Walls for -Hilb via Reids Recipe
The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein 3-fold quotient singularities ³/ with the representation theory of the group . The first crepant resolution studied in depth was the -Hilbert scheme -HilbA3, which is also a moduli space...
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| description | The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein 3-fold quotient singularities ³/ with the representation theory of the group . The first crepant resolution studied in depth was the -Hilbert scheme -HilbA3, which is also a moduli space of θ-stable representations of the McKay quiver associated to . As the stability parameter θ varies, we obtain many other crepant resolutions. In this paper, we focus on the case where is abelian, and compute explicit inequalities for the chamber of the stability space defining -Hilb ³ in terms of a marking of exceptional subvarieties of -Hilb ³ called Reid's recipe. We further show which of these inequalities define walls. This procedure depends only on the combinatorics of the exceptional fibre and has applications to the birational geometry of other crepant resolutions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 106, 38 pages
Walls for G-Hilb via Reid’s Recipe
Ben WORMLEIGHTON
Department of Mathematics and Statistics, Washington University in St. Louis,
MO 63130, USA
E-mail: benw@wustl.edu
URL: https://sites.google.com/view/benw/
Received November 14, 2019, in final form October 14, 2020; Published online October 24, 2020
https://doi.org/10.3842/SIGMA.2020.106
Abstract. The three-dimensional McKay correspondence seeks to relate the geometry of
crepant resolutions of Gorenstein 3-fold quotient singularities A3{G with the representation
theory of the group G. The first crepant resolution studied in depth was the G-Hilbert
scheme G-HilbA3, which is also a moduli space of θ-stable representations of the McKay
quiver associated to G. As the stability parameter θ varies, we obtain many other crepant
resolutions. In this paper we focus on the case where G is abelian, and compute explicit
inequalities for the chamber of the stability space defining G-HilbA3 in terms of a marking
of exceptional subvarieties of G-HilbA3 called Reid’s recipe. We further show which of
these inequalities define walls. This procedure depends only on the combinatorics of the
exceptional fibre and has applications to the birational geometry of other crepant resolutions.
Key words: wall-crossing; McKay correspondence; Reid’s recipe; quivers
2020 Mathematics Subject Classification: 14E16; 14M25; 16G20
1 Introduction
Let G � SLnpCq be a finite subgroup. When n � 2 there is a famous ADE classification of such
subgroups that matches the classification of Du Val or modality zero singularities by taking
a subgroup G to the quotient singularity 0 P A2{G. This observation and the surrounding deep
interactions of the geometry of A2{G and its resolutions, and the representation theory of G
are known as the two-dimensional McKay correspondence [1, 2, 13, 16, 17, 21, 23]. In this
case, the unique minimal or crepant resolution has a modular interpretation as the G-Hilbert
scheme G-HilbA2. The moduli space G-HilbM for M a variety and G � AutpMq a finite
subgroup parameterises G-clusters in M : zero-dimensional G-invariant subschemes M of A2
with H0pOZq � CrGs as G-modules. This was generalised to three dimensions for finite abelian
subgroups of SL3pCq by Nakamura [22] who showed that G-HilbA3 is a crepant resolution
of A3{G and then to all subgroups G by the celebrated work of Bridgeland–King–Reid [5]. They
moreover established an equivalence of categories
Db
�
G-HilbA3
�
� Db
G
�
A3
�
, (1.1)
which also holds if G-HilbA3 is replaced by any projective crepant resolution of A3{G. Compare
also the results of Bridgeland [4].
