect
matchings, but for any reference matching 0 there are only 10 distinct height
changes $h(\Pi,{}_{0})$. The characteristic polygon $\Delta(\Gamma)$ is shown
in Figure 1.

112333444566778910111112121313141515161717181819192020212223242425262728$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

12345678910

Figure 1. (a) A dimer model and its dual quiver $Q$; (b) The characteristic
polygon $\Delta(\Gamma)$.

### 2.2. Consistency

A dimer model determines a graph in $\mathbb{T}$ with nodes 0 and edges 1. A
_zig-zag path_ on is a path in this graph that turns maximally right at every
white node and maximally left at every black node. Each zig-zag path $\gamma$
is necessarily periodic, so each determines a class $[\gamma]\in
H_{1}(\mathbb{T},\mathbb{Z})$. Each edge $\mathfrak{e}\in{}_{1}$ determines
two zig-zag paths: one traversing the edge in each direction. Following
Ishii–Ueda [IU11, Definition 3.5], we say that is _consistent_ if: there is no
homologically trivial zig-zag path; no zig-zag path intersects itself on the
universal cover; and no pair of zig-zag paths intersect each other on the
universal cover in the same direction more than once. For alternative
definitions of consistency and for a comparison between them, see _ibid._ or
Bocklandt [Boc12]; for a single paragraph that summarises the situation, see
[BCQ15, Remark 2.1].

Combining the work of Broomhead [Bro12, Lemma 5.6] with results by Ishii–Ueda
[IU08, Proposition 6.3] and Craw–Quintero-Vélez [CQ12, Theorem 3.15] leads to
the following result:

######  Proposition 2.2.

Let be a consistent dimer model. The centre $Z(A)$ of the Jacobian algebra of
is isomorphic to the semigroup algebra $\mathbb{C}[\sigma_{0}^{\vee}\cap M]$.
In particular, the Gorenstein toric threefold determined by the characteristic
polygon satisfies $X\cong\operatorname{Spec}Z(A)$.

### 2.3. Moduli of quiver representations

A _representation of the quiver_ $Q$ is a collection $\\{V_{i}\mid i\in
Q_{0}\\}$ of finite dimensional $\mathbb{C}$-vector spaces and a collection
$\\{v_{a}\colon V_{\operatorname{\mathsf{t}}(a)}\to
V_{\operatorname{\mathsf{h}}(a)}\mid a\in Q_{1}\\}$ of $\mathbb{C}$-linear
maps. A representation $V$ of $Q$ is said to satisfy the relations
$\\{p_{a}^{+}-p_{a}^{-}\mid a\in Q_{1}\\}$ from (2.1) if and only if the
corresponding linear combination of $\mathbb{C}$-linear maps from
$V_{\operatorname{\mathsf{h}}(a)}$ to $V_{\operatorname{\mathsf{t}}(a)}$ is
zero. A morphism $f\colon V\to V^{\prime}$ of representations of $Q$ is a
collection $\\{f_{i}\colon V_{i}\to V^{\prime}_{i}\mid i\in Q_{0}\\}$ of
$\mathbb{C}$-linear maps such that
$v^{\prime}_{a}f_{\operatorname{\mathsf{t}}(a)}=f_{\operatorname{\mathsf{h}}(a)}v_{a}$
for all $a\in Q_{1}$. The category of representations of $Q$ satisfying the
relations from (2.1) is equivalent to the abelian category
$\operatorname{mod-}\\!A$ of finite-dimensional left modules over the Jacobian
algebra $A$. The dimension vector of a representation $V$ is the vector
$\dim(V)\in\mathbb{N}^{Q_{0}}$ whose $i$th component is $\dim(V_{i})$ for
$i\in Q_{0}$. Each vertex $i\in Q_{0}$ defines a simple object
$S_{i}=\mathbb{C}e_{i}$ in $\operatorname{mod-}\\!A$ called the _vertex
simple_ for $i\in Q_{0}$; as a representation of $Q$, this has
$V_{i}=\mathbb{C}$ and $V_{j}=0$ for $j\neq i$, where the maps $v_{a}$ are
zero for all $a\in Q_{1}$.

We are interested primarily in representations of dimension vector
$\mathbf{v}:=(1,1,\dots,1)\in\mathbb{N}^{Q_{0}}$. Consider the rational vector
space

$\Theta=\Big{\\{}\theta\in\operatorname{Hom}\big{(}\mathbb{Z}^{Q_{0}},\mathbb{Q}\big{)}\mid\theta(\mathbf{v})=0\Big{\\}}.$

Given a representation $V$ of $Q$, define $\theta(V):=\theta(\dim V)$ for
$\theta\in\Theta$. A representation $V$ is _$\theta$ -semistable_ if
$\theta(V)=0$ and every subrepresentation $V^{\prime}\subset V$ satisfies
$\theta(V^{\prime})\geq 0$, and it is _$\theta$ -stable_ if these inequalities
are strict for every nonzero, proper subrepresentation $V^{\prime}\subset V$.
These notions apply to representations satisfying the relations from (2.1), so
the equivalence of abelian categories mentioned above gives us the notion of
$\theta$-(semi)stability for $A$-modules. For any $\theta\in\Theta$, King
[Kin94, Proposition 5.2] constructs the coarse moduli space
$\overline{\mathcal{M}_{\theta}}$ of S-equivalence classes of
$\theta$-semistable $A$-modules of dimension vector $\mathbf{v}$ using GIT. We
say that $\theta\in\Theta$ is _generic_ if every $\theta$-semistable
$A$-module of dimension vector $v$ is $\theta$-stable, in which case
$\overline{\mathcal{M}_{\theta}}$ coincides with the fine moduli space
$\mathcal{M}_{\theta}$ of isomorphism classes of $\theta$-stable $A$-modules
of dimension vector $\mathbf{v}$. The universal family of $\theta$-stable
$A$-modules on $\mathcal{M}_{\theta}$ is a tautological locally free sheaf

$T=\bigoplusop\displaylimits_{i\in Q_{0}}L_{i},$

where $L_{i}$ has rank one for all $i\in Q_{0}$, together with a tautological
$\mathbb{C}$-algebra homomorphism

(2.3) $\phi\colon
A\rightarrow\operatorname{End}_{\mathscr{O}_{\mathcal{M}_{\theta}}}(T).$

We fix once and for all a vertex of the quiver that we denote $0\in Q_{0}$,
and we normalise the tautological bundle by fixing
$L_{0}\cong\mathscr{O}_{\mathcal{M}_{\theta}}$. For each closed point
$y\in\mathcal{M}_{\theta}$, the fibre $T_{y}=T\otimes\mathcal{O}_{y}$ of the
tautological bundle $T$ over $y$ is a $\theta$-stable $A$-module of dimension
vector $\mathbf{v}$, where the $A$-module structure is obtained by restriction
from the tautological maps $\phi(a)\colon L_{\operatorname{\mathsf{t}}(a)}\to
L_{\operatorname{\mathsf{h}}(a)}$ for $a\in Q_{1}$.

### 2.4. Toric VGIT for consistent dimer models

Assume from now on that is a consistent dimer model in $\mathbb{T}$. In this
case, the moduli spaces $\mathcal{M}_{\theta}$ can be constructed using toric
geometry. To describe the relevant toric GIT quotients, note that the homology
$H_{*}(\mathbb{T},\mathbb{Z})$ is computed by the complex

$\mathbb{Z}^{Q_{2}}\stackrel{{\scriptstyle\partial_{2}}}{{\longrightarrow}}\mathbb{Z}^{Q_{1}}\stackrel{{\scriptstyle\partial_{1}}}{{\longrightarrow}}\mathbb{Z}^{Q_{0}}$

with maps given by $\partial_{2}(f)={}_{a\subseteq\partial f}a$ for $f\in
Q_{2}$ and
$\partial_{1}(a)=\operatorname{\mathsf{h}}(a)-\operatorname{\mathsf{t}}(a)$
for $a\in Q_{1}$. Since $Q$ is connected, the sublattice
$B:=\operatorname{Image}(\partial_{1})\subset\mathbb{Z}^{Q_{0}}$ has corank
one. Mozgovoy–Reineke [MR10, Lemma 3.3] show that

$\Lambda:=\mathbb{Z}^{Q_{1}}/\langle\partial_{2}(f)-\partial_{2}(f^{\prime})\mid
f,f^{\prime}\in Q_{2}\rangle$

is a free abelian group (when admits a perfect matching which is the case
since is consistent); an explicit list of generators of is given in [CQ12,
Lemma 3.14]. The incidence map $\partial_{1}\colon\mathbb{Z}^{Q_{1}}\to B$ of
the quiver $Q$ factors through the quotient map
$\operatorname{wt}\colon\mathbb{Z}^{Q_{1}}\to\Lambda$, giving rise to a
commutative diagram

(2.4)
${\mathbb{Z}^{Q_{1}}}$${0}$${M}$${B}$${0}$$\scriptstyle{\operatorname{wt}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{d}$

of lattices. It follows that the map of semigroup algebras
$\mathbb{C}[\mathbb{Z}^{Q_{1}}]\to\mathbb{C}[\Lambda]$ induced by the quotient
map $\operatorname{wt}$ is compatible with the $B$-gradings induced by
$\partial_{1}$ and $d$. In geometric terms, this means the inclusion of
algebraic tori
$\operatorname{Spec}\mathbb{C}[\Lambda]\rightarrow\operatorname{Spec}\mathbb{C}[\mathbb{Z}^{Q_{1}}]$
induced by $\operatorname{wt}$ is equivariant with respect to the action of
the algebraic torus $T_{B}:=\operatorname{Hom}(B,\mathbb{C}^{\times})$ whose
weights are encoded by $\partial_{1}$ and $d$. If we write + for the image
under $\operatorname{wt}$ of the subsemigroup
$\mathbb{N}^{Q_{1}}\subset\mathbb{Z}^{Q_{1}}$, then the induced closed
immersion

(2.5)
$V:=\operatorname{Spec}\mathbb{C}[{}^{+}]\hookrightarrow\mathbb{C}^{Q_{1}}:=\operatorname{Spec}\mathbb{C}[\mathbb{N}^{Q_{1}}]$

is equivariant with respect to the $T_{B}$-action. For any character
$\theta\in B=\operatorname{Hom}(T_{B},\mathbb{C}^{\times})$, write

$V\operatorname{\\!/\\!\\!/\\!}_{\theta}T_{B}:=\operatorname{Proj}\left(\bigoplusop\displaylimits_{j\geq
0}\mathbb{C}[{}^{+}]_{j\theta}\right)$

for the GIT quotient, where $\mathbb{C}[{}^{+}]_{j\theta}$ denotes the vector
space of $j\theta$-semi-invariant elements of $\mathbb{C}[{}^{+}]$. Since is
consistent, the study of the moduli spaces $\mathcal{M}_{\theta}$ by
Ishii–Ueda [IU08] can be interpreted directly as variation of GIT quotient for
the toric varieties $V\operatorname{\\!/\\!\\!/\\!}_{\theta}T_{B}$ as in
Mozgovoy [Mog09, Section 4] or Craw–Quintero-Vélez [CQ12]; for a one paragraph
summary of the argument, see [BCQ15, Remark 2.4].

######  Proposition 2.3.

Assume that is consistent. For any generic $\theta\in\Theta$, there is a
commutative diagram

(2.6)
${\mathcal{M}_{\theta}}$${V\operatorname{\\!/\\!\\!/\\!}_{\theta}T_{B}}$${X}$${V\operatorname{\\!/\\!\\!/\\!}_{0}T_{B}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

of toric varieties in which the horizontal arrows are isomorphisms, the
vertical arrows are projective, crepant resolutions, and where the right-hand
arrow is obtained by variation of GIT quotient.

### 2.5. The (dual) tautological bundle is a tilting bundle

To state an important result linking the geometry of the moduli spaces
$\mathcal{M}_{\theta}$ with the Jacobian algebra $A$ of a consistent dimer
model, let $D^{b}(\operatorname{coh}(\mathcal{M}_{\theta}))$ denote the
bounded derived category of coherent sheaves on $\mathcal{M}_{\theta}$ and
$D^{b}(\operatorname{mod-}\\!A)$ the bounded derived category of finitely-
generated left $A$-modules.

######  Theorem 2.4 (Ishii–Ueda [IU15]).

Let be a consistent dimer model. For generic $\theta\in\Theta$, let $T$ denote
the tautological bundle on $\mathcal{M}_{\theta}$. The tautological
$\mathbb{C}$-algebra homomorphism $\phi\colon A\to\operatorname{End}(T)$ is an
isomorphism, and the functor

(2.7)
${}_{\theta}(-)=\mathbf{R}\\!\operatorname{Hom}_{\mathscr{O}_{\mathcal{M}_{\theta}}}(T^{\vee},-)\colon
D^{b}(\operatorname{coh}(\mathcal{M}_{\theta}))\longrightarrow
D^{b}(\operatorname{mod-}\\!A)$

induced by the dual bundle
$T^{\vee}:=\mathcal{H}om(T,\mathcal{O}_{\mathcal{M}_{\theta}})$ is an exact
equivalence of triangulated categories.

In fact, [IU15] show that $\mathbf{R}\\!\operatorname{Hom}(T,-)$ is an
equivalence, _i. e._ that $T$ itself is a tilting bundle. However, the fact
that $T^{\vee}$ is tilting follows as in [BCQ15, Lemma 3.2], and the functor θ
from (2.7) has some especially nice properties. For example, for each point
$y\in\mathcal{M}_{\theta}$, the skyscraper sheaf $\mathcal{O}_{y}$ is sent by
θ to the $A$-module

(2.8)
${}_{\theta}(\mathcal{O}_{y})=\mathbf{R}\Gamma\circ\mathbf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{M}_{\theta}}}(\mathcal{O}_{Y},T\otimes\mathcal{O}_{y})=\mathbf{R}\Gamma(T\otimes\mathcal{O}_{y})=\Gamma(T\otimes\mathcal{O}_{y})\cong
T_{y}$

obtained as the fibre of the tautological bundle $T$ over $y$ (here and here
alone, denotes global sections).

Theorem 2.4 also facilitates our understanding of the tautological line
bundles $\\{L_{i}\mid i\in Q_{0}\\}$ on $\mathcal{M}_{\theta}$. Indeed, for
each $i,j\in Q_{0}$, the tautological isomorphism $\phi\colon
A\to\operatorname{End}(T)$ induces an isomorphism between the space
$e_{j}Ae_{i}$ spanned by classes of paths in $Q$ from $i$ to $j$ and the
$\mathbb{C}$-vector space $\operatorname{Hom}(L_{i},L_{j})$. Under this
isomorphism, each $a\in Q_{1}$ is assigned to an effective torus-invariant
divisor $\operatorname{div}(a)$ obtained as the divisor of zeroes of a section
of $L_{\operatorname{\mathsf{h}}(a)}\otimes
L_{\operatorname{\mathsf{t}}(a)}^{-1}$. Explicitly, for each ray
$\rho\in{}_{\theta}(1)$ in the toric fan of $\mathcal{M}_{\theta}$, if we
write ρ and $D_{\rho}$ for the corresponding $\theta$-stable perfect matching
in and torus-invariant divisor in $\mathcal{M}_{\theta}$ respectively, then
[BM09, Theorem 4.2] and [IU15, Lemma 4.1] give that $\operatorname{div}(a)$ is
the divisor of zeroes of the section

(2.9)
$t^{\operatorname{div}(a)}:=\prodop\displaylimits_{\\{\rho\in{}_{\theta}(1)\mid\mathfrak{e}_{a}\in{}_{\rho}\\}}t_{\rho}\in
H^{0}(L_{\operatorname{\mathsf{h}}(a)}\otimes
L_{\operatorname{\mathsf{t}}(a)}^{-1}),$

where $\mathfrak{e}_{a}$ is the edge in the dimer model dual to the arrow $a$;
here, we write the section in the variables of the Cox ring
$\mathbb{C}[t_{\rho}\mid\rho\in{}_{\theta}(1)]$ of $\mathcal{M}_{\theta}$. For
$i=0$, if we choose any path $p$ in the quiver $Q$ from $0$ to $j\in Q_{0}$
and write $\operatorname{div}(p)$ for the sum of the divisors
$\operatorname{div}(a)$ indexed by arrows $a$ in the support of $p$, then
$\operatorname{div}(p)$ is the divisor of zeroes of the section
$t^{\operatorname{div}(p)}:=\prodop\displaylimits_{a\in\operatorname{supp}(p)}t^{\operatorname{div}(a)}\in
H^{0}(L_{j})$ and hence
$L_{j}\cong\mathcal{O}_{\mathcal{M}_{\theta}}\big{(}\operatorname{div}(p)\big{)}$.