Using the GIT approach of King [19] to constructing moduli of quiver representations the
G-Hilbert scheme can also be realised as a moduli space of θ-stable quiver representations
MθpQ,dq, where Q is the McKay quiver of G and d � pdiq is a given dimension vector. In this
situation the stability parameter θ lives in the stability space
Θ :�
#
θ P HomZ
�
ZQ0 ,Q
�
:
¸
iPQ0
diθpiq � 0
+
,
mailto:email@address
https://sites.google.com/view/benw/
https://doi.org/10.3842/SIGMA.2020.106
2 B. Wormleighton
where Q0 is the set of vertices of Q. By definition, the vertices of the McKay quiver biject
with the irreducible representations IrrpGq of G and so one can view ZQ0 as the abelian group
underlying the representation ring of G. As θ varies, it is possible that one obtains many
different crepant resolutions of A3{G; in the case that G is abelian, Craw–Ishii [10] show that all
projective crepant resolutions arise in this way. The stability space Θ has a wall-and-chamber
structure such that the moduli space MθpQ,dq is constant so long as θ remains inside a given
chamber. We denote the moduli space MC :� MθpQ,dq for any generic θ in a chamber C.
Denote the chamber corresponding to G-HilbA3 by C0. The positive orthant
Θ� :�
θ P Θ: θpρq ¡ 0 for all nontrivial ρ P IrrpGq
(
lies inside C0 however it is not usually equal to it. The primary purpose of this paper is to provide
explicit combinatorial inequalities defining C0 and identify precisely which of these define walls
of C0. We remark that such equations were computed for a group of order 11 in [10, Example 9.6].
Assume that G is abelian. In this context [10, Theorem 9.5] gives an abstract description of
such inequalities, however it is difficult to perform explicit calculations or deduce general state-
ments from their presentation. One can view some of the results herein as a combinatorialisation
of [10, Theorem 9.5], which turn out to be very amenable to applications.
We briefly outline the context and notation of [10] that we will also use. For a chamber C � Θ
the equivalence from (1.1) induces an isomorphism ϕC : K0pMCq Ñ KG
�
A3
�
� ReppGq. Here
K0pMCq denotes the K-group of sheaves supported on the preimage of the G-orbit for the origin
under the resolution MC Ñ A3{G. Walls in Θ are cut out by hyperplanes
�°
i αi � θpχiq � 0
�
for some characters χi P IrrpGq and integers αi P Z, though in general not all θ on such
a hyperplane will be non-generic. The inequalities in [10] have three different forms, each
coming from exceptional subvarieties. Firstly, each exceptional curve C � G-HilbA3 gives an
inequality of the form
θ
�
ϕC0pOCq
�
¡ 0.
The characters appearing in these inequalities are packaged in collections of monomials associ-
ated to exceptional curves that were named by Nakamura [22] in a different context as G-igsaw
pieces. Our first result is to pin down which characters lie in G-igsaw pieces. In general there
are several G-igsaw pieces corresponding to a single curve C; the union of all the pieces that
do not include the trivial character is the set of characters that appear in the inequality for C.
We call this union the total G-igsaw piece.
As G is abelian the singularity A3{G and its crepant resolutions are toric. There is a method
of marking the exceptional subvarieties of G-Hilb – the edges and vertices in the triangulation –
by characters of G known as “Reid’s recipe”. This was used to explicitly describe the McKay
correspondence in the classical terms of providing a basis of H�
�
G-HilbA3,Z
�
indexed by char-
acters by Craw [9]. It was later categorified by Logvinenko [20], Cautis–Logvinenko [7], and
Cautis–Craw–Logvinenko [6], who expressed the locus labelled by a character χ in terms of the
support of an object associated to χ in the derived category of G-Hilb. We will discuss this
in more detail in Sections 2.2 and 2.4.
Theorem 1.1 (Algorithm 3.3). There is a combinatorial procedure that we call the unlocking
procedure for computing the characters appearing in the total G-igsaw piece for an exceptional
curve in G-HilbA3. The input of this procedure is the data of Reid’s recipe and the combinatorics
of the exceptional fibre.
To briefly illustrate how the procedure works, we consider the example of G � 1
30p25, 2, 3q.