######  Example 2.5.

Consider again the consistent dimer model from Example 2.1. The stability
parameter $\theta=(\theta_{i})\in\Theta$ satisfying $\theta_{i}>0$ for all
$i>0$ is generic. After listing all 10 of the $\theta$-stable perfect
matchings (see [Tap15, Figure 1.5]), it is straightforward to label each arrow
in $Q$ with the section from Equation (2.9) as shown in Figure 2(a). The
triangulation of the characteristic polygon $\Delta(\Gamma)$ that determines
the toric fan θ of $\mathcal{M}_{\theta}$ is shown in Figure 2(b).

1,2,61,2,63,44,54,54,53,8,9,103,8,9,103,8,9,1061,5,6,7,8,91,5,6,7,8,91,2,3,8,9,101,2,3,8,9,103,4,5,7,8,9,1041,94,5,74,5,75,6,75,6,72,3,102,3,101,6,7,93,4,103,4,101,2,9,101,2,6,7,8,9,101,2,6,7,8,9,104,5,6,7,84,5,6,7,82,3,4,8,102,3,4,8,10552,105,7,8,92336,71,23,4,5,81,6$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

12345678910

Figure 2. (a) Divisors labelling arrows in $Q$; (b) Triangulation of
$\Delta(\Gamma)$ defining a fan θ.

To illustrate how to compute with the tautological line bundles, let
$\sigma_{+}$ and $\sigma_{-}$ denote the 3-dimensional cones in θ generated by
the rays $\\{\rho_{7},\rho_{8},\rho_{9}\\}$ and
$\\{\rho_{8},\rho_{9},\rho_{10}\\}$ respectively, and consider
$\tau:=\sigma_{+}\cap\sigma_{-}\in{}_{\theta}(2)$. Let $p_{+}$ denote the path
in $Q$ from vertex 0 to vertex 9 labelled with the section

$t^{\operatorname{div}(p_{+})}=t_{2}t_{3}t_{4}t_{10}\in
H^{0}(U_{\sigma_{+}},L_{9}).$

The primitive vector $m\in M$ orthogonal to $\tau$ that is positive on
$\sigma_{+}$ determines the Laurent monomial
$t^{m}=t_{5}t_{6}t_{7}/t_{2}t_{3}t_{10}$ in the variables of the Cox ring, and

$t^{m}\cdot t^{\operatorname{div}(p_{+})}=t_{4}t_{5}t_{6}t_{7}\in
H^{0}(U_{\sigma_{-}},L_{9}).$

The path $p_{-}$ from vertex 0 to 9 labelled with
$t^{\operatorname{div}(p_{-})}=t_{4}t_{5}t_{6}t_{7}$ is the generating section
of $H^{0}(U_{\sigma_{-}},L_{9})$, and since $t^{m}$ appears with exponent 1 in
the equation $t^{\operatorname{div}(p_{-})}=t^{m}\cdot
t^{\operatorname{div}(p_{+})}$, it follows that $L_{9}$ has degree one on the
torus-invariant rational curve $C_{\tau}$ in $\mathcal{M}_{\theta}$ defined by
the cone $\tau$.

### 2.6. Classical Reid’s recipe

The first example in the literature of a consistent dimer model in a real
2-torus is the hexagonal tiling for a finite abelian subgroup
$G\subset\operatorname{SL}(3,\mathbb{C})$ studied by Reid [Rei97]; a more
formal construction in the context of dimer models was given by Ueda–Yamazaki
[UY11, Section 5]. In this case:

  * •

the quiver $Q$ is the McKay quiver of $G$ with vertex set
$Q_{0}=\operatorname{Irr}(G)$ given by the set of isomorphism classes of
irreducible representations of $G$;

  * •

the Jacobian algebra $A=\mathbb{C}[x,y,z]\rtimes G$ is the skew group algebra
of $G$;

  * •

the characteristic polygon $\Delta(\Gamma)$ is the junior simplex and the
Gorenstein toric threefold is $X\cong\mathbb{C}^{3}/G$.

Let $0\in Q_{0}$ denote the vertex corresponding to the trivial representation
of $G$. For the generic stability condition $\theta\in\Theta$ satisfying
$\theta_{i}>0$ for $i\neq 0$, the fine moduli space $\mathcal{M}_{\theta}$ is
isomorphic to the $G$-Hilbert scheme $G\operatorname{-Hilb}$ that parametrises
$G$-invariant subschemes $Z\subset\mathbb{C}^{3}$ such that
$H^{0}(\mathcal{O}_{Z})$ is isomorphic as a $G$-module to $\mathbb{C}[G]$. The
fan of the toric variety $G\operatorname{-Hilb}$ can be constructed as in
Craw–Reid [CR02].

_Reid’s recipe_ was introduced in several examples by Reid [Rei97] and proved
in general by Craw [Cra05]. The recipe marks interior line segments and
lattice points of the triangulation with certain nontrivial irreducible
representations $i\in Q_{0}\setminus\\{0\\}$. To state the result, let
$S_{i}:=\mathbb{C}e_{i}$ denote the vertex simple $A$-module where $e_{i}\in
A$ is the idempotent corresponding to the trivial path at vertex $i$.

######  Recipe 2.6 (Classical version of Reid’s recipe).

  1. (1)

Let $\tau\in\Sigma(2)$ be an interior line segment and let
$m=(m_{1},m_{2},m_{3})\in M$ be the primitive vector in either direction
normal to the hyperplane spanned by $\tau$. The line segment $\tau$ is marked
with the vertex $i\in Q_{0}$ such that the numerator and denominator of the
$G$-invariant Laurent monomial
$x^{m_{1}}y^{m_{2}}z^{m_{3}}\in\mathbb{C}[x^{\pm 1},y^{\pm 1},z^{\pm 1}]$ lie
in the $i$-character space.

  2. (2)

Let $\rho\in\Sigma(1)$ be an interior lattice point. A vertex $i\in Q_{0}$
marks $\rho$ if and only if $S_{i}$ lies in the socle of every $G$-cluster
defined by a torus-invariant point of the divisor $D_{\rho}$ in
$G\operatorname{-Hilb}$.

Surprisingly, every nonzero vertex marks either a unique lattice point or a
connected chain of edges that branches at most once [Cra05, Corollary 4.5]. As
an application, a complete set of minimal relations in
$\operatorname{Pic}(G\operatorname{-Hilb})$ between the tautological lines
bundles $\\{L_{i}\mid i\in Q_{0}\\}$ of $G\operatorname{-Hilb}$ can be given
[Cra05, Theorem 6.1]. The description of the marking of lattice points
presented in Recipe 2.6(2) above is equivalent to the recipe from the original
construction, see Craw–Ishii [CI04, Proposition 9.1].

Logvinenko [Log10] opened up the study of a geometric version of Reid’s recipe
when he proved a pair of conjectures from Cautis–Logvinenko [CL09] in the
special case when $\mathbb{C}^{3}/G$ has a single isolated singularity. To
state these results, consider $\mathcal{M}_{\theta}=G\operatorname{-Hilb}$ and
the functor θ from (2.7). Write the quasi-inverse to θ as

(2.10) ${}_{\theta}(-):=T^{\vee}\overset{\mathbf{L}}{\otimes}_{A}(-)\;\;\colon
D^{b}(\operatorname{mod-}\\!A)\longrightarrow
D^{b}\big{(}\operatorname{coh}(\mathcal{M}_{\theta})\big{)}$

rather than $-\overset{\mathbf{L}}{\otimes}_{A^{\textrm{op}}}T^{\vee}$. It is
immediate that θ sends the indecomposable projective $A$-module $Ae_{i}$ to
the line bundle $L_{i}^{-1}$ for each $i\in Q_{0}$. Cautis–Logvinenko [CL09]
studied the image under θ of the vertex simple $A$-modules $S_{i}$, and showed
that the object ${}_{\theta}(S_{i})$, which a priori is a complex of coherent
sheaves, is in fact quasi-isomorphic to the shift (by 0 or 1) of a coherent
sheaf. Logvinenko [Log10] went further by establishing the following striking
relationship between Reid’s recipe and the (support of the) object
${}_{\theta}(S_{i})$:

######  Theorem 2.7 (Logvinenko [Log10]).

Let $\mathbb{C}^{3}/G$ have a single isolated singularity and let $i\in Q_{0}$
be any non-zero vertex, _i. e._ $i$ is not the trivial representation.
According to Reid’s recipe, vertex $i$ marks either:

  1. $(\mathrm{i})$

a unique interior lattice point $\rho\in\Sigma(1)$, and ${}_{\theta}(S_{i})$
is quasi-isomorphic to $L_{i}^{-1}\otimes\mathcal{O}_{D_{\rho}}$;

  2. $(\mathrm{ii})$

a unique interior line segment $\tau\in\Sigma(2)$, and ${}_{\theta}(S_{i})$ is
quasi-isomorphic to $L_{i}^{-1}\otimes\mathcal{O}_{C_{\tau}}$;

  3. $(\mathrm{iii})$

two or more interior line segments in $\Sigma(2)$, and ${}_{\theta}(S_{i})$ is
quasi-isomorphic to an object $\mathcal{F}[1]$, where $\mathcal{F}$ is a
coherent sheaf whose support is the union of all torus-invariant divisors
$D_{\rho}$ such that two line segments $\tau$ in  containing $\rho$ are marked
by vertex $i$.

Cautis–Craw–Logvinenko [CCL17] later described the mysterious sheaves
$\mathcal{F}$ that appear in Theorem 2.7$(\mathrm{iii})$ in terms of Reid’s
recipe and they generalised Theorem 2.7 to every finite abelian subgroup
$G\subset\operatorname{SL}(3,\mathbb{C})$, _i. e._ $\mathbb{C}^{3}/G$ need not
have a single isolated singularity. This complete description of the objects
${}_{\theta}(S_{i})$ is called _Derived Reid’s recipe_. Even for this derived
category statement, the classical Recipe 2.6 forms an important part of the
result and it is essential in defining the notion of a CT-subdivision (see
[CCL17, Definition 4.2]) that provides the key technical tool for the proofs.

## 3\. Nakamura’s jigsaw transformations

This section studies fundamental hexagons and provides a dimer model analogue
of the ‘$G$-igsaw transformations’ introduced originally by Nakamura [Nak01]
for a finite abelian subgroup $G\subset\operatorname{SL}(3,\mathbb{C})$.

### 3.1. Distinguished $\theta$-stable $A$-modules

For any generic stability parameter $\theta\in\Theta$, we now describe how to
associate a $\theta$-stable $A$-module $M_{\sigma}$ to each cone in the fan θ
of the toric variety $\mathcal{M}_{\theta}$.

First, for any ray $\rho\in{}_{\theta}(1)$, let
$O_{\rho}\subset\mathcal{M}_{\theta}$ denote the corresponding two-dimensional
torus-orbit. For any point $y\in O_{\rho}$, let $(v_{a})\in\mathbb{C}^{Q_{1}}$
be any quiver representation that determines the $\theta$-stable $A$-module
$T_{y}$ obtained as the fibre of $T$ over the point $y$. Ishii–Ueda [IU08,
Lemma 6.1] prove that the set of edges

(3.1) ${}_{\rho}:=\\{\mathfrak{e}_{a}\in{}_{1}\mid v_{a}=0\\}$

in is a perfect matching that is independent of the choice of point $y\in
O_{\rho}$ and the choice of representative $(v_{a})$ of $T_{y}$, and moreover,
every $\theta$-stable perfect matching of arises in this way.

Using these perfect matchings, we associate a $\theta$-stable $A$-module to
each cone of the fan θ as follows. For $\sigma\in{}_{\theta}$, write
$\sigma(1):=\\{\rho\in{}_{\theta}(1)\mid\rho\subseteq\sigma\\}$, and for each
$a\in Q_{1}$, define scalars

(3.2) $v_{a}:=\left\\{\begin{array}[]{cl}0&\text{if
}\mathfrak{e}_{a}\in\bigcupop\displaylimits_{\rho\in\sigma(1)}{}_{\rho}\\\
1&\text{otherwise}\end{array}\right..$

For example, the zero cone $\sigma$ contains no rays, so $v_{a}=1$ for all
$a\in Q_{1}$ in that case. For each cone $\sigma\in{}_{\theta}$, the point
$(v_{a})\in\mathbb{C}^{Q_{1}}$ is a representation of the quiver $Q$ of
dimension vector $\mathbf{v}$ that depends only on $\sigma$. Let $M_{\sigma}$
denote the corresponding $\mathbb{C}Q$-module of dimension vector $v$. In fact
we can say more about $M_{\sigma}$:

######  Lemma 3.1.

For $\sigma\in{}_{\theta}$, there exists a unique point $y\in
O_{\sigma}\subseteq\mathcal{M}_{\theta}$ such that $M_{\sigma}\cong T_{y}$. In
particular, $M_{\sigma}$ is a $\theta$-stable $A$-module of dimension vector
$v$.

###### Proof.

This is essentially a restatement of results from Ishii–Ueda [IU08] and
Mozgovoy [Mog09]. Indeed, for $\sigma\in{}_{\theta}$, we claim that the point
$(v_{a})\in\mathbb{C}^{Q_{1}}$ from (3.2) is a $\theta$-stable point of the
subscheme $V\subseteq\mathbb{C}^{Q_{1}}$ from (2.5) and, moreover, the point
$y:=[(v_{a})]\in\mathcal{M}_{\theta}$ lies in $O_{\sigma}$. The proof is a
case-by-case analysis according to the dimension of $\sigma$ as follows. For
$0\in{}_{\theta}$, we have $v_{a}=1$ for all $a\in Q_{1}$, so $(v_{a})\in
V\cap(\mathbb{C}^{\times})^{Q_{1}}$. Every point in
$(\mathbb{C}^{\times})^{Q_{1}}$ is $\theta$-stable for all $\theta\in\Theta$,
so $y=[(v_{a})]$ satisfies
$y\in\bigl{(}V\cap(\mathbb{C}^{\times})^{Q_{1}}\bigl{)}/(\mathbb{C}^{\times})^{Q_{0}}=O_{0}\subset\mathcal{M}_{\theta}$
as required. For a cone $\sigma\in{}_{\theta}(d)$ of dimension $d>0$, the
result was established by [IU08, Lemma 6.2] when $d=1$, by [Mog09, Corollary
4.19] when $d=2$ and by both [IU08, Section 4] and [Mog09, Corollary 4.18]
when $d=3$. ∎

### 3.2. Fundamental hexagons

We now recall the combinatorial description of the torus-invariant
$\theta$-stable $A$-modules $M_{\sigma}$ for $\sigma\in{}_{\theta}(3)$ by
Ishii–Ueda [IU08, Lemma 4.5]. Let
$\pi\colon\mathbb{R}^{2}\rightarrow\mathbb{T}$ denote the universal cover of
the two-dimensional torus. The preimage of defines a $\mathbb{Z}^{2}$-periodic
regular cell decomposition $\widetilde{}\Gamma$ of $\mathbb{R}^{2}$.
Similarly, the preimage of the quiver $Q$ dual to is a
$\mathbb{Z}^{2}$-periodic quiver $\widetilde{Q}$ in $\mathbb{R}^{2}$, such
that each vertex $\widetilde{i}$ of $\widetilde{Q}$ lies inside a unique
polygonal tile of $\widetilde{}\Gamma$.