This notation means that G is the subgroup of SL3pCq generated by
g �
�
�ε25 ε2
ε3
�
,
Walls for G-Hilb via Reid’s Recipe 3
where ε is a primitive 30th root of unity. Crepant resolutions correspond to triangulations of
the simplex at height 1 – the “junior simplex” – with vertices in the lattice Z3�Z �
�
25
30 ,
2
30 ,
3
30
�
.
The triangulation for G-Hilb is shown in Fig. 1 along with Reid’s recipe. We often denote the
character χa defined by χapgq � εa by the integer a.
e1
e2e3
1
26
21
16
11
6
6
6
6
6
6
23
13
3
3
3
25
28
15
18
5
8
19
17
29
22
14
720
20
10
10
15
15
15
2
2
4
4
4
4
5 5
9
12
12
10
20
20
25
27
27
24
Figure 1. G-Hilb and Reid’s recipe for G � 1
30 p25, 2, 3q.
Throughout this paper we use the convention of ordering vertices as shown in Fig. 1. We will
demonstrate the unlocking procedure for the curve C shown on the left side of Fig. 2 marked
with the character 5; that is, the character taking g ÞÑ ε5. On the right side of Fig. 2 we illustrate
the unlocking procedure. Roughly, we consider all the curves (or edges) marked with 5, add
one character marking each divisor containing two such curves (or vertices between two edges
marked with 5), and finally add the characters appearing in G-igsaw pieces for certain curves
cohabiting a divisor with a curve marked with 5. In this case, the only such extra curve is
marked with 9 and the G-igsaw piece for this curve consists just of the character 9 itself. Some
recursion will be necessary in general to compute the smaller G-igsaw pieces of such curves.
It follows that the total G-igsaw piece for C has characters 5, 7, 9, 11. Observe that this total
G-igsaw piece only picked one of the two characters 7, 14 marking a divisor containing two
5-curves. We will elaborate in Section 3.1 how the unlocking procedure identifies which of the
two characters should be added.
Following [25], walls inside Θ are of various types denoted 0-III depending on the birational
geometry of the moduli spaces near the wall. Walls of Type I correspond to flops induced by
curves. By [10, Theorem 9.12], every flop in a single exceptional p�1,�1q-curve can be realised
by a wall-crossing of Type I directly from C0, which is very much not true for other resolutions;
see [10, Example 9.13]. Walls of Type III – that contract a divisor to a curve – arise from
exceptional p�2, 0q-curves corresponding to certain “boundary” edges in the triangulation for G-
Hilb. We will define this terminology in Section 2.2. The final possibility is that an exceptional
curve is a p1,�3q-curve that, if the corresponding inequality was irredundant, would produce
a wall of Type II where by definition a divisor is contracted to a point. [10, Proposition 3.8]
shows that there are in fact no Type II walls in Θ.
We denote the set of characters appearing in the total G-igsaw piece for an exceptional
curve C by G-igpCq. The following result is implied by [10, Corollary 5.2, Proposition 9.7 and
Theorem 9.12] and Theorem 1.1.
Proposition 1.2 (Propositions 4.1 and 4.2). Suppose C � G-HilbA3 is an exceptional curve
marked with character χ by Reid’s recipe. The inequality corresponding to C is given by
θpϕC0pOCqq �
¸
χPG-igpCq
degpRχ|Cqθpχq ¡ 0.
4 B. Wormleighton
5
11
75 5 5
9
Figure 2. Unlocking for a 5-curve.
If C is a p�1,�1q-curve then the necessary inequality corresponding to C that defines a Type I
wall of C0 is given by
θ
�
ϕC0pOCq
�
�
¸
χPG-igpCq
θpχq ¡ 0.
If C is a p1,�3q-curve then the inequality corresponding to C is given by
θ
�
ϕC0pOCq
�
� 2 � θ
�
χb2
�
�
¸
χPG-igpCqztχb2u
θpχq � 0.
In all three cases G-igpCq is computed by the unlocking procedure.