For a cone $\sigma\in{}_{\theta}(3)$, let
$\rho_{0},\rho_{1},\rho_{2}\in{}_{\theta}(1)$ denote the rays in $\sigma$, and
write ${}_{0},{}_{1},{}_{2}$ for the corresponding $\theta$-stable perfect
matchings. Let $Q^{\sigma}$ denote the subquiver of $Q$ with vertex set
$Q_{0}$ and with arrow set comprising those $a\in Q_{1}$ for which the
corresponding scalar from (3.2) satisfies $v_{a}\neq 0$; in terms of perfect
matchings, this is the set of arrows $a\in Q_{1}$ such that
$\mathfrak{e}_{a}\notin\bigcupop\displaylimits_{0\leq i\leq 2}{}_{i}$. Fix a
vertex $i\in Q_{0}$. Then for any choice of $\widetilde{i}\in\pi^{-1}(i)$,
there is a uniquely defined subquiver $\widetilde{Q^{\sigma}}$ of
$\widetilde{Q}$ that has $\widetilde{i}$ as a vertex such that $\pi$
identifies the quivers $\widetilde{Q^{\sigma}}$ and $Q^{\sigma}$ [IU08, Lemma
4.1].

######  Definition 3.2.

For $\sigma\in{}_{\theta}(3)$, the _fundamental hexagon_
$\operatorname{Hex}(\sigma)$ is the subset of $\mathbb{R}^{2}$ covered by the
tiles of $\widetilde{\Gamma}$ dual to the vertices of
$\widetilde{Q^{\sigma}}$. Let $\operatorname{Graph}(\sigma)$ denote the
$\mathbb{Z}^{2}$-periodic graph obtained as the union of all
$\mathbb{Z}^{2}$-translates of the boundary of $\operatorname{Hex}(\sigma)$.

Note that $\operatorname{Hex}(\sigma)$ encodes the same information as the
quiver $Q^{\sigma}$, so it allows one to reconstruct the $A$-module
$M_{\sigma}$ for $\sigma\in{}_{\theta}(3)$. Also, since the boundary of
$\operatorname{Hex}(\sigma)$ is a subset of $\operatorname{Graph}(\sigma)$, it
is sometimes convenient to regard an edge of the boundary of
$\operatorname{Hex}(\sigma)$ as an edge of the dimer model in the 2-torus
$\mathbb{T}$.

######  Example 3.3.

Consider the dimer model from Example 2.1, and let $\sigma\in{}_{\theta}(3)$
be the cone generated by the rays $\rho_{8},\rho_{9},\rho_{10}$ in the fan θ
shown in Figure 2(b). The dashed arrows in green in Figure 3 illustrate one
lift $\widetilde{Q^{\sigma}}$ to $\mathbb{R}^{2}$ of the quiver $Q^{\sigma}$,
while the dashed edges in red illustrate the edges in the boundary of
$\operatorname{Hex}(\sigma)$. The set $\bigcupop\displaylimits_{0\leq\rho\leq
2}{}_{\rho}$ is the union of all edges shown in black and all edges shown in
dashed red; note that $\operatorname{Graph}(\sigma)$ is obtained from this set
by removing each connected component comprising only a single edge.

11233359111112122020232424252628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

11233359111112122020232424252628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

11233359111112122020232424252628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

11233359111112122020232424252628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

Figure 3. The red dashed lines form the boundary of the fundamental hexagon.

Since the tiles in $\operatorname{Hex}(\sigma)$ are obtained by lifting each
tile of in $\mathbb{T}$, we see that $\operatorname{Hex}(\sigma)$ forms a
fundamental domain for the action of $\mathbb{Z}^{2}$ on $\mathbb{R}^{2}$, and
the boundary of $\operatorname{Hex}(\sigma)$ coincides with
$\operatorname{Graph}(\sigma)$ when both are viewed as subsets of
$\mathbb{T}$. To fully justify the terminology from Definition 3.2 and to show
how the boundary of $\operatorname{Hex}(\sigma)$ can be calculated directly
from perfect matchings, we bring together results of Ishii–Ueda and Mozgovoy
in the following statement.

######  Proposition 3.4.

For $\sigma\in{}_{\theta}(3)$, the boundary of $\operatorname{Hex}(\sigma)$
contains precisely six trivalent nodes of $\operatorname{Graph}(\sigma)$ that,
when listed cyclically, alternate between white nodes of one
$\mathbb{Z}^{2}$-orbit and black nodes in a second $\mathbb{Z}^{2}$-orbit; all
other nodes of $\operatorname{Graph}(\sigma)$ have valency two. Moreover,
$\operatorname{Graph}(\sigma)$ is the unique connected component of the locus
$\bigcupop\displaylimits_{0\leq\rho\leq 2}{}_{\rho}$ in $\mathbb{T}$
comprising more than a single edge.

###### Proof.

The first statement follows from [IU08, Lemma 4.4] and the proof of [IU08,
Lemma 4.5]. For the second statement, Mozgovoy [Mog09, Corollary 4.18] shows
that each edge $\mathfrak{e}\in\bigcupop\displaylimits_{0\leq\rho\leq
2}{}_{\rho}$ lies either:

  1. $(\mathrm{i})$

strictly in the interior of $\operatorname{Hex}(\sigma)$, in which case
$\mathfrak{e}\in{}_{0}\cap{}_{1}\cap{}_{2}$. Such edges do not touch any other
edge of $\bigcupop\displaylimits_{0\leq\rho\leq 2}{}_{\rho}$, because no two
edges of the same perfect matching touch; or

  2. $(\mathrm{ii})$

on the boundary of $\operatorname{Hex}(\sigma)$, where $\mathfrak{e}$ is an
edge in a chain linking adjacent trivalent points of
$\operatorname{Graph}(\sigma)$. These six chains are identified pairwise by
the quotient map $\pi$. Each chain comprises an odd number of edges, and for
$0\leq\rho\leq 2$, the edges in each chain belong alternately to either a
single perfect matching ρ or to a pair of perfect matchings ρ±1, where
addition of indices is taken modulo $3$ as shown in Figure 4.

Therefore edges of $\operatorname{Graph}(\sigma)$ are precisely those of type
$(\mathrm{ii})$, and these edges form a connected component of
$\bigcupop\displaylimits_{0\leq\rho\leq 2}{}_{\rho}$. Every other edge in
$\bigcupop\displaylimits_{0\leq\rho\leq 2}{}_{\rho}$ is of type
$(\mathrm{i})$, and every such edge lies in a connected component comprising
only that single edge. This completes the proof. ∎

110220${}_{0}\cap{}_{1}\cap{}_{2}$${}_{0}\cap{}_{1}\cap{}_{2}$${}_{0}\cap{}_{1}\cap{}_{2}$${}_{0}\cap{}_{1}\cap{}_{2}$1${}_{0}\cap{}_{2}$1${}_{0}\cap{}_{2}$…$\mathfrak{m}_{0,1}\cap\mathfrak{m}_{1,2}$2${}_{0}\cap{}_{1}$${}_{0}\cap{}_{1}$2...$\mathfrak{m}_{0,2}\cap\mathfrak{m}_{1,2}$0${}_{1}\cap{}_{2}$${}_{1}\cap{}_{2}$0...$\mathfrak{m}_{0,1}\cap\mathfrak{m}_{0,2}$1${}_{0}\cap{}_{2}$1${}_{0}\cap{}_{2}$…$\mathfrak{m}_{0,1}\cap\mathfrak{m}_{1,2}$2${}_{0}\cap{}_{1}$${}_{0}\cap{}_{1}$2...$\mathfrak{m}_{0,2}\cap\mathfrak{m}_{1,2}$0${}_{1}\cap{}_{2}$${}_{1}\cap{}_{2}$0...$\mathfrak{m}_{0,1}\cap\mathfrak{m}_{0,2}$
Figure 4. The perfect matchings and meandering walks (see Section 3.3) for
$\operatorname{Hex}(\sigma)$.

### 3.3. Meandering walks

We now generalise the notion of a zig-zag path in by associating a walk in a
consistent dimer model to any line segment $\tau\in{}_{\theta}(2)$, where θ is
the toric fan of $\mathcal{M}_{\theta}$ for any generic stability parameter
$\theta\in\Theta$. In fact, the content of this section requires only that is
non-degenerate [IU08].

######  Definition 3.5.

Let i, j be $\theta$-stable perfect matchings of , and let
$\rho_{i},\rho_{j}\in{}_{\theta}(1)$ be the corresponding rays. The _symmetric
difference_ of i and j is the set
${}_{i}\ominus{}_{j}:=({}_{i}\cup{}_{j})\setminus({}_{i}\cap{}_{j})$. For
$\tau\in{}_{\theta}(2)$, the _meandering walk_ of $\tau$ is the set
$\mathfrak{m}_{\tau}:=\mathfrak{m}_{i,j}={}_{i}\ominus{}_{j}$, where
$\rho_{i},\rho_{j}$ are the ray generators of $\tau$.

To explain the terminology, we’ll see that the edges in a meandering walk form
a cycle that does not always turn maximally right at white nodes and maximally
left at black nodes, but rather, it meanders. Meandering walks provide a
common generalisation of the notions of a zig-zag path as in Broomhead [Bro12]
and a $\sigma$-Strand as in Logvinenko [Log03, Definition 6.51].

The proof of Proposition 3.4, and specifically statement $(\mathrm{ii})$ of
that proof, implies that if $\rho_{i}$ and $\rho_{j}$ are the endpoints of a
side $\tau$ in a triangle $\sigma\in{}_{\theta}(3)$, then the edges of the
chains that form two adjacent sides of the boundary of
$\operatorname{Hex}(\sigma)$ belong alternately to the corresponding perfect
matchings i and j. Therefore, the edges in a meandering walk
$\mathfrak{m}_{\tau}$ form a cycle in , and conversely, the edges of the
chains that form any two adjacent sides of $\operatorname{Hex}(\sigma)$ form
the edges in a meandering walk $\mathfrak{m}_{\tau}$ for some two-dimensional
cone $\tau\subset\sigma$. We illustrate this in Figure 4 where, for example,
the meandering walk $\mathfrak{m}_{1,2}$ traverses edges that lie alternately
in perfect matchings 1 and 2. Note that the chain of edges along any single
side of the boundary of $\operatorname{Hex}(\sigma)$ form the intersection of
a pair of meandering walks.

Traversing the edges in a meandering walk $\mathfrak{m}_{\tau}$ around the
torus defines a homology class $[\mathfrak{m}_{\tau}]\in
H_{1}(\mathbb{T},\mathbb{Z})$. Changing the direction in which we traverse the
edges would change the sign of this class, but this sign is irrelevant for the
next result.

######  Lemma 3.6.

Let $\tau\in{}_{\theta}(2)$. For either choice of direction on the meandering
walk $\mathfrak{m}_{\tau}$, the resulting homology class
$[\mathfrak{m}_{\tau}]\in H_{1}(\mathbb{T},\mathbb{Z})$ is orthogonal to the
line segment $\tau$ in the characteristic polygon $\Delta(\Gamma)\subset
H^{1}(\mathbb{T},\mathbb{R})$ with respect to (the extension over $\mathbb{R}$
of) the perfect pairing $H_{1}(\mathbb{T},\mathbb{Z})\otimes
H^{1}(\mathbb{T},\mathbb{Z})\to\mathbb{Z}$.

###### Proof.

Let $\rho_{i},\rho_{j}\in{}_{\theta}(1)$ be the ray generators of $\tau$, and
choose $\sigma\in{}_{\theta}(3)$ satisfying $\tau\subset\sigma$. Traversing
the edges of $\mathfrak{m}_{\tau}$ in $\mathbb{T}$ defines a cycle obtained as
the image under $\pi\colon\mathbb{R}^{2}\to\mathbb{T}$ of a path that follows
two adjacent sides of the fundamental hexagon $\operatorname{Hex}(\sigma)$. If
we choose coordinates on $\mathbb{R}^{2}$ such that this path travels from
$(0,0)$ to $(0,1)$, then tracing the cycle
$\mathfrak{m}_{\tau}\subset\mathbb{T}$ in this direction defines the class
$(0,1)\in H_{1}(\mathbb{T},\mathbb{Z})$. Figure 5 depicts part of
$\operatorname{Graph}(\sigma)$ in the universal cover: the thicker edges of
$\operatorname{Graph}(\sigma)$ show lifts of $\mathfrak{m}_{\tau}$ via the
universal cover $\pi$; the dashed lines delineate a fundamental domain for the
action of $\mathbb{Z}^{2}$ on $\mathbb{R}^{2}$.

$(0,0)$$(0,1)$$(1,0)$ Figure 5. Lifts of the meandering walk
$\mathfrak{m}_{\tau}$ with homology class $[\mathfrak{m}_{\tau}]=(0,1)$.

For the $\theta$-stable perfect matchings i and j at the lattice points
$\rho_{i}$ and $\rho_{j}$ respectively, we saw in Section 2.1 that the height
change $h({}_{i},{}_{j})\in\mathbb{Z}^{2}$ is a vector parallel to the line
segment $\tau$ in the characteristic polygon $\Delta(\Gamma)$. More
invariantly, we regard the height change as a cohomology class
$h({}_{i},{}_{j})\in H^{1}(\mathbb{T},\mathbb{Z})$ that we now compare to the
homology class of the corresponding meandering walk $\mathfrak{m}_{\tau}$;
here, the walk $\mathfrak{m}_{\tau}$ up two sides of the hexagon at the
extreme left of Figure 5 corresponds to the walk $\mathfrak{m}_{0,2}$ in
Figure 4, so for the purposes of comparison we set $i=0$ and $j=2$. Every edge
in the perfect matchings 0 and 2 is shown in Figure 4. Comparing this with the
top-left hexagon in Figure 5, we see that it is possible to travel from
$p=(\varepsilon,\varepsilon)$ for small $\varepsilon>0$ to $p+(0,1)$ in Figure
5 without crossing a single edge in either 0 or 2, so according to (2.2), the
$y$-coordinate of $h({}_{i},{}_{j})$ equals 0. On the other hand, when
travelling from $p$ to $p+(1,0)$, one is forced to cross once the walk
$\mathfrak{m}_{0,2}$ whose edges lie alternately in 0 and 2 (as shown in
Figure 4), so the $x$-coordinate of $h({}_{i},{}_{j})$ equals 1. Therefore
$h({}_{i},{}_{j})=(1,0)\in\mathbb{Z}^{2}=H^{1}(\mathbb{T},\mathbb{Z})$, and
the natural pairing gives
$h({}_{i},{}_{j})\cdot[\mathfrak{m}_{i,j}]=(1,0)\cdot(0,1)=0$ as required. ∎

######  Remark 3.7.

Ishii–Ueda [IU15, Theorem 11.1] implies that the homology class of each zig-
zag in is orthogonal to the corresponding line segment in the boundary of
$\Delta(\Gamma)$. Lemma 3.6 provides a generalisation of this statement to any
line segment in the triangulation of $\Delta(\Gamma)$ determined by the
crepant resolution $\tau_{\theta}\colon\mathcal{M}_{\theta}\to X$ for any
generic $\theta\in\Theta$. Lemma 3.6 also extends the result of Logvinenko
[Log03, Proposition 6.57] for $\sigma$-Strands beyond the McKay quiver case.

### 3.4. Generalised Nakamura jigsaw transformations

For any finite abelian subgroup $G\subset\operatorname{SL}(3,\mathbb{C})$,
Nakamura [Nak01] introduced an algorithm to construct the $G$-Hilbert scheme
where the key step in each iteration of the algorithm was a combinatorial
procedure called a ‘$G$-igsaw transformation’ from one torus-invariant
$G$-cluster to another. In this section we generalise Nakamura’s $G$-igsaw
transformations to any generic stability condition $\theta$ and any consistent
dimer model in $\mathbb{T}$.

First we recall Nakamura’s $G$-igsaw transformation using the notation from
Section 2.6. Let be the fan of $G\operatorname{-Hilb}$. For each
$\sigma_{+}\in\Sigma(3)$, there is a $G$-invariant monomial ideal
$I_{+}\subset\mathbb{C}[x,y,z]$ such that the torus-invariant $G$-cluster
$\mathbb{C}[x,y,z]/I_{+}$ is the fibre of the universal family on
$G\operatorname{-Hilb}$ over the origin in the chart $U_{\sigma_{+}}$. The set
$\mathcal{S}_{+}$ of monomials in $\mathbb{C}[x,y,z]\setminus I_{+}$ provides
an eigenbasis for $\mathbb{C}[x,y,z]/I_{+}$, _i. e._ each monomial in
$\mathcal{S}_{+}$ lies in a different character space of the $G$-action.
Following Nakamura [Nak01], we obtain the _$G$ -graph_ of
$\mathbb{C}[x,y,z]\setminus I_{+}$ by introducing directed edges between
monomials in $\mathcal{S}_{+}$ to encode the $\mathbb{C}[x,y,z]$-module
structure on $\mathbb{C}[x,y,z]/I_{+}$; in particular, each directed edge in
the $G$-graph is naturally labelled by a variable $x$, $y$ or $z$. To describe
the $G$-igsaw transformations of this monomial $G$-cluster, let
$\tau\in\Sigma(2)$ be any interior line segment satisfying
$\tau=\sigma_{+}\cap\sigma_{-}$ for some $\sigma_{-}\in\Sigma(3)$ and let
$m=(m_{1},m_{2},m_{3})\in M$ be the primitive vector in the normal direction
to the hyperplane spanned by $\tau$ such that $\langle m,n\rangle\geq 0$ for
all $n\in\sigma_{+}$.

######  Proposition 3.8.

 _(Nakamura_ [Nak01, Lemma 2.8]_)._ Let $S_{-}$ be the $G$-graph of the
torus-invariant $G$-cluster obtained as the fibre of the universal family on
$G\operatorname{-Hilb}$ over the origin in $U_{\sigma_{-}}$. For each monomial
$x^{a}y^{b}z^{c}\in\mathcal{S}_{+}$, the unique element of $\mathcal{S}_{-}$
in the same character space is obtained by multiplying $x^{a}y^{b}z^{c}$ by
the highest power of the $G$-invariant Laurent monomial
$x^{m_{1}}y^{m_{2}}z^{m_{3}}$ such that the product lies in
$\mathbb{C}[x,y,z]$, _i. e._

$\mathcal{S}_{-}=\Big{\\{}x^{a}y^{b}z^{c}(x^{m_{1}}y^{m_{2}}z^{m_{3}})^{d(a,b,c)}\in\mathbb{C}[x,y,z]\mid
x^{a}y^{b}z^{c}\in\mathcal{S}_{+}\Big{\\}}$

for $d(a,b,c):=\max\\{d\in\mathbb{N}\mid a+dm_{1}\geq 0,b+dm_{2}\geq
0,c+dm_{3}\geq 0\\}$.

######  Remarks 3.9.

  1. (1)

At least one of $m_{1},m_{2},m_{3}\in\mathbb{Z}$ is negative because $\tau$ is
an interior line segment, so $d(a,b,c)$ is well-defined for each
$x^{a}y^{b}z^{c}\in\mathcal{S}_{+}$.

  2. (2)

Since $G\subset\operatorname{SL}(3,\mathbb{C})$, the result of [Nak01, Lemma
2.8] is correct; compare [CMT07, Remark 4.13].

  3. (3)

Worked examples of jigsaw transformations illustrating Proposition 3.8 appear
in [Nak01, Section 5].

We now work towards the generalisation of this statement. Let
$\theta\in\Theta$ be generic and write θ for the toric fan of
$\mathcal{M}_{\theta}$. Let $\sigma_{\pm}\in{}_{\theta}(3)$ be adjacent cones
in θ, with $\tau=\sigma_{+}\cap\sigma_{-}\in{}_{\theta}(2)$. Let
$\rho_{0},\rho_{1},\rho_{2}$ and $\rho_{1},\rho_{2},\rho_{3}$ denote the rays
in $\sigma_{+}$ and $\sigma_{-}$ respectively as shown in Figure 6, and for
$0\leq i\leq 3$ we write i for the perfect matching associated to the ray
$\rho_{i}$ as in (3.1).

$\tau$$\tau$$\sigma_{+}$$\sigma_{-}$$\rho_{0}$$\rho_{3}$$\rho_{1}$$\rho_{2}$
Figure 6. Cones $\sigma_{+}$ and $\sigma_{-}$ in the triangulation .

######  Definition 3.10.

Let $\tau\in{}_{\theta}(2)$. A _jigsaw piece_ for $\tau$ is the closure of any
connected component of the locus
$\mathbb{T}\setminus\bigcupop\displaylimits_{0\leq i\leq 3}{}_{i}$.

Each jigsaw piece for $\tau$ is the union of a collection of tiles of . Tiles
can be lifted from to $\widetilde{\Gamma}$, and we draw the jigsaw pieces in
the universal cover. The terminology is chosen to suggest that one can move
jigsaw pieces around in $\widetilde{\Gamma}$ by deliberately choosing
different lifts.

######  Lemma 3.11.

Let $\sigma_{\pm}\in{}_{\theta}(3)$ satisfy $\tau=\sigma_{+}\cap\sigma_{-}$ as
in Figure 6 above, and regard

(3.3) $\mathfrak{c}_{-}:=\mathfrak{m}_{1,3}\cap\mathfrak{m}_{2,3}$

as a subset of the edges in the boundary of $\operatorname{Hex}(\sigma_{-})$.
The closure of any connected component of
$\operatorname{Hex}(\sigma_{+})\setminus\mathfrak{c}_{-}$ is a jigsaw piece of
$\tau$, and every jigsaw piece of $\tau$ arises in this way. In particular, if
we cut $\operatorname{Hex}(\sigma_{+})$ along the edges of $\mathfrak{c}_{-}$
then we obtain precisely the jigsaw pieces of $\tau$.

###### Proof.

The rays $\rho_{1},\rho_{2}$ lie in $\sigma_{+}\cap\sigma_{-}$, so the
fundamental hexagons $\operatorname{Hex}(\sigma_{\pm})$ share four of their
six boundary sides, namely those along the lifts of $\mathfrak{m}_{1,2}$ in
Figure 4. We also know $\mathfrak{m}_{0,1}\cap\mathfrak{m}_{0,2}$ traverses
the other boundary sides of $\operatorname{Hex}(\sigma_{+})$, and cutting
$\operatorname{Hex}(\sigma_{+})$ along the set $\mathfrak{c}_{-}$ requires
that we cut along the edges of $\mathfrak{m}_{1,3}\cap\mathfrak{m}_{2,3}$. All
together then, the edges that cut out the connected components of
$\operatorname{Hex}(\sigma_{+})\setminus\mathfrak{c}_{-}$ are those in the set

(3.4)
$\mathfrak{C}:=\mathfrak{m}_{1,2}\cup(\mathfrak{m}_{0,1}\cap\mathfrak{m}_{0,2})\cup(\mathfrak{m}_{1,3}\cap\mathfrak{m}_{2,3})$

that forms the union of all edges in the boundaries of
$\operatorname{Hex}(\sigma)$ and $\operatorname{Hex}(\sigma_{-})$. Regarding
$\operatorname{Graph}(\sigma_{\pm})$ as subsets of $\mathbb{T}$, we have that

$\pi(\mathfrak{C})=\pi\big{(}\partial\operatorname{Hex}(\sigma_{+})\big{)}\cup\pi\big{(}\partial\operatorname{Hex}(\sigma_{-})\big{)}=\operatorname{Graph}(\sigma_{+})\cup\operatorname{Graph}(\sigma_{-}).$

Proposition 3.4 implies that this subset of $\mathbb{T}$ is the unique
connected component of the locus $\bigcupop\displaylimits_{0\leq i\leq
3}{}_{i}$ comprising more than a single edge. Since each jigsaw piece for
$\tau$ is the closure of a union of tiles in , we may ignore isolated edges of
$\bigcupop\displaylimits_{0\leq i\leq 3}{}_{i}$ when computing the jigsaw
pieces. Thus, jigsaw pieces for $\tau$ are precisely the (images under $\pi$
of the) closures of the connected components of
$\operatorname{Hex}(\sigma_{+})\setminus\mathfrak{c}_{-}$. ∎

######  Theorem 3.12 (Generalised jigsaw transformation).

Let $M_{\sigma_{+}}$ be the torus-invariant $\theta$-stable $A$-module
associated to $\sigma_{+}\in{}_{\theta}(3)$. By cutting the fundamental
hexagon $\operatorname{Hex}(\sigma_{+})$ in $\mathbb{R}^{2}$ along the edges
of $\mathfrak{c}_{-}$ and rearranging the resulting jigsaw pieces (i. e.
translating each piece by a carefully chosen element of $\mathbb{Z}^{2}$), one
obtains $\operatorname{Hex}(\sigma_{-})$, thereby determining the
$\theta$-stable $A$-module $M_{\sigma_{-}}$ for the adjacent cone
$\sigma_{-}\in{}_{\theta}(3)$.

###### Proof.

By lifting the result of Lemma 3.11 to the universal cover, we may cut the
interior of $\operatorname{Hex}(\sigma_{+})$ in $\mathbb{R}^{2}$ along
$\mathfrak{c}_{-}$ in order to decompose $\operatorname{Hex}(\sigma_{+})$ into
the union of all jigsaw pieces for $\tau$, each a particular lift of the
corresponding jigsaw piece in $\mathbb{T}^{2}$. By symmetry, a similar
statement holds for the cut of $\operatorname{Hex}(\sigma_{-})$ along the
chain of edges $\mathfrak{c}_{+}:=\mathfrak{m}_{0,1}\cap\mathfrak{m}_{0,2}$ in
the boundary of $\operatorname{Hex}(\sigma_{+})$. The result follows since any
two lifts of a jigsaw piece differ only in translation by an element of
$\mathbb{Z}^{2}$. ∎

To see that Nakamura’s $G$-igsaw transformations from Proposition 3.8 can be
recovered as a special case of Theorem 3.12, we adopt the notation from
Section 2.6. In particular, for the stability parameter $\theta$ satisfying
$\theta_{i}>0$ for each $i\neq 0$, the fine moduli space
$\mathcal{M}_{\theta}$ is isomorphic to $G\operatorname{-Hilb}$.

######  Corollary 3.13.

Let $G\subset\operatorname{SL}(3,\mathbb{C})$ be a finite abelian subgroup.
Nakamura’s $G$-igsaw transformations from Proposition 3.8 are a special case
of those from Theorem 3.12.

###### Proof.

Let be the fan of $G\operatorname{-Hilb}$ and $\sigma_{\pm}\in\Sigma(3)$
adjacent cones satisfying $\tau=\sigma_{+}\cap\sigma_{-}$. The isomorphism
$\mathcal{M}_{\theta}\cong G\operatorname{-Hilb}$ identifies the torus-
invariant $\theta$-stable $A$-modules $M_{\sigma_{\pm}}$ with the torus-
invariant $G$-clusters $\mathbb{C}[x,y,z]/I_{\pm}$ obtained as the fibres of
the universal family on $G\operatorname{-Hilb}$ over the origins in
$U_{\sigma_{\pm}}$. Under this identification, the quivers $Q^{\sigma_{\pm}}$
correspond to the $G$-graphs $\mathcal{S}_{\pm}$ of
$\mathbb{C}[x,y,z]/I_{\pm}$. Theorem 3.12 cuts the quiver $Q^{\sigma_{+}}$
(equivalently, the $A$-module $M_{\sigma_{+}}$) into pieces and rearranges
them to produce $Q^{\sigma_{-}}$ (equivalently $M_{\sigma_{-}}$), just as
Nakamura’s $G$-igsaw transformation cuts $\mathcal{S}_{+}$ (equivalently
$\mathbb{C}[x,y,z]/I_{+}$) into pieces and rearranges them to produce
$\mathcal{S}_{-}$ (equivalently $\mathbb{C}[x,y,z]/I_{-}$) as in Proposition
3.8. ∎

######  Remark 3.14.

Each edge of the $G$-graphs $\mathcal{S}_{\pm}$ is labelled with one of the
variables $x,y,z$ according to the $\mathbb{C}[x,y,z]$-module structure of
$\mathbb{C}[x,y,z]/I_{\pm}$. Contrast this with our labelling of arrows in
$Q^{\sigma_{\pm}}$ with monomials $t^{\operatorname{div}(a)}$ in the Cox ring
$\mathbb{C}[t_{\rho}\mid\rho\in\Sigma(1)]$ following (2.9). To compare these
labels, identify $x,y,z$ with the variables in the Cox ring indexed by the
lattice points $\rho\in\Sigma(1)$ defined by corners of the junior simplex
$\Delta(\Gamma)$. If we substitute $t_{\rho}=1$ in $t^{\operatorname{div}(a)}$
for each lattice point that is not a corner of the polygon $\Delta(\Gamma)$,
then we obtain a single variable $x,y$ or $z$. Indeed, labelling each arrow by
$t^{\operatorname{div}(a)}$ encodes the tautological isomorphism $\phi\colon
A\to\operatorname{End}(T)$, whereas labelling each edge in a $G$-graph by a
variable $x,y,z$ encodes the description of $A$ as the endomorphism algebra of
the direct sum of the maximal Cohen–Macaulay modules on $\mathbb{C}^{3}/G$.
The process of substituting $t_{\rho}=1$ for all non-corner lattice points
corresponds to pushing forward each tautological line bundle on
$G\operatorname{-Hilb}$ to the corresponding maximal Cohen–Macaulay module on
$\mathbb{C}^{3}/G$.

######  Example 3.15.

Continuing Examples 2.1 and 3.3, consider the cones $\sigma_{+}$ and
$\sigma_{-}$ in Figure 2 generated by the rays
$\\{\rho_{7},\rho_{8},\rho_{9}\\}$ and $\\{\rho_{8},\rho_{9},\rho_{10}\\}$
respectively. Figure 7 illustrates several copies of the fundamental hexagons
$\operatorname{Hex}(\sigma_{\pm})$, where edges in
$\bigcupop\displaylimits_{7\leq i\leq 10}{}_{i}$ are coloured as follows:
edges in $\mathfrak{m}_{1,2}$, $\mathfrak{c}_{+}$ and $\mathfrak{c}_{-}$ are
blue, green and red respectively.

1123335920202324242628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

1123335920202324242628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

1123335920202324242628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

1123335920202324242628$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$

Figure 7. The edges of $\bigcupop\displaylimits_{7\leq i\leq 10}{}_{i}$ shown
in the universal cover of $\mathbb{T}$.

The boundary of $\operatorname{Hex}(\sigma_{-})$ comprises the edges in red
and blue (compare Figure 3); notice that the green edges of $\mathfrak{c}_{+}$
split this hexagon into two jigsaw pieces. If we fix the position of the
jigsaw piece containing the tile dual to the vertex of the quiver labelled
$0$, and translate the second jigsaw piece along the direction of the blue
edges until it sits on the opposite side of the first jigsaw piece, then we
cover precisely $\operatorname{Hex}(\sigma_{+})$ whose boundary edges are blue
and green (with the red edges of $\mathfrak{c}_{-}$ cutting it in two).

For each cone $\sigma_{\pm}\in\Sigma(3)$ from Figure 6, recall the subquiver
$Q^{\sigma_{\pm}}$ of $Q$ that has vertex set $Q_{0}$ and arrow set
$Q^{\sigma_{\pm}}_{1}$ comprising the arrows $a\in Q_{1}$ that support the
corresponding $A$-module $M_{\sigma_{\pm}}$.

######  Corollary 3.16.

Suppose that the head and tail of $a\in Q_{1}$ are dual to tiles in different
jigsaw pieces for $\tau$. If $a\in Q^{\sigma_{+}}_{1}$ then
$\mathfrak{e}_{a}\in{}_{3}$, and similarly, if $a\in Q^{\sigma_{-}}_{1}$ then
$\mathfrak{e}_{a}\in{}_{0}$

###### Proof.

For $a\in Q^{\sigma_{+}}_{1}$, Lemma 3.11 shows that
$\mathfrak{e}_{a}\in\mathfrak{c}_{-}=\mathfrak{m}_{1,3}\cap\mathfrak{m}_{2,3}$,
so either $\mathfrak{e}_{a}\in{}_{3}$ or
$\mathfrak{e}_{a}\in{}_{1}\cap{}_{2}$. However,
$\mathfrak{e}_{a}\notin\bigcupop\displaylimits_{i=0}^{2}{}_{i}$ because $a\in
Q^{\sigma_{+}}_{1}$, giving $\mathfrak{e}_{a}\in{}_{3}$ as required. The case
$a\in Q^{\sigma_{-}}_{1}$ is similar. ∎

Let $\operatorname{Hex}(\sigma_{+})^{\circ}$ and
$\partial\operatorname{Hex}(\sigma_{+})$ denote the interior and the boundary
of $\operatorname{Hex}(\sigma_{+})$ respectively.