We can also use the unlocking procedure to compute inequalities that do not come from
exceptional curves. The other two kinds of inequality come from exceptional divisors. For each
character ψ marking a divisor, we obtain an inequality θpψq ¡ 0. The second kind of inequality
coming from divisors is more complicated.
Proposition 1.3 (Proposition 2.6). Suppose D1 is a (not necessarily prime) exceptional divisor
in G-HilbA3. Then any θ P C0 satisfies
θ
�
ϕC0pω
_
D1q
�
�
¸
C�D1
¸
χPG-igpCq
θpχq ¡ 0,
where C ranges over exceptional curves inside D1.
As a result of Propositions 1.2 and 1.3 we can immediately deduce the conclusion [10, Propo-
sition 3.8] for C0.
Corollary 1.4 (Corollary 4.3). C0 has no Type II walls.
We can similarly reprove [10, Theorem 9.12] by combinatorial means.
Proposition 1.5 (Proposition 4.4). Each flop in a p�1,�1q-curve in G-HilbA3 is induced by
a wall-crossing from C0.
We can use these formulae to show exactly which inequalities are necessary to define C0.
Theorem 1.6 (Theorem 4.17). Suppose G � SL3pCq is a finite abelian subgroup. The walls of
the chamber C0 for G-HilbA3 and their types are as follows:
� a Type I wall for each exceptional p�1,�1q-curve,
� a Type III wall for each generalised long side,
� a Type 0 wall for each irreducible exceptional divisor,
Walls for G-Hilb via Reid’s Recipe 5
� the remaining walls are of Type 0 coming from divisors as in Proposition 1.3. We discuss
which of these are necessary and how to reconstruct the divisor D1 in Section 4.7.
We will define the term “generalised long side” in Definition 4.12, which is an entirely com-
binatorial notion. We complete the description of wall-crossing behaviours from [25] by calling
a wall Type 0 if the corresponding contraction is a isomorphism.
This explicit and malleable description of the walls for C0 has applications to studying the
birational geometry of other crepant resolutions of A3{G. In forthcoming work [18], the author
and Y. Ito use this description of C0 to study the geometry of another Hilbert scheme-like
resolution introduced in [14] called the “iterated G-Hilbert scheme” or “Hilb of Hilb”.
2 Resolutions of A3{G
2.1 Setup
Let G � SL3pCq be a finite abelian subgroup. We will assume that G is cyclic for notational
simplicity during this preliminary section however we will shed this assumption from Section 3
onwards. We will denote by 1
r pa, b, cq the cyclic subgroup of SL3pCq generated by the matrix
g �
�
�εa εb
εc
�
,
where ε is a primitive rth root of unity. The first resolution of the quotient singularity A3{G
to be studied was the G-Hilbert scheme G-HilbA3, the fine moduli space of G-clusters: zero-
dimensional G-invariant subschemes Z � A3 with H0pOZq � CrGs as G-modules. G-Hilb was
shown to be smooth for abelian G by Nakamura [22], and subsequently shown to be smooth –
and hence a resolution – for all finite G � SL3pCq by Bridgeland–King–Reid [5].
From the work of King [19], Ito–Nakajima [16], and Craw [8] one can reinterpret G-Hilb as
a moduli space of quiver representations. The quiver in question is the McKay quiver with
vertices indexed by irreducible representations of G and the number of arrows between ρ and ρ1
defined to be
dim HomGpρ
1 b ρstd, ρq,
where ρstd is the standard representation of G acting on C3. We choose the dimension vector
d � pdim ρqρ and a stability parameter θ P Θ as defined above. We define Mθ :�MθpQ,dq to be
the fine moduli space of θ-stable representations of the McKay quiver with dimension vector d
subject to certain relations coming from the associated preprojective algebra. When θpρq ¡ 0
for all nontrivial ρ one has that MθpQ,dq � G-HilbA3.
It was apparent from [5] that their smoothness result and equivalence of categories (1.1) holds
for any generic θ and so one obtains potentially many different resolutions of the form MθpQ,dq
and also the corresponding equivalences of categories
Φθ : DbpMθq Ñ Db
G
�
A3
�
.
As discussed above, the stability space Θ has a wall-and-chamber structure such that any θ, θ1
from the same open chamber C � Θ produce isomorphic moduli spaces: Mθ � Mθ1 . For sim-
plicity we denote by MC and ΦC the moduli space and equivalence of categories for any generic
θ P C.
When G is abelian, each resolution MC is toric. Fix the lattice N � Z3 � Z �
�
a
r ,
b
r ,
c
r
�
. The
singularity A3{G is described by the cone σ � Conepe1, e2, e3q inside NR � N bZ R � R3
xx1,x2,x3y
6 B. Wormleighton
and crepant resolutions correspond to triangulations of the slice σXpx1�x2�x3 � 1q – usually
referred to as the “junior simplex” – such that the vertices of each triangle lies in N , and each
triangle is smooth (its vertices form a Z-basis of N). In pictures we will always draw only the
slice to produce two-dimensional pictures.
2.2 Reid’s recipe
Let us focus on the case MC � G-HilbA3. We will denote the universal G-cluster by Z and the
chamber of Θ corresponding to G-Hilb by C0. Craw–Reid [12] present an entertaining algorithm
to construct the triangulation for G-Hilb that, after commenting on some of the salient features,
we will use without comment.
We call the triangulation for G-Hilb the Craw–Reid triangulation. It divides the junior
simplex into so-called “regular triangles” of equal side length that fall into one of two cases:
� corner triangles, which have one of the vertices e1, e2, e3 of the junior simplex as a vertex,
� meeting of champions, for which none of the vertices of the junior simplex are vertices.
Craw–Reid show that there is at most one meeting of champions triangle (possibly of side
length zero, in which case it is a point). After dividing the junior simplex into such triangles,
one subdivides them further into smooth triangles as depicted in Fig. 3: the resulting unimodal
triangulation describes the resolution G-Hilb. We call line segments on the boundary of a regular
triangle boundary edges and the corresponding toric curves boundary curves.
Figure 3. A regular triangle and its triangulation.
In early versions of the McKay correspondence [23] one of the chief aims was to supply
a bijection from irreducible characters of G to a basis of cohomology on a crepant resolution.
This was explicitly computed for G-Hilb by Craw [9] when G is abelian using “Reid’s recipe”:
a labelling of exceptional subvarieties by characters of G. Reid’s recipe is one of the main tools
we will use to compute walls and so we will describe it in some detail.
An exceptional curve C in G-Hilb corresponds to an edge in the Craw–Reid triangulation,
which in turn corresponds to a two-dimensional cone in the fan for G-Hilb. A primitive normal
vector pα, β, γq to this cone defines a G-invariant ratio of monomials
xαyβzγ � m1{m2,
where x, y, z are eigencoordinates on C3 for G. Mark the curve C with the character by
which G acts on m1 (or m2). We define the χ-chain to be the collection of all exceptional curves
(or edges in the Craw–Reid triangulation) marked by the character χ. We say that a triangle
in the Craw–Reid triangulation is a χ-triangle if one of its edges is marked with the character χ.
After marking all curves, there is a procedure for labelling the compact exceptional divisors,
or interior vertices of the triangulation. Let D be such a divisor corresponding to a vertex v.
There are three cases:
Walls for G-Hilb via Reid’s Recipe 7
� v is trivalent: D � P2 and the three exceptional curves in D are all marked with the same
character χ. Mark D with χb2.
� v is 4- or 5-valent, or 6-valent and not inside a regular triangle: D is a Hirzebruch surface
blown up in valency-4 points. There are two pairs of exceptional curves in D each marked
with the same character χ and χ1. Mark D with χb χ1.