######  Lemma 3.17.

The cuts $\mathfrak{c}_{-}$ of $\operatorname{Hex}(\sigma_{+})$ and
$\mathfrak{c}_{+}$ of $\operatorname{Hex}(\sigma_{-})$ have the following
properties.

  1. $(\mathrm{i})$

Neither $\mathfrak{c}_{-}$ nor $\mathfrak{c}_{+}$ intersects itself;

  2. $(\mathrm{ii})$

If an edge lies in either
$\mathfrak{c}_{-}\cap\partial\operatorname{Hex}(\sigma_{+})$ or
$\mathfrak{c}_{+}\cap\partial\operatorname{Hex}(\sigma_{-})$, then this edge
lies in $\mathfrak{c}_{-}\cap\mathfrak{c}_{+}$, and the connected component of
$\mathfrak{c}_{-}\cap\mathfrak{c}_{+}$ containing this edge comprises an odd
number of edges;

  3. $(\mathrm{iii})$

If we follow $\mathfrak{c}_{-}$ $($resp. $\mathfrak{c}_{+})$ in either
direction around the torus, then the list of nodes where we depart from and
arrive at $\partial\operatorname{Hex}(\sigma_{+})$ _(_ resp.
$\partial\operatorname{Hex}(\sigma_{-}))$ have colours that alternate between
black and white.

  4. $(\mathrm{iv})$

The set of nodes that are simultaneously an endnode of an edge in
$\mathfrak{c}_{-}$ $($resp. $\mathfrak{c}_{+})$ and an endnode of an edge in
$\mathfrak{m}_{1,2}$ comprises one black node and one white node; these are
the trivalent nodes of $\operatorname{Graph}(\sigma_{-})$ $($resp.
$\operatorname{Graph}(\sigma_{+}))$.

###### Proof.

We prove each statement for $\mathfrak{c}_{-}$ and $\sigma_{+}$; swap signs
throughout to obtain the proof for $\mathfrak{c}_{+}$ and $\sigma_{-}$. For
$(\mathrm{i})$, if $\mathfrak{c}_{-}$ intersects itself at a node
$\mathfrak{n}\in{}_{0}$, then at least three edges of $\mathfrak{c}_{-}$ touch
$\mathfrak n$. Since edges in $\mathfrak{c}_{-}$ belong alternately to 3 and
${}_{1}\cap{}_{2}$, it follows that two of the three edges in
$\mathfrak{c}_{-}$ touching $\mathfrak n$ belong to the same perfect matching,
a contradiction. For $(\mathrm{ii})$, let $\mathfrak{e}$ be an edge of
$\mathfrak{c}_{-}\cap\partial\operatorname{Hex}(\sigma_{+})$. Edges of
$\mathfrak{c}_{-}$ belong alternately to 3 and ${}_{1}\cap{}_{2}$, and Figure
4 illustrates that $\mathfrak{e}$ is forced to lie in
$\mathfrak{c}_{+}\subset\partial\operatorname{Hex}(\sigma_{+})$. Moreover,
since edges of $\mathfrak{c}_{+}$ belong alternately to 0 and
${}_{1}\cap{}_{2}$, the connected component of
$\mathfrak{c}_{-}\cap\mathfrak{c}_{+}$ containing $\mathfrak{e}$ must begin
and end with an edge in ${}_{1}\cap{}_{2}$, so this component comprises an odd
number of edges. For $(\mathrm{iii})$, we deduce from $(\mathrm{ii})$ that the
endnodes of each connected component of $\mathfrak{c}_{-}\cap\mathfrak{c}_{+}$
must be of different colours. It suffices therefore to prove that each
connected component of
$\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$ also contains an
odd number of edges, as again the endnodes will be of different colours. Every
node in $\partial\operatorname{Hex}(\sigma_{+})$ is an endnode of an edge in
$\partial\operatorname{Hex}(\sigma_{+})$ that lies in either 1 or 2 (or both);
see Figure 4. Since the edges of $\mathfrak{c}_{-}$ alternate between edges of
3 and ${}_{1}\cap{}_{2}$, an edge in
$\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$ that has an
endnode in $\partial\operatorname{Hex}(\sigma_{+})$ must therefore lie in 3.
It follows that each connected component of
$\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$ contains an odd
number of edges. Statement $(\mathrm{iv})$ follows by combining statement
$(\mathrm{iii})$ with an examination of the endnodes of $\mathfrak{c}_{-}$. ∎

$\mathfrak{m}_{1,2}$$\mathfrak{m}_{1,2}$$\mathfrak{m}_{1,2}$$\mathfrak{m}_{1,2}$
Figure 8. $\operatorname{Hex}(\sigma_{+})$ and one of its translates are both
pictured with the cut $\mathfrak{c}_{-}$ (in dashed green); we highlight the
edges of $\mathfrak{c}_{+}$ (in red) to illustrate the statements of Lemma
3.17.

## 4\. Combinatorial Reid’s recipe

### 4.1. The $0$-generated stability condition

Our normalisation of the tautological bundle $T$ is chosen so that
$L_{0}\cong\mathcal{O}_{\mathcal{M}_{\theta}}$ for the vertex $0\in Q_{0}$; we
refer to the tile of dual to $0$ as the _zero tile_. Choose a lift of $0\in
Q_{0}$ to the universal cover that we also denote $0\in\widetilde{Q}$; again,
we refer to the tile of $\widetilde{\Gamma}$ dual to $0$ as the _zero tile_.
For each cone $\sigma\in{}_{\theta}(3)$, choose the lift
$\widetilde{Q^{\sigma}}$ of the subquiver $Q^{\sigma}\subset Q$ so that
$0\in\widetilde{Q^{\sigma}}$. In particular, the zero tile in
$\widetilde{\Gamma}$ is contained in the fundamental hexagon
$\operatorname{Hex}(\sigma)$ for every $\sigma\in{}_{\theta}(3)$.

Fix once and for all a _0-generated_ stability parameter, _i. e._ a parameter
of the form

(4.1) $\theta=(\theta_{i})\in\Theta\text{ such that }\theta_{i}>0\text{ for
all }i\neq 0.$

It is well known that $0$-generated stability conditions are generic.
Furthermore, we have the following:

######  Lemma 4.1.

A quiver representation $(v_{a})\in\mathbb{C}^{Q_{1}}$ is $\theta$-stable if
and only if for every vertex $i\in Q_{0}\setminus\\{0\\}$, there is a path
$p=a_{\ell}\cdots a_{1}$ in $Q$ from $0$ to $i$ such that
$\prodop\displaylimits_{1\leq j\leq\ell}v_{a_{j}}\neq 0$.

###### Proof.

See, for example, [BCQ15, Lemma 3.1(ii)]. ∎

We now study jigsaw transformations of torus-invariant $\theta$-stable
$A$-modules for a 0-generated stability parameter $\theta\in\Theta$. As in
Figure 6, choose adjacent three-dimensional cones
$\sigma_{\pm}\in{}_{\theta}(3)$ and let $\tau:=\sigma_{+}\cap\sigma_{-}$ be
the common face of dimension two. As above, the subquivers $Q^{\sigma_{\pm}}$
of $Q$ from $\mathbb{T}$ lift to define subquivers
$\widetilde{Q^{\sigma_{\pm}}}$ of $\widetilde{Q}$ in $\mathbb{R}^{2}$ such
that $\operatorname{Hex}(\sigma_{\pm})$ both contain the zero tile.

######  Definition 4.2.

The _zero jigsaw piece_ for $\tau$, denoted $\operatorname{J}_{0}$, is the
jigsaw piece in $\widetilde{\Gamma}$ containing the zero tile.

Both $\operatorname{Hex}(\sigma_{+})$ and $\operatorname{Hex}(\sigma_{-})$ can
be obtained by gluing jigsaw pieces of $\tau$ along parts of their boundaries,
so the zero jigsaw piece $\operatorname{J}_{0}$ lies in
$\operatorname{Hex}(\sigma_{+})\cap\operatorname{Hex}(\sigma_{-})$. The next
result records an important consequence of Lemma 4.1 for certain edges in the
boundary of $\operatorname{J}_{0}$. Recall that we write
$\operatorname{Hex}(\sigma_{+})^{\circ}$ and
$\partial\operatorname{Hex}(\sigma_{+})$ for the interior and the boundary
respectively of $\operatorname{Hex}(\sigma_{+})$.

######  Lemma 4.3.

Let $\mathfrak{e}_{a}\in{}_{1}$ be an edge in the boundary of
$\operatorname{J}_{0}$ that lies in the interior of
$\operatorname{Hex}(\sigma_{+})$. The dual arrow $a\in Q_{1}^{\sigma_{+}}$ is
such that $\operatorname{\mathsf{t}}(a)$ is dual to a tile in
$\operatorname{J}_{0}$ and $\operatorname{\mathsf{h}}(a)$ is dual to a tile
that does not lie in $\operatorname{J}_{0}$.

###### Proof.

Every edge in the boundary of $\operatorname{J}_{0}$ is contained in
$\partial\operatorname{Hex}(\sigma_{+})$ or
$\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$ by Lemma 3.11.
The boundary of $\operatorname{J}_{0}$ is not contained in
$\partial\operatorname{Hex}(\sigma_{+})$, otherwise there would only be one
jigsaw piece; nor is it contained in
$\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$, because
$\mathfrak{c}_{-}$ does not intersect itself by Lemma 3.17$(\mathrm{i})$.
Therefore, the boundary of $\operatorname{J}_{0}$ comprises edges from both
$\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$ and
$\partial\operatorname{Hex}(\sigma_{+})$. Let $\gamma$ be any connected
component of $\mathfrak{c}_{-}\cap\operatorname{Hex}(\sigma_{+})^{\circ}$ that
lies in the boundary of $\operatorname{J}_{0}$. Note that $\gamma$ cuts
$\operatorname{Hex}(\sigma_{+})$ into two connected components, say $C_{0}$
and $C_{1}$; assume without loss of generality that the interior of
$\operatorname{J}_{0}$ is contained in $C_{0}$. Suppose for a contradiction
that the arrow $a$ dual to an edge $\mathfrak{e}_{a}$ in the path $\gamma$ is
such that $\operatorname{\mathsf{t}}(a)$ is dual to a tile in $C_{1}$ and
$\operatorname{\mathsf{h}}(a)$ is dual to a tile in $\operatorname{J}_{0}$.
Since edges of $\gamma$ belong alternately to 3 and ${}_{1}\cap{}_{2}$, every
arrow of $Q^{\sigma_{+}}$ dual to an edge in $\gamma$ also has tail dual to a
tile in $C_{1}$ and head dual to a tile in $\operatorname{J}_{0}$. But then
for any vertex $i$ dual to a tile in $C_{1}$, there does not exist a path $p$
in $Q^{\sigma_{+}}$ from $0\in\operatorname{J}_{0}$ to $i\in C_{1}$; this
contradicts Lemma 4.1. ∎

The statement of Lemma 4.3 also holds if we replace $+$ by $-$ throughout. We
now construct an $A$-submodule $N(\tau)_{+}$ of the torus-invariant $A$-module
$M_{\sigma_{+}}$.

######  Definition 4.4.

Let $Q(\tau)^{+}$ denote the subquiver of $Q^{\sigma_{+}}$ with vertex set and
arrow set satisfying

$\displaystyle Q(\tau)_{0}^{+}$ $\displaystyle:=$ $\displaystyle\big{\\{}i\in
Q_{0}\mid\text{the tile dual to }i\text{ does not lie in
}\operatorname{J}_{0}\big{\\}}$ $\displaystyle Q(\tau)_{1}^{+}$
$\displaystyle:=$ $\displaystyle\big{\\{}a\in
Q_{1}^{\sigma_{+}}\mid\operatorname{\mathsf{t}}(a),\operatorname{\mathsf{h}}(a)\text{
are dual to tiles that do not lie in }\operatorname{J}_{0}\big{\\}}.$

Consider the dimension vector $\underline{d}:=(d_{i})\in\mathbb{N}^{Q_{0}}$
with $d_{i}=1$ for $i\in Q(\tau)_{0}^{+}$ and $d_{i}=0$ otherwise, and define
a representation of $Q$ of dimension vector $\underline{d}$ by associating to
each arrow $a\in Q_{1}$ the scalar

$v_{a}=\left\\{\begin{array}[]{cr}1&\text{if }a\in Q(\tau)_{1}^{+};\\\
0&\text{otherwise.}\end{array}\right.$

Let $N(\tau)_{+}$ denote the corresponding $\mathbb{C}Q$-module of dimension
vector $\underline{d}$.

######  Proposition 4.5.

The $\mathbb{C}Q$-module $N(\tau)_{+}$ is a proper, nonzero $A$-submodule of
$M_{\sigma_{+}}$.

###### Proof.

The $\mathbb{C}Q$-module $M_{\sigma_{+}}$ is an $A$-module because it
satisfies the relations (2.1). If we prove that $N(\tau)_{+}$ is a
$\mathbb{C}Q$-submodule of $M_{\sigma_{+}}$, then it will be an $A$-submodule
because it will satisfy the same relations. For this, it suffices to show
every arrow $a\in Q^{\sigma_{+}}_{1}$ with $\operatorname{\mathsf{t}}(a)\in
Q(\tau)_{0}^{+}$ also has $\operatorname{\mathsf{h}}(a)\in Q(\tau)_{0}^{+}$.
If we suppose otherwise, then there exists $a\in Q_{1}^{\sigma_{+}}$ with
$\operatorname{\mathsf{t}}(a)\in Q(\tau)_{0}^{+}$ and
$\operatorname{\mathsf{h}}(a)\notin Q(\tau)_{0}^{+}$. The dual edge
$\mathfrak{e}_{a}\in{}_{1}$ lies in the boundary of $\operatorname{J}_{0}$ and
in the interior of $\operatorname{Hex}(\sigma_{+})$, so Lemma 4.3 gives a
contradiction. It follows that $N(\tau)_{+}$ is an $A$-submodule of
$M_{\sigma_{+}}$. Since $\operatorname{Hex}(\sigma_{+})$ and
$\operatorname{Hex}(\sigma_{-})$ are distinct, there are at least two jigsaw
pieces for $\tau$, so $N(\tau)_{+}$ is nonzero. It’s a proper submodule of
$M_{\sigma_{+}}$ because $\operatorname{J}_{0}$ contains the zero tile. ∎

######  Remark 4.6.

If we replace + by $-$ throughout Definition 4.4 and Proposition 4.5, then we
obtain a proper nonzero submodule $N(\tau)_{-}$ of $M_{\sigma_{-}}$. The
$A$-modules $N(\tau)_{+}$ and $N(\tau)_{-}$ are isomorphic if and only if
there are precisely two jigsaw pieces.

### 4.2. Tautological bundles of degree one on the curve

Recall that for each vertex $i\in Q_{0}$, the tautological line bundle $L_{i}$
on $\mathcal{M}_{\theta}$ is of the form
$L_{i}\cong\mathcal{O}_{\mathcal{M}_{\theta}}(\operatorname{div}(p))$ where
$p$ is any path in the quiver $Q$ from vertex $0$ to vertex $i$ and where
$\operatorname{div}(p)$ is the divisor of zeroes of the section
$t^{\operatorname{div}(p)}=\prodop\displaylimits_{a\in\operatorname{supp}(p)}t^{\operatorname{div}(a)}$
determined by the sections (2.9) that label arrows $a$ in the path $p$.

######  Lemma 4.7.

For $i\in Q_{0}$, let $p$ be a path in $Q^{\sigma_{+}}$ from vertex 0 to
vertex $i$. Then $t_{\rho_{3}}$ appears with multiplicity one in
$t^{\operatorname{div}(p)}$ if and only if vertex $i$ is dual to a tile in a
jigsaw piece that lies adjacent to $\operatorname{J}_{0}$ in
$\operatorname{Hex}(\sigma_{+})$.