� v is 6-valent and lies in the interior of a regular triangle: D is a del Pezzo surface of
degree 6, and there are three pairs of exceptional curves each marked with the same
character χ, χ1, χ2. D has two G-invariant maps to P2, mark D by the two characters
arising from the monomials constituting these two maps. These two characters φ1, φ2
satisfy
χb χ1 b χ2 � φ1 b φ2.
For more detail see [9, Lemmas 3.1–3.4]. We will frequently refer to a curve or a divisor marked
with a character χ as a χ-curve or a χ-divisor.
Example 2.1. In Fig. 1 with G � 1
30p25, 2, 3q, the leftmost curve marked with the character 20
has normal p�2, 25, 0q giving the G-invariant ratio y25{x2. G acts on the numerator and denom-
inator by the character ε ÞÑ ε20, hence the marking. The divisor marked with 23 incident to
the previous curve marked with 20 has two pairs of curves with characters 20 and 3 and a fifth
curve with character 15. Thus, the divisor is correctly marked by 20� 3 � 23.
We refer to divisors of the first two types – that is, all divisors not isomorphic to a del Pezzo
surface of degree 6 – as Hirzebruch divisors, and to divisors isomorphic to a del Pezzo surface of
degree 6 as del Pezzo divisors. We ask the reader to have grace on the slight abuse of terminology
as P2 is also a del Pezzo surface. For a character χ marking a curve, we denote by Hirzpχq the
set of characters marking Hirzebruch divisors in the interior of the χ-chain and by dPpχq the set
of characters marking del Pezzo divisors in the interior of the χ-chain. We will often say “along
the χ-chain” in place of “in the interior of the χ-chain”.
2.3 G-igsaw pieces
Consider the G-clusters at torus-fixed points of G-Hilb, or triangles in the Craw–Reid trian-
gulation. The ideal defining such a cluster is a monomial ideal and one can draw a Newton
polygon in the hexagonal lattice Z3{Z � p1, 1, 1q to illustrate the monomial basis. An example
of a torus-fixed G-cluster for the group G � 1
6p1, 2, 3q is shown in Fig. 4. Notice that there is
exactly one monomial in each character space for G as desired.
y
yz 1 x
z xz
Figure 4. A G-cluster for G � 1
6 p1, 2, 3q corresponding to a torus-fixed point of G-Hilb.
The monomial ideal in Crx, y, zs defining this cluster is@
x2, y2, z2, xy
D
.
G-clusters corresponding to adjacent triangles separated by an exceptional curve C differ by
taking a subset of the monomials basing one G-cluster and moving them to other monomials
in the same character space; that is, multiplying by G-invariant ratios of monomials. This
process was studied in [22] and called a G-igsaw transformation.
8 B. Wormleighton
Definition 2.2. Let C � G-HilbA3 be an exceptional curve corresponding to the common
edge of two adjacent triangles τ , τ 1. Denote by Zτ , Zτ 1 the two G-clusters corresponding to the
torus-fixed points of G-HilbA3 for τ , τ 1. We call the set of characters labelling monomials in Zτ
(or in Zτ 1) that partake in the G-igsaw transformation the total G-igsaw piece for C and denote
it by G-igpCq.
We will often also refer to the set of monomials underlying G-igpCq for one of the triangles
either side of C as a total G-igsaw piece. There is a single monomial that divides all others
in the total G-igsaw piece, and this is the monomial in the G-cluster in the character space for
the character marking C. Indeed, an alternative definition of a total G-igsaw piece for C is the
set of all monomials in the G-graph for one of the triangles adjacent to C that are divisible by
the monomial in this eigenspace.
Example 2.3. We continue the example of G � 1
6p1, 2, 3q. The Craw–Reid triangulation and
Reid’s recipe for this group is shown in Fig. 5. The triangle labelled by � is the triangle
corresponding to the G-cluster from Fig. 4.
�
5
32
2
1
3
4
Figure 5. Triangulation and Reid’s recipe for 1
6 p1, 2, 3q.