###### Proof.

We claim that the multiplicity of $t_{\rho_{3}}$ in
$t^{\operatorname{div}(p)}$ is equal to the number of arrows $a$ in the
support of the path $p$ such that the dual edge satisfies
$\mathfrak{e}_{a}\in\mathfrak{c}_{-}$. To prove the claim, let $a\in
Q^{\sigma_{+}}_{1}$ and consider two cases:

  1. $(\mathrm{i})$

if $\mathfrak{e}_{a}\in\mathfrak{c}_{-}$, then Corollary 3.16 gives
$\mathfrak{e}_{a}\in{}_{3}$, in which case equation (2.9) shows that
$t_{\rho_{3}}$ appears with multiplicity one in $t^{\operatorname{div}(a)}$;

  2. $(\mathrm{ii})$

if $\mathfrak{e}_{a}\notin\mathfrak{c}_{-}$, then we must show that
$t_{\rho_{3}}$ does not divide $t^{\operatorname{div}(a)}$ or, equivalently,
that $\mathfrak{e}_{a}\notin{}_{3}$. Since $a\in Q^{\sigma_{+}}_{1}$, the edge
$\mathfrak{e}_{a}$ lies in neither 1 nor 2 (nor $\mathfrak{c}_{-}$), so
$\mathfrak{e}_{a}$ does not lie in the boundary of
$\operatorname{Hex}(\sigma_{-})$. The proof of Proposition 3.4 applied to
$\operatorname{Hex}(\sigma_{-})$ shows that
$\mathfrak{e}_{a}\notin\bigcupop\displaylimits_{1\leq i\leq 3}{}_{\rho_{i}}$
as required.

This proves the claim. Since the edges of $\mathfrak{c}_{-}$ provide the cuts
that separate jigsaw pieces in $\operatorname{Hex}(\sigma_{+})$, and since $p$
has tail at vertex $0$, it follows that the number of arrows $a$ in the
support of $p$ satisfying $\mathfrak{e}_{a}\in\mathfrak{c}_{-}$ is equal to
one if and only if the head of $p$ lies in a jigsaw piece that lies adjacent
to $\operatorname{J}_{0}$ in $\operatorname{Hex}(\sigma_{+})$. ∎

######  Remark 4.8.

Similarly, if $p$ is a path in $Q^{\sigma_{-}}$ from vertex 0 to vertex $i$,
then $t_{\rho_{0}}$ appears with multiplicity one in
$t^{\operatorname{div}(p)}$ iff vertex $i$ is dual to a tile in a jigsaw piece
that lies adjacent to $\operatorname{J}_{0}$ in
$\operatorname{Hex}(\sigma_{-})$.

######  Proposition 4.9.

Let $C_{\tau}\subseteq\mathcal{M}_{\theta}$ be the rational curve determined
by the cone $\tau=\sigma_{+}\cap\sigma_{-}\subseteq{}_{\theta}$. A tile of
lies in a jigsaw piece adjacent to $\operatorname{J}_{0}$ in
$\operatorname{Hex}(\sigma_{+})$ iff the vertex $i\in Q_{0}$ dual to the tile
satisfies $\deg(L_{i}|_{C_{\tau}})=1$.

###### Proof.

We first compute the coordinate function on the toric chart in
$\mathcal{M}_{\theta}$ corresponding to $\tau$. Let $m\in M$ be the primitive
vector that is perpendicular to $\tau$ such that $\langle m,n\rangle\geq 0$
for all $n\in\sigma_{+}$. Since $\rho_{1},\rho_{2}\subset\tau$, we have
$\langle m,v_{\rho_{1}}\rangle=\langle m,v_{\rho_{2}}\rangle=0$, where
$v_{\rho_{i}}\in N=\operatorname{Hom}(M,\mathbb{Z})$ is the primitive lattice
point on the ray $\rho_{i}\in{}_{\theta}(1)$. Our choice of $m$ gives $\langle
m,v_{\rho_{0}}\rangle>0$ and $\langle m,v_{\rho_{3}}\rangle<0$, see Figure 6.
Since $m$ is a primitive generator and both cones $\sigma_{\pm}$ are basic, we
have $\langle m,v_{\rho_{0}}\rangle=1$ and $\langle m,v_{\rho_{3}}\rangle=-1$.
Thus, if we use the natural inclusion
$M\hookrightarrow\mathbb{Z}^{{}_{\theta}(1)}$ to identify $\mathbb{C}[M]$ with
a subring of the ring of Laurent monomials $\mathbb{C}[t_{\rho}^{\pm
1}\mid\rho\in{}_{\theta}(1)]$ in the variables of the Cox ring of
$\mathcal{M}_{\theta}$, then

(4.2) $t^{m}=\frac{t_{\rho_{0}}}{t_{\rho_{3}}}\cdot t^{\ell}$

where $\langle\ell,v_{\rho_{i}}\rangle=0$ for $0\leq i\leq 3$, _i. e._
$t^{\ell}$ is independent of
$t_{\rho_{0}},t_{\rho_{1}},t_{\rho_{2}},t_{\rho_{3}}$.

Let $i\in Q_{0}$ be any vertex, let $p_{\pm}$ be any path in
$Q^{\sigma_{\pm}}$ from 0 to $i$ and write
$U_{\sigma_{\pm}}:=\operatorname{Spec}\mathbb{C}[\sigma_{\pm}^{\vee}\cap M]$
for the toric chart in $\mathcal{M}_{\theta}$. Given the section
$t^{\operatorname{div}(p_{+})}$ that generates $H^{0}(U_{\sigma_{+}},L_{i})$
as a $\mathbb{C}[\sigma_{+}^{\vee}\cap M]$-module, we obtain the generating
section $t^{\operatorname{div}(p_{-})}\in H^{0}(U_{\sigma_{-}},L_{i})$
directly using the transition function from $U_{\sigma_{+}}$ to
$U_{\sigma_{-}}$. Explicitly, $t^{\operatorname{div}(p_{-})}$ is obtained by
multiplying $t^{\operatorname{div}(p_{+})}$ by the highest power of the above
toric coordinate function $t^{m}$ such that the resulting Laurent monomial
lies in $\mathbb{C}[\sigma_{-}^{\vee}\cap M]$, _i. e._

(4.3) $t^{\operatorname{div}(p_{-})}=(t^{m})^{d}\cdot
t^{\operatorname{div}(p_{+})};$

here $d=\deg(L_{i}|_{C_{\tau}})$ is the greatest positive integer such that
$(t^{m})^{d}\cdot
t^{\operatorname{div}(p_{+})}\in\mathbb{C}[\sigma_{-}^{\vee}\cap M]$. Since
$p_{-}$ is a path in the quiver $Q^{\sigma_{-}}$ and since any arrow $a\in
Q_{1}$ satisfying $\mathfrak{e}_{a}\in{}_{3}$ does not appear in
$Q^{\sigma_{-}}$ by construction, it follows that the left hand side of (4.3)
is independent of $t_{\rho_{3}}$. The Laurent monomial $t^{\ell}$ is also
independent of $t_{\rho_{3}}$, so Equation (4.2) implies that the multiplicity
of the variable $t_{\rho_{3}}$ in $t^{\operatorname{div}(p_{+})}$ must be
$d=\deg(L_{i}|_{C_{\tau}})$. The result follows immediately from Lemma 4.7. ∎

######  Remark 4.10.

The statement of Proposition 4.9 holds with $-$ replacing $+$ because the
analogue of Lemma 4.7 holds in this case, see Remark 4.8. In particular, the
question of whether a tile lies in a jigsaw piece adjacent to
$\operatorname{J}_{0}$ is independent of whether one views the jigsaw piece in
$\operatorname{Hex}(\sigma_{+})$ or in $\operatorname{Hex}(\sigma_{-})$. We
may therefore refer to a jigsaw piece as being _adjacent to
$\operatorname{J}_{0}$_ without further qualification.

######  Corollary 4.11.

The source vertices of the quivers $Q(\tau)^{+}$ and $Q(\tau)^{-}$ from
Definition 4.4 and Remark 4.6 coincide.

###### Proof.

It is enough to prove that the source vertices of the quivers $Q(\tau)^{\pm}$
are dual to tiles that lie in jigsaw pieces adjacent to
$\operatorname{J}_{0}$, because then the result follows from Remark 4.10. Let
$i\in Q(\tau)_{0}^{+}=Q(\tau)_{0}^{-}$ be any vertex. Lemma 4.1 shows that
there exist paths $p_{+}$ and $p_{-}$ in the quivers $Q^{\sigma_{+}}$ and
$Q^{\sigma_{-}}$ respectively, such that
$\operatorname{\mathsf{t}}(p_{+})=0=\operatorname{\mathsf{t}}(p_{-})$ and
$\operatorname{\mathsf{h}}(p_{+})=i=\operatorname{\mathsf{h}}(p_{-})$. The
tile dual to the vertex $i$ does not lie in $\operatorname{J}_{0}$ by
Definition 4.4, so both $p_{+}$ and $p_{-}$ must each traverse precisely one
arrow $a_{+}$ and $a_{-}$ respectively such that the vertices
$\operatorname{\mathsf{t}}(a_{+}),\operatorname{\mathsf{t}}(a_{-})$ are dual
to tiles in $\operatorname{J}_{0}$ and
$\operatorname{\mathsf{h}}(a_{+}),\operatorname{\mathsf{h}}(a_{-})$ are dual
to tiles in jigsaw pieces that lie adjacent to $\operatorname{J}_{0}$. Thus,
after arriving at the vertices $\operatorname{\mathsf{h}}(a_{+})$ and
$\operatorname{\mathsf{h}}(a_{-})$, the paths $p_{+}$ and $p_{-}$ go on to
traverse paths in $Q(\tau)^{+}$ and $Q(\tau)^{-}$ respectively that end at
vertex $i$. In particular, for every vertex $i\in
Q(\tau)^{+}_{0}=Q(\tau)^{+}_{0}$, there exists a path in $Q(\tau)^{+}$ and a
path in $Q(\tau)^{-}$ that start at a vertex dual to a tile in a jigsaw piece
adjacent to $\operatorname{J}_{0}$ and that end at vertex $i$. Thus, the
source vertices of $Q(\tau)^{+}$ and $Q(\tau)^{-}$ are those dual to tiles in
jigsaw pieces adjacent to $\operatorname{J}_{0}$. ∎

######  Example 4.12.

Suppose that a given cone $\tau=\sigma_{+}\cap\sigma_{-}$ determined three
jigsaw pieces. Figure 9 illustrates (lifts to the universal cover of) the
jigsaw pieces in $\operatorname{Hex}(\sigma_{+})$, where they are denoted
$\operatorname{J}_{0},\operatorname{J}_{1},\operatorname{J}_{2}$, each
separated from the others by lifts of the cut $\mathfrak{c}_{-}$ shown in
green. Figure 9 also illustrates how different lifts of the jigsaw pieces form
$\operatorname{Hex}(\sigma_{-})$, where they are denoted
$\operatorname{J}_{0},\operatorname{J}_{1}^{\prime},\operatorname{J}_{2}^{\prime}$.
Since $\operatorname{J}_{1}$ and $\operatorname{J}_{1}^{\prime}$ are different
lifts of the same jigsaw piece, we see that only one jigsaw piece lies
_adjacent to $\operatorname{J}_{0}$_ in the sense of Remark 4.10.

$\iddots$$\iddots$$\mathfrak{m}_{1,2}$$\mathfrak{m}_{1,2}$$\mathfrak{m}_{1,2}$$\mathfrak{m}_{1,2}$$\operatorname{J}_{0}$$\operatorname{J}_{1}$$\operatorname{J}_{2}$$\operatorname{J}_{1}^{\prime}$$\operatorname{J}_{2}^{\prime}$
Figure 9. Translates of $\operatorname{Hex}(\sigma_{+})$ and several lifts of
the cut $\mathfrak{c}_{-}$ illustrate how $\operatorname{Hex}(\sigma_{-})$ is
formed from jigsaw pieces $\operatorname{J}_{0}$ and the translates
$\operatorname{J}_{1}^{\prime}$ and $\operatorname{J}_{2}^{\prime}$ of
$\operatorname{J}_{1}$ and $\operatorname{J}_{2}$ respectively.

### 4.3. Combinatorial Reid’s recipe

We now introduce Reid’s recipe for a consistent dimer model . We begin by
introducing the recipe for marking interior lattice points of θ with vertices
of $Q$. For $i\in Q_{0}$, write $S_{i}:=\mathbb{C}e_{i}$ for the vertex simple
$A$-module corresponding to the vertex $i$. For any $0$-generated $A$-module
$M$, the module $S_{i}$ is contained in the socle of $M$ if and only if
$S_{i}$ is a submodule of $M$.

######  Definition 4.13 (Marking interior lattice points).

Let $\rho\in{}_{\theta}(1)$ be an interior lattice point. A vertex $i\in
Q_{0}$ _marks_ $\rho$ if $S_{i}$ lies in the socle of the $A$-module
$M_{\sigma}$ for every cone $\sigma\in{}_{\theta}(3)$ satisfying
$\rho\subset\sigma$.

######  Remark 4.14.

The vertex $0$ never marks a lattice point since $\theta$ is a $0$-generated
stability parameter.

######  Lemma 4.15.

Reid’s recipe marks every interior lattice point of θ with at least one
nonzero vertex of $Q$.

###### Proof.

Let $\rho\in{}_{\theta}(1)$ be an interior lattice point. We claim that the
$A$-module $M_{\rho}$ is nilpotent, _i. e._ there exists $n\in\mathbb{N}$ such
that any element $a\in A$ represented by a path of length greater than $n$ in
$Q$ satisfies $a\cdot m=0$ for all $m\in M_{\rho}$. Indeed, any path in $Q$ of
length $n:=|Q_{0}|+1$ contains a cycle $\gamma$, and it suffices to show that
$\gamma\cdot m=0$ for all $m\in M_{\rho}$. Le Bruyn–Procesi [LP90, Theorem 1]
associates to $\gamma$ a polynomial function
$f_{\gamma}\in\mathbb{C}[V]^{T_{B}}$ on the affine variety $X\cong
V\operatorname{\\!/\\!\\!/\\!}_{0}T_{B}$. Since $\rho$ is interior, the point
$y\in\mathcal{M}_{\theta}$ associated to the $A$-module $M_{\rho}$ by Lemma
3.1 satisfies $\tau_{\theta}(y)=x_{0}$, where
$\tau_{\theta}\colon\mathcal{M}_{\theta}\to X$ is the crepant resolution from
(2.6) and $x_{0}\in X$ is the unique torus-invariant point. It follows that
the value of $f_{\gamma}$ on the quiver representation $(v_{a})$ associated to
$M_{\rho}$ by (3.2) is zero. In particular, there exists a nonzero vertex
$i\in Q_{0}$ such that $M_{\rho}$ contains the vertex simple $A$-module
$S_{i}$ as a submodule. This means that $S_{i}$ lies in the socle of
$M_{\rho}$.

Let $\sigma\in{}_{\theta}(3)$ satisfy $\rho\subset\sigma$, in which case the
module structure on $M_{\sigma}$ is obtained from $M_{\rho}$ by setting some
of the scalars $v_{a}$ from (3.2) to zero, _i. e._ $M_{\sigma}$ is a quotient
module of $M_{\rho}$. Since $M_{\sigma}$ is isomorphic to $M_{\rho}$ as a
$\mathbb{C}$-vector space, it follows that having $S_{i}$ in the socle of
$M_{\rho}$ implies that $S_{i}$ is in the socle of $M_{\sigma}$. As a result,
vertex $i$ marks the lattice point $\rho$ by Definition 4.13. ∎

######  Definition 4.16 (Marking interior line segments).

Let $\tau\in{}_{\theta}(2)$ be an interior line segment. A vertex $i\in Q_{0}$
_marks_ $\tau$ if $i$ is one of the common source vertices of the quivers
$Q(\tau)^{\pm}$ from Corollary 4.11.

######  Remarks 4.17.

  1. (1)

Vertex $0$ never marks an interior line segment, because $0$ cannot be a
vertex of $Q(\tau)^{\pm}$.

  2. (2)

The quivers $Q(\tau)^{\pm}$ are contained in the acyclic quivers
$Q^{\sigma_{\pm}}$, so $Q(\tau)^{\pm}$ are themselves acyclic and hence have
at least one source. It follows that each interior line segment
$\tau\in{}_{\theta}(2)$ is marked by at least one nonzero vertex of $Q$.

######  Corollary 4.18.

Let $G$ be a finite abelian subgroup of $\operatorname{SL}(3,\mathbb{C})$.
Reid’s recipe for the fan of $G\operatorname{-Hilb}$ as given by Definitions
4.13 and 4.16 agrees with the classical Recipe 2.6.