Passing through the 4-curve C adjacent to the triangle � performs a G-igsaw transformation
with total G-igsaw piece centred on the monomial with character 4, which in this case is xz.
The G-igsaw transformation switches xz for y2 – since the G-invariant ratio for C is xz{y2 –
producing the new G-cluster
y2
y
yz 1 x
z
If we pass through the 2-curve bordering � then the total G-igsaw piece contains the monomials
y, yz and produces the G-cluster
1 x x2
z xz x2z
As the two total G-igsaw pieces for a given curve are related by multiplying by G-invariant
ratios it is clear that they each have the same set of characters represented by their monomials.
We denote the set of characters in either total G-igsaw piece for a curve C by G-igpCq. For
convenience we will also denote by χpmq the character by which G acts on a monomial m.
2.4 Tautological bundles
The sheaf R � π�OZ is locally free with fibre H0pOZq above Z P G-HilbA3. It splits into
eigensheaves
R �
à
χPIrrG
Rχ
Walls for G-Hilb via Reid’s Recipe 9
and these summands are called tautological bundles. Since G is abelian, the Rχ are line bun-
dles. [9] gives relations between these line bundles in the Picard group, which translate to
divisibility relations between eigenmonomials. For a triangle τ in the Craw–Reid triangulation,
denote the generator of Rχ on the affine piece corresponding to τ by rχ,τ . We usually omit
reference to τ so long as the context is clear
Theorem 2.4 ([9, Theorem 6.1]). The relations between pgenerators ofq tautological line bundles
are described by Reid’s recipe in the following way.
� If three lines marked with the same character χ meet at a vertex marked with ψ � χb2
then
r2χ � rψ.
� If four or five or six lines consisting of two pairs marked by characters χ, χ1 and zero
or one or two extra lines marked with further characters meet at a vertex marked with
ψ � χb χ1 then
rχ � rχ1 � rψ.
� If six lines consisting of three pairs marked by characters χ, χ1, χ2 meet at a vertex marked
with φ, φ1 then
rχ � rχ1 � rχ2 � rφ � rφ1 .
The claim is that these relations hold and generate all relations between tautological bundles.
We will make heavy use of these divisibility relations between eigenmonomials to study G-igsaw
pieces for exceptional curves.
As alluded to, the work of Logvinenko [20], Cautis–Logvinenko [7], and Craw–Cautis–Logvi-
nenko [6] categorifies Reid’s recipe via the tautological bundles. Many of the constructions in [6,
Sections 3–4] resemble constructions made in Section 3 below, however the computations they
make are for the inverse equivalence of (1.1) to that utilised in [10] and here.
Evident from [6, 10] and below, characters marking a divisor or a single curve are special.
They are termed “essential characters” and have been further examined in [11, 24].
2.5 Abstract inequalities for C0
In [10, Section 9] Craw–Ishii provide an abstract description of sufficiently many inequalities to
carve out the chamber C0. These inequalities arise from the linearisation map
LC0 : Θ Ñ PicpMC0q,
which takes θ to the ample Q-divisor on Mθ arising from GIT; note that we have identified
the Picard groups of different resolutions. By construction LC0pC0q � AmppMC0q and so one
obtains inequalities
θ
�
ϕC0pOCq
�
¡ 0
for all exceptional curves C � G-Hilb. If such an inequality is necessary to define C0, the
geometry of C determines the type of the corresponding wall as follows:
� If C is a p�1,�1q-curve – that is, it corresponds to an interior edge inside a regular
triangle – then pθpϕC0pOCqq � 0q X C0 is a Type I wall.
10 B. Wormleighton
� If C is a p1,�3q-curve – that is, it corresponds to one of the edges incident to a trivalent
vertex – then pθpϕC0pOCqq � 0q X C0 is a Type II wall.
� If C is contained in a Hirzebruch divisor but it is not in either of the previous cases, then
pθpϕC0pOCqq � 0q X C0 is a Type III wall.