###### Proof.

Recall from Corollary 3.13 that the isomorphism $\mathcal{M}_{\theta}\cong
G\operatorname{-Hilb}$ identifies the torus-invariant quiver representations
$M_{\sigma_{\pm}}$ with the $G$-clusters $\mathbb{C}[x,y,z]/I_{\pm}$ obtained
as the fibres of the universal family on $G\operatorname{-Hilb}$ over the
origins in the charts $U_{\sigma_{\pm}}$. It follows that the marking of an
interior lattice point $\rho\in\Sigma(1)$ according to Recipe 2.6(2) agrees
with the marking from Definition 4.13.

To show that the recipes agree for an interior line segment
$\tau\in\Sigma(2)$, consider first the classical recipe. Let
$\sigma_{\pm}\in\Sigma(3)$ satisfy $\tau=\sigma_{+}\cap\sigma_{-}$ and let
$m=(m_{1},m_{2},m_{3})\in M$ denote the primitive vector in the normal
direction to the hyperplane spanned by $\tau$ satisfying $\langle
m,n\rangle\geq 0$ for all $n\in\sigma_{+}$. Nakamura [Nak01, Lemma 1.8]
implies that the denominator of the $G$-invariant Laurent monomial
$x^{m_{1}}y^{m_{2}}z^{m_{3}}$ is one of the monomials of $\mathcal{S}_{+}$,
say $x^{a}y^{b}z^{c}$, and the classical Recipe 2.6(1) marks $\tau$ with the
vertex $i\in Q_{0}\setminus\\{0\\}$ such that $x^{a}y^{b}z^{c}$ lies in the
$i$-character space. Crucially, Proposition 3.8 shows that a monomial in
$\mathcal{S}_{+}$ moves during the $G$-igsaw transformation across $\tau$ if
and only if it is divisible by $x^{a}y^{b}z^{c}$, _i. e._ the directed
subgraph of $\mathcal{S}_{+}$ comprising monomials that move in the $G$-igsaw
transformation has a unique source given by the monomial $x^{a}y^{b}z^{c}$.

To compare this with the recipe from Definition 4.16, identify the $G$-graphs
$\mathcal{S}_{\pm}$ with the quivers $Q^{\sigma_{\pm}}$ as in Corollary 3.13
and let $x,y,z$ denote the variables in the Cox ring
$\mathbb{C}[t_{\rho}\mid\rho\in\Sigma(1)]$ indexed by the lattice points
$\rho\in\Sigma(1)$ defined by corners of the junior simplex as in Remark 3.14.
The jigsaw transformation from $M_{\sigma_{+}}$ to $M_{\sigma_{-}}$ as in
Theorem 3.12 moves the tile dual to a vertex $j\in Q_{0}$ if and only if the
section $t^{\operatorname{div}(p)}$ labelling a path from $0$ to $j$ in
$Q^{\sigma_{+}}$ is divisible by $x^{a}y^{b}z^{c}$. This jigsaw transformation
coincides with that of Nakamura by Corollary 3.13, so the subquiver
$Q(\tau)^{+}$ of $Q^{\sigma_{+}}$ whose vertex set comprises vertices dual to
tiles that move in the jigsaw transformation across $\tau$ has a unique source
vertex $i$ corresponding to the eigencharacter of the monomial
$x^{a}y^{b}z^{c}$. Definition 4.16 marks $\tau$ with vertex $i$, so the
recipes agree. ∎

### 4.4. Examples

We now present two examples where Combinatorial Reid’s recipe from Definitions
4.13 and 4.16 is used to mark the interior lattice points and line segments of
the toric fan θ with vertices of the quiver $Q$ dual to a consistent dimer
model.

######  Example 4.19.

Consider the dimer model from Example 2.1. The stability parameter
$\theta=(-9,1,1,\ldots,1)$ is $0$-generated and the fan θ of
$\mathcal{M}_{\theta}$ in Figure 2(b) shows how we list the lattice points
$\rho_{1},\dots,\rho_{10}\in{}_{\theta}(1)$.

To implement Reid’s recipe, first compute the quiver $Q^{\sigma}$ for each
cone $\sigma\in{}_{\theta}(3)$. One such quiver $Q^{\sigma}$ is shown in green
in Figure 3: notice that vertices 2 and 5 are both sinks of that quiver, so
both vertices are candidates to mark the interior lattice point $\rho_{8}$
according to Definition 4.13. In fact, both 2 and 5 are sinks in every quiver
$Q^{\sigma}$ determined by a cone $\sigma\in{}_{\theta}(3)$ such that
$\rho_{8}\subset\sigma$, so both 2 and 5 mark lattice point $\rho_{8}$ in
Reid’s recipe. A similar calculation shows that vertex 2 also marks the
lattice point $\rho_{9}$ as shown in Figure 10.

93,93746688412, 52 Figure 10. Marking of θ using combinatorial Reid’s recipe
for the dimer model from Example 2.1.

As for the interior line segments, consider the cone $\tau$ generated by the
rays $\rho_{8}$ and $\rho_{9}$. Figure 7 illustrates the jigsaw pieces for
$\tau$: there is precisely one jigsaw piece $\operatorname{J}_{1}$ that lies
adjacent to $\operatorname{J}_{0}$, and the quivers $Q(\tau)^{\pm}$ supported
in $\operatorname{J}_{1}$ coincide. The vertices 3 and 9 are the sources of
this quiver and, according to Definition 4.16, Reid’s recipe marks the
interior line segment $\tau$ with both 3 and 9. More generally, Combinatorial
Reid’s recipe marks all interior lattice points and interior line segments of
θ with vertices as shown in Figure 10.

######  Example 4.20.

Consider the consistent dimer model shown in black in Figure 11(a); the dual
quiver $Q$ is illustrated in grey in the same figure. The stability condition
$\theta=(-25,1,1,\ldots,1)$ is 0-generated and the labelling sections
$t^{\operatorname{div}(a)}$ from equation (2.9) that determine the
tautological isomorphism $\phi\colon A\to\operatorname{End}(T)$ are written
explicitly on (or can be deduced from) [Tap15, Figures 4.2(a), 4.3]. The toric
fan θ is shown in Figure 11(b), where we have marked every interior line
segment and interior lattice point of θ with vertices of the quiver $Q$
according to Combinatorial Reid’s recipe.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$19$$20$$21$$22$$23$$24$$25$123445567888910111112131415151617181920212223232425262728292930313132333334353637383940414242434444454647474849505152535455565758596060616162636364646566676768696970717172

77, 212120415151111111110101010777333112222323617921191588,2225516,2418121314

Figure 11. (a) A consistent dimer model; (b) the fan θ marked according to
Reid’s recipe.

######  Remark 4.21.

These examples exhibit several new phenomena that are not present in the
classical Reid’s recipe [Rei97, Cra05] for the toric fan of the $G$-Hilbert
scheme for a finite abelian subgroup $G$ in $\operatorname{SL}(3,\mathbb{C})$.
Indeed:

  1. (1)

distinct interior lattice points can be marked with the same vertex: in Figure
10, vertex $2$ marks both interior lattice points; while in Figure 11(b),
vertex 8 marks a pair of lattice points. Contrast this with the classical
case, where any vertex of the McKay quiver that marks an interior lattice
point of does not mark another lattice point or line segment in (see [Cra05,
Corollary 4.6]);

  2. (2)

interior line segments can be marked with more than one vertex: in Figure 10,
vertices 3 and 9 both mark the same line segment; in Figure 11(b), vertices 7
and 21 mark the same line segment. In the classical case, every interior line
segment of is marked with a unique vertex of the McKay quiver.

  3. (3)

the marking of an interior line segment $\tau\in{}_{\theta}(2)$ is not
determined by the hyperplane containing $\tau$: in Figure 10, three coplanar
cones in ${}_{\theta}(2)$ are marked with 3, with 3 and 9 and with 9
respectively. In the classical case, the marking of an interior line segment
$\tau$ is determined by the normal vector of the hyperplane containing the
cone, so each cone in a given hyperplane is marked identically.

  4. (4)

the marking of an interior lattice point $\rho\in{}_{\theta}(1)$ is not
determined by the geometry of the toric surface $D_{\rho}$: in Figure 11(b),
one del Pezzo surface of degree six is marked by a pair of vertices 16 and 24,
while a second del Pezzo surface of degree six is marked by the single vertex
5. In the classical case, the marking is determined by the geometry of the
surface (see [Cra05, Section 3]).

  5. (5)

the Euler number of an irreducible component of the exceptional divisor is not
bounded above by six: in Figure 11(b), the unique interior lattice point of θ
marked by vertex 25 determines a toric surface with Euler number 7. In the
classical case, a result of Craw–Reid [CR02, Corollary 1.4] shows that the
Euler number of every torus-invariant projective surface $D_{\rho}$ in
$G\operatorname{-Hilb}$ is at most six.

## 5\. Compatibility with Geometric Reid’s Recipe

Motivated by Logvinenko’s geometric version of Reid’s recipe (see Theorem
2.7), we now study the relation between Reid’s recipe for a consistent dimer
model as in Definitions 4.13 and 4.16 and the statement of Geometric Reid’s
recipe given by Bocklandt–Craw–Quintero Vélez [BCQ15, Theorem 1.4]. We
conclude with a pair of natural conjectures.

### 5.1. Compatibility with Geometric Reid’s Recipe

To begin we generalise a statement from Cautis–Craw–Logvinenko [CCL17,
Proposition 4.8] to consistent dimer models.

######  Lemma 5.1.

Let $\sigma\in{}_{\theta}(3)$ and $i\in Q_{0}$. Assume that the vertex simple
$A$-module $S_{i}$ lies in the socle of the torus-invariant $\theta$-stable
$A$-module $M_{\sigma}$. According to Reid’s recipe, vertex $i$ marks either:

  1. $(\mathrm{i})$

an edge of the triangle $\sigma$, _i. e._ there exists $\tau\in{}_{\theta}(2)$
with $\tau\subset\sigma$ such that $i$ marks $\tau$. In this case, $S_{i}$
lies in the socle of every $\theta$-stable $A$-module in the torus-invariant
curve $C_{\tau}$; or

  2. $(\mathrm{ii})$

a node of the triangle $\sigma$, _i. e._ there exists $\rho\in{}_{\theta}(1)$
with $\rho\subset\sigma$ such that $i$ marks $\rho$. In this case, $S_{i}$
lies in the socle of every $\theta$-stable $A$-module in the torus-invariant
divisor $D_{\rho}$.

###### Proof.

Let $y\in\mathcal{M}_{\theta}$ be the torus-invariant point defined by the
cone $\sigma\in{}_{\theta}(3)$, so $M_{y}=M_{\sigma}$ by Lemma 3.1. Since
$S_{i}$ lies in the socle of $M_{\sigma}$, the vertex $i$ cannot be the zero
vertex (unless has only one node, in which case $X=\mathbb{C}^{3}$ and Reid’s
recipe says nothing of interest). We may therefore apply several results of
Bocklandt–Craw–Quintero-Vélez [BCQ15, Propositions 3.4, 4.7, Lemma 4.8,
Correction] to deduce that the intersection of

$Z_{i}:=\big{\\{}y\in\mathcal{M}_{\theta}\mid
S_{i}\subseteq\operatorname{soc}(M_{y})\big{\\}}$

with the locus $\tau_{\theta}^{-1}(x_{0})$ is either a single $(-1,-1)$-curve
in $\mathcal{M}_{\theta}$, a single $(0,-2)$-curve in $\mathcal{M}_{\theta}$
or a connected union of compact torus-invariant divisors in
$\mathcal{M}_{\theta}$. We have $S_{i}\subseteq\operatorname{soc}(M_{y})$ by
assumption, hence $y\in Z_{i}$. We consider each case in turn.

If this locus is a $(-1,-1)$-curve in $\mathcal{M}_{\theta}$ then there exists
$\tau\in{}_{\theta}(2)$ such that $Z_{i}=C_{\tau}$, so $S_{i}$ lies in the
socle of every $\theta$-stable $A$-module parametrised by points of
$C_{\tau}$. Also, $y\in Z_{i}=C_{\tau}$, so the inclusion-reversing
correspondence between orbit-closures in $\mathcal{M}_{\theta}$ and cones in
the fan θ gives $\tau\subset\sigma$. To see that Reid’s recipe marks $\tau$
with vertex $i$, the result of [BCQ15, Lemma 4.10] implies that the jigsaw
transformation across $\tau$ defines precisely two jigsaw pieces, where the
jigsaw piece adjacent to $\operatorname{J}_{0}$ comprises a single tile dual
to vertex $i$. In particular, $i$ is the source vertex of the quivers
$Q^{\pm}$ associated to $\tau$, so $i$ marks $\tau$ according to Definition
4.16.

The case of a $(0,-2)$-curve in $\mathcal{M}_{\theta}$ is almost identical,
except now the curve $C_{\tau}$ defined by the cone $\tau\in{}_{\theta}(2)$
coincides with the intersection $Z_{i}\cap\tau_{\theta}^{-1}(x_{0})$ as noted
in [BCQ15, Correction]. Otherwise, by applying the analogue of [BCQ15, Lemma
4.10] as stated in [BCQ15, Correction, Lemma 3], the argument from the
previous paragraph goes through verbatim to give that vertex $i$ marks the
cone $\tau$ as required.

Otherwise, $Z_{i}$ is a compact torus-invariant divisor. Since $y\in Z_{i}$,
there exists $\rho\in{}_{\theta}(1)$ such that $y\in D_{\rho}\subseteq Z_{i}$,
so $S_{i}$ lies in the socle of every $\theta$-stable $A$-module parametrised
by a point in $D_{\rho}$. It follows from Definition 4.13 that vertex $i$
marks $\rho$. Applying the inclusion-reversing correspondence between orbit
closures in $\mathcal{M}_{\theta}$ and cones in θ to the inclusion $y\in
D_{\rho}$ gives $\rho\subset\sigma$ as required. ∎

Recall that the derived equivalence θ from (2.10) sends the vertex simple
$A$-module $S_{i}$ for $i\in Q_{0}$ to an object ${}_{\theta}(S_{i})$ in the
bounded derived category of coherent sheaves on $\mathcal{M}_{\theta}$. In
particular, for each $k\in\mathbb{Z}$ we obtain a coherent sheaf
$H^{k}({}_{\theta}(S_{i}))$.

######  Proposition 5.2.

Let $i\in Q_{0}$ be a nonzero vertex and assume $H^{0}({}_{\theta}(S_{i}))\neq
0$. According to Reid’s recipe, vertex $i$ marks either:

  1. $(\mathrm{i})$

a unique interior line segment $\tau\in{}_{\theta}(2)$, in which case the
corresponding torus-invariant curve $C_{\tau}$ satisfies
${}_{\theta}(S_{i})\cong L_{i}^{-1}|_{C_{\tau}}$; or

  2. $(\mathrm{ii})$

at least one interior lattice point $\rho\in{}_{\theta}(1)$, in which case the
corresponding torus-invariant divisor $D_{\rho}$ is contained in the support
of ${}_{\theta}(S_{i})$. In fact, if $Z_{i}$ denotes the union of all torus-
invariant divisors marked by vertex $i$, then ${}_{\theta}(S_{i})\cong
L_{i}^{-1}|_{Z_{i}}$.

###### Proof.

Since $i$ is nonzero and $H^{0}(\Psi(S_{i}))\neq 0$, [BCQ15, Proposition 4.7]
gives $\sigma\in{}_{\theta}(3)$ such that the corresponding torus-invariant
$\theta$-stable $A$-module $M_{\sigma}$ contains $S_{i}$ in its socle. Now
Lemma 5.1 applies, giving two cases. If Lemma 5.1$(\mathrm{i})$ occurs, then
there exists an interior line segment $\tau\in{}_{\theta}(2)$ such that vertex
$i$ marks $\tau$, and $S_{i}$ lies in the socle of every $\theta$-stable
$A$-module in the curve $C_{\tau}=\\{y\in\mathcal{M}_{\theta}\mid
S_{i}\subseteq\operatorname{soc}(M_{y})\\}\cap\tau_{\theta}^{-1}(x_{0})$. In
this case, [BCQ15, Proposition 1.3, Correction] gives ${}_{\theta}(S_{i})\cong
L_{i}^{-1}|_{C_{\tau}}$ as required. Otherwise Lemma 5.1$(\mathrm{ii})$
occurs, giving an interior lattice point $\rho\in{}_{\theta}(1)$ such that
vertex $i$ marks $\rho$, and $S_{i}$ lies in the socle of every
$\theta$-stable $A$-module corresponding to a point of $D_{\rho}$. This holds
if and only if $D_{\rho}$ is contained in the locus
$Z_{i}=\\{y\in\mathcal{M}_{\theta}\mid
S_{i}\subseteq\operatorname{soc}(M_{y})\\}$. In particular, $Z_{i}$ is non-
empty, so [BCQ15, Proposition 1.3] applies to show that $Z_{i}$ is equal to
the support of ${}_{\theta}(S_{i})$ and in fact ${}_{\theta}(S_{i})\cong
L_{i}^{-1}|_{Z_{i}}$. This completes the proof. ∎

######  Remarks 5.3.

  1. (1)

The support of ${}_{\theta}(S_{i})$ in Proposition 5.2$(\mathrm{ii})$ is known
to be connected [BCQ15, Lemma 4.8]. Therefore Proposition 5.2$(\mathrm{ii})$
implies that the union of all divisors $D_{\rho}$ in $\mathcal{M}_{\theta}$
such that $\rho$ is marked by vertex $i$ is connected. This statement is not
an obvious consequence of Definition 4.13.

  2. (2)

The examples from Section 4.4 show that $Z_{i}$ from Proposition
5.2$(\mathrm{ii})$, may be reducible. That the support of ${}_{\theta}(S_{i})$
may be reducible was surely known to Ishii and Ueda, see [IU16, Lemma 11.2].

As a result, we may enhance slightly the statement of Geometric Reid’s recipe
for consistent dimer models from [BCQ15, Theorem 1.4] to one that makes
reference to the marking of lattice points and _some_ line segments of the fan
θ according to Definitions 4.13 and 4.16:

######  Corollary 5.4 (Geometric Reid’s recipe for dimer models).

Let be a consistent dimer model and let $i\in Q_{0}$ be a vertex in the dual
quiver. Precisely one of the following statements holds:

  1. $(\mathrm{i})$

vertex $i$ marks at least one interior lattice point in ${}_{\theta}(1)$, in
which case ${}_{\theta}(S_{i})\cong L_{i}^{-1}\otimes\mathcal{O}_{Z_{i}}$,
where $Z_{i}$ is the connected union of torus-invariant divisors indexed by
lattice points $\rho$ marked with $i$;

  2. $(\mathrm{ii})$

vertex $i$ marks a unique interior line segment $\tau\in{}_{\theta}(2)$, in
which case ${}_{\theta}(S_{i})\cong L_{i}^{-1}\otimes\mathcal{O}_{C_{\tau}}$;

  3. $(\mathrm{iii})$

${}_{\theta}(S_{i})\cong\mathcal{F}[1]$, where $\mathcal{F}$ is a coherent
sheaf whose support is a connected union of compact torus-invariant divisors;
or

  4. $(\mathrm{iv})$

$i=0$, in which case ${}_{\theta}(S_{0})$ is quasi-isomorphic to the dualising
complex of the locus $\tau_{\theta}^{-1}(x_{0})$.

###### Proof.

The only enhancement to [BCQ15, Theorem 1.4] is in parts $(\mathrm{i})$ and
$(\mathrm{ii})$, but for convenience we explain the logic of the proof. If
$i=0$, then case $(\mathrm{iv})$ holds by [BCQ15, Proposition 3.7]. Otherwise,
$i\neq 0$ and [BCQ15, Theorem 1.1, Proposition 3.4] gives a unique
$k\in\\{-1,0\\}$ such that $H^{k}({}_{\theta}(S_{i}))\neq 0$. If $k=-1$, then
case $(\mathrm{iii})$ holds by [BCQ15, Corollary 5.2, Proposition 5.6].
Otherwise $k=0$, in which case [BCQ15, Proposition 4.11, Correction] gives
${}_{\theta}(S_{i})\cong L_{i}^{-1}|_{Z_{i}^{c}}$ for
$Z_{i}^{c}:=\big{\\{}y\in\tau_{\theta}^{-1}(x_{0})\mid
S_{i}\subseteq\operatorname{soc}(M_{y})\big{\\}}$. Proposition 5.2 implies
that $Z_{i}^{c}$ is determined by the interior lattice points and line
segments of θ marked by $i$ according to Definitions 4.13 and 4.16. Note that
$Z_{i}^{c}$ is equal to $Z_{i}=\\{y\in\mathcal{M}_{\theta}\mid
S_{i}\subseteq\operatorname{soc}(M_{y})\\}$ in case $(\mathrm{i})$, and to
$C_{\tau}$ in case $(\mathrm{ii})$. ∎

### 5.2. Open questions

After comparing Corollary 5.4 with the statement of Theorem 2.7, it is natural
to suggest the following.

######  Conjecture 5.5.

Let $i\in Q_{0}$ be nonzero. Then $\mathcal{F}:=H^{-1}({}_{\theta}(S_{i}))\neq
0$ if and only if $i$ marks two or more interior line segments in θ according
to Definition 4.16, in which case the support of ${}_{\theta}(S_{i})$ is the
union of all torus-invariant divisors $D_{\rho}$ such that two line segments
$\tau$ in θ containing $\rho$ are marked by vertex $i$.

######  Remarks 5.6.

  1. (1)

A proof of Conjecture 5.5 would imply that every nonzero vertex of $Q$ appears
‘once’ on the fan θ in a sense similar to the statement from [Cra05, Corollary
4.6]. In particular, each nonzero vertex $i\in Q$ would either mark lattice
points in θ (this happens when $H^{0}({}_{\theta}(S_{i}))\neq 0$) or line
segments in θ (this happens when $H^{-1}({}_{\theta}(S_{i}))\neq 0$). Note
that $i$ would not be able to mark both lattice points and line segments,
because [BCQ15, Theorem 1.1, Proposition 3.4] gives a unique $k\in\\{-1,0\\}$
such that $H^{k}({}_{\theta}(S_{i}))\neq 0$. The examples from Section 4.4
illustrate this dichotomy.

  2. (2)

When $H^{-1}({}_{\theta}(S_{i}))\neq 0$, the support of the object
${}_{\theta}(S_{i})$ is known to be connected [BCQ15, Corollary 5.2]. A proof
of Conjecture 5.5 would therefore also imply that the union of line segments
marked by $i$ is a connected graph in the triangulation. Such a statement
would be similar to that from Remark 5.3(1) which applies when $i$ marks
lattice points.

Conjecture 5.5 is not strong enough to be called _Derived Reid’s recipe_ for
consistent dimer models because it does not describe explicitly the objects
${}_{\theta}(S_{i})$ in terms of the labelling of lattice points and line
segments in θ as was done for classical Reid’s recipe in [CCL17, Theorem 1.2].
Such an explicit description would require that one computes the cohomology of
the _wheel_ associated to $L_{i}^{-1}$ as defined by Craw–Quintero-Vélez
[CQ15]. This calculation becomes increasingly complicated as the number of
arrows with head at vertex $i$ increases; see Example 3.5 from _ibid._ for one
such example.

We also anticipate that Combinatorial Reid’s recipe will enable one to compute
a minimal set of relations in $\operatorname{Pic}(\mathcal{M}_{\theta})$
between the tautological line bundles on $\mathcal{M_{\theta}}$. A proof of
the following would extend the result of [Cra05, Theorem 6.1] to any
consistent dimer model.

######  Conjecture 5.7.

The Picard group of $\mathcal{M}_{\theta}$ is generated by the nontrivial
tautological bundles $\\{L_{i}\\}_{i\neq 0}$, modulo the subgroup generated by
one relation for each interior lattice point $\rho\in{}_{\theta}(1)$, namely

$\bigotimesop\displaylimits_{i\text{ marks
}\rho}L_{i}\cong\bigotimesop\displaylimits_{i\in
Q_{0}}L_{i}^{\otimes(n(i,\rho)-1)},$

where $n(i,\rho)$ is the number of interior line segments
$\tau\in{}_{\theta}(2)$ marked by vertex $i$ such that $\rho\subset\tau$.

######  Example 5.8.

In Example 4.19, one computes explicitly that there are two such relations:

$L_{2}\otimes L_{5}\cong L_{3}\otimes L_{6}\otimes L_{8}\quad\text{and}\quad
L_{2}\cong L_{4}\otimes L_{9}.$

Similarly, in Example 4.20 there are nine such relations:

$L_{14}\cong L_{10}\otimes L_{11};\quad L_{18}\cong L_{10}\otimes L_{15};\quad
L_{25}\cong L_{10}\otimes L_{21};\quad L_{13}\cong L_{7}\otimes L_{11};\quad
L_{8}\cong L_{1}\otimes L_{7};$ $L_{16}\otimes L_{24}\cong L_{7}\otimes
L_{15}\otimes L_{23};\quad L_{8}\otimes L_{22}\cong L_{2}\otimes L_{7}\otimes
L_{21};\quad L_{12}\cong L_{3}\otimes L_{11};\quad L_{5}\cong L_{2}\otimes
L_{3}.$

Additional examples where Conjectures 5.5 and 5.7 have been verified can be
found in [Tap15].

A proof of Conjecture 5.7 would enable one to cook-up virtual bundles in the
spirit of the classical case (see Reid [Rei97] and Craw [Cra05, Section 7]),
leading to a $\mathbb{Z}$-basis of the integral cohomology
$H^{*}(\mathcal{M}_{\theta},\mathbb{Z})$ indexed by the vertices of the quiver
$Q$. This would provide the analogue for consistent dimer models of the
geometric McKay correspondence for a finite subgroup of
$\operatorname{SL}(2,\mathbb{C})$ given by Gonzalez-Sprinberg–Verdier [GSV84].

## References

  * [BM09] M. Bender and S. Mozgovoy,  _Crepant resolutions and brane tilings II: Tilting bundles_ ,  preprint arXiv:0909.2013 (2009).
  * [Boc12] R. Bocklandt,  _Consistency conditions for dimer models_ ,  Glasg. Math. J. 54(2012), no. 2, 429–447.
  * [Boc13] by same author,  _Calabi-Yau algebras and weighted quiver polyhedra_ ,  Math. Z. 273 (2013), no. 1-2, 311–329.
  * [BCQ15] R. Bocklandt, A. Craw, and A. Quintero Vélez,  _Geometric Reid’s recipe for dimer models_ ,  Math. Ann. 361 (2015), no. 3-4, 689–723.  See also the _Correction to: Geometric Reid’s recipe for dimer models_. Math. Ann. (2021). https://doi.org/10.1007/s00208-020-02127-w.
  * [BKR01] T. Bridgeland, A. King, and M. Reid,  _The McKay correspondence as an equivalence of derived categories_ ,  J. Amer. Math. Soc. 14 (2001), no. 3, 535–554.
  * [Bro12] N. Broomhead,  _Dimer models and Calabi-Yau algebras_ ,  Mem. Amer. Math. Soc. 215 (2012) (1011):viii+86.
  * [CCL17] S. Cautis, A. Craw, and T. Logvinenko,  _Derived Reid’s recipe for abelian subgroups of ${\rm SL}_{3}(\mathbb{C})$_,  J. Reine Angew. Math. 727 (2017), 1–48.
  * [CL09] S. Cautis and T. Logvinenko,  _A derived approach to geometric McKay correspondence in dimension three_ ,  J. Reine Angew. Math. 636 (2009), 193–236.
  * [Cra05] A. Craw,  _An explicit construction of the McKay correspondence for $A$-Hilb $\mathbb C^{3}$_,  J. Algebra 285 (2005), no. 2, 682–705.
  * [CI04] A. Craw and A. Ishii,  _Flops of $G$-Hilb and equivalences of derived categories by variation of GIT quotient_,  Duke Math. J. 124 (2004), no. 2, 259–307.
  * [CMT07] A. Craw, D. Maclagan, and R.R. Thomas,  _Moduli of McKay quiver representations. II. Gröbner basis techniques_ J. Algebra 316 (2007), no. 2, 514–535.
  * [CQ12] A. Craw and A. Quintero Vélez,  _Cellular resolutions of noncommutative toric algebras from superpotentials_ ,  Adv. Math. 229 (2012), no. 3, 1516–1554.
  * [CQ15] by same author,  _Cohomology of wheels on toric varieties_ ,  Hokkaido Math. J. 44 (2015), no. 1, 47–79.
  * [CR02] A. Craw and M. Reid,  _How to calculate $A$-Hilb $\mathbb{C}^{3}$_,  in: Geometry of toric varieties, volume 6 of Sémin. Congr., 129–154. Soc. Math. France, Paris, 2002.
  * [GSV84] G. Gonzalez-Sprinberg and J.-L. Verdier,  _Construction géométrique de la correspondance de McKay_ ,  Ann. Sci. École Norm. Sup. (4) 16 (1984), no. 3, 409–449.
  * [IU08] A. Ishii and K. Ueda,  _On moduli spaces of quiver representations associated with dimer models_ ,  in: Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, B9, 127–141. Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.
  * [IU11] by same author,  _A note on consistency conditions on dimer models_ ,  in: Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, B24, 143–164. Res. Inst. Math. Sci. (RIMS), Kyoto, 2011.
  * [IU15] by same author,  _Dimer models and the special McKay correspondence_ ,  Geom. Topol. 19 (2015), no. 6, 3405–3466.
  * [IU16] by same author,  _Dimer models and crepant resolutions_ Hokkaido Math. J. 45 (2016), no. 1, 1–42.
  * [IN99] Y. Ito and I. Nakamura,  _Hilbert schemes and simple singularities_ ,  in: New trends in algebraic geometry (Warwick, 1996), volume 264 of London Math. Soc. Lecture Note Ser., 151–233. Cambridge Univ. Press, Cambridge, 1999.
  * [KV00] M. Kapranov and E. Vasserot,  _Kleinian singularities, derived categories and Hall algebras_ ,  Math. Ann. 316 (2000), no. 3, 565–576.
  * [Kin94] A.D. King,  _Moduli of representations of finite-dimensional algebras_ ,  Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 4, 515–530.
  * [LP90] L. Le Bruyn and C. Procesi,  _Semisimple representations of quivers_ ,  Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.
  * [Log03] T. Logvinenko,  Families of $G$-constellations parametrised by resolutions of quotient singularities,  Ph.D. Thesis. University of Bath, UK, 2004. Available from arXiv:math/0305194.
  * [Log10] by same author,  _Reid’s recipe and derived categories_ ,  J. Algebra 324 (2010), no. 8, 2064–2087.
  * [McK80] J. McKay,  _Graphs, singularities, and finite groups_ ,  in: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), volume 37 of Proc. Sympos. Pure Math., 183–186. Amer. Math. Soc., Providence, R.I., 1980\.
  * [Mog09] S. Mozgovoy,  _Crepant resolutions and brane tilings I: Toric realization_.  preprint arXiv:0908.3475 (2009).
  * [MR10] S. Mozgovoy and M. Reineke,  _On the noncommutative Donaldson-Thomas invariants arising from brane tilings_ ,  Adv. Math. 223 (2010), no. 5, 1521–1544.
  * [Nak01] I. Nakamura,  _Hilbert schemes of abelian group orbits_ ,  J. Algebraic Geom. 10 (2001), no. 4, 757–779.
  * [Rei97] M. Reid,  _Mckay correspondence_ ,  in: Proc. of Algebraic Geom. Symposium (Kinosaki, November 1996), pages 14–41, 1997. Available from arXiv:alg-geom/9702016.
  * [Tap15] J. Tapia Amador,  Combinatorial Reid’s recipe for consistent dimer models.  PhD thesis, University of Bath, 2015. Available from https://purehost.bath.ac.uk/portal/files/187939640/CRRCDM2.pdf.
  * [UY11] K. Ueda and M. Yamazaki,  _A note on dimer models and McKay quivers_ ,  Comm. Math. Phys. 301 (2011), no. 3, 723–747.
  * [Wor20] B. Wormleighton,  _Walls for $G$-Hilb via Reid’s Recipe_,  SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), 106, 38 pages.