One can express the inequality θpϕCpOCqq ¡ 0 abstractly via [10, Corollary 5.2], a consequence
of which is
θ
�
ϕCpOCq
�
�
¸
ρ
degpRρ|Cqθpρq.
Any character ρ not in G-igpCq has Rρ|C � OC and so it doesn’t appear in the sum above.
It follows that
θ
�
ϕCpOCq
�
�
¸
ρPG-igpCq
degpRρ|Cqθpρq.
To complete the description by Craw–Ishii, their remaining inequalities – which if they are
necessary inequalities will define walls of Type 0 – are obtained from divisors in two ways outlined
in Lemma 2.5 and Proposition 2.6.
Lemma 2.5 ([10, Corollary 5.6 and Theorem 9.3]). Suppose that D � G-HilbA3 is an irreducible
exceptional divisor marked with a character ψ. Then all θ P C0 satisfy the inequality
θ
�
ϕC0
�
R�1
ψ |D
��
� θpψq ¡ 0.
Moreover, the inequalities of this form are necessary and hence define walls of C0.
We call the inequalities coming from Lemma 2.5 subsheaf inequalities. The reason for this is
the following. Suppose that OZ is a G-cluster parameterised by a point in an irreducible excep-
tional divisor D � G-Hilb. Let ψ be a character marking D. It follows from [10, Corollary 4.6
and Lemma 9.1] that rψ is in the socle of H0pOZq and is constant on D, hence defines a subsheaf
S � O0 bψ of OZ for all Z P D. By definition we must have θpSq � θpψq ¡ 0 for all θ P C0 and
[10, Proposition 9.3] shows that this is indeed a wall of C0.
There is a dual version of this using quotients instead of subsheaves. [10, Lemma 5.7 and
Theorem 9.5] shows that
θ
�
ϕC0pωD1q
�
0
for all θ P C0 and for every possibly reducible but connected exceptional divisor D1. From [10,
Theorem 9.5] this inequality corresponds to evaluating θ on the minimal rigid quotient Q of OZ
for all Z P D1.
Proposition 2.6. Suppose D1 � G-HilbA3 is a possibly reducible but connected exceptional
divisor. Then all θ P C0 satisfy the inequality¸
C�D1
¸
χPG-igpCq
θpχq ¡ 0.
We call the inequalities from Proposition 2.6 quotient inequalities.
Proof. For the sheaf Q to be trivial on D1 it means that Rρ|D1 is trivial for all ρ � Q. Equi-
valently, all torus-invariant G-clusters in D1 share the same eigenmonomial rρ for each ρ � Q
or, also equivalently, ρ R G-igpCq for any C � D1. Reversing the inequality θpQq 0 gives
θpCrGs{Qq ¡ 0 and CrGs{Q contains exactly the characters in the statement of the result
since G is abelian and so all irreducible representations have multiplicity 1 in CrGs. �
The value of Proposition 2.6 is that it reduces computing inequalities from divisors to com-
puting various total G-igsaw pieces, which is the topic of the next section.
Walls for G-Hilb via Reid’s Recipe 11
3 Computing characters in total G-igsaw pieces
Our motivating question for this section is the following: how can one determine the characters
that appear in a total G-igsaw piece for a curve purely from the data of Reid’s recipe? As we
shall see, the answer depends somewhat on how C sits inside G-Hilb though it is still completely
combinatorial.
3.1 Combinatorial definitions
We start by making some combinatorial definitions. For two points u, v P R3 we denote by
ru, vs, pu, vq, ru, vq, pu, vs the closed, open, and half-open line segments with endpoints u and v.
For example, u P ru, vq but v R ru, vq.
Let C be a p�1,�1q-curve marked with χ. Let α be the edge corresponding to C in the
Craw–Reid triangulation. Pick a point u0 in the interior of α. We will use u0 to view the
χ-chain as an (embedded) quiver Ξχ,u0 as follows. Let the vertices of the quiver be the vertices
corresponding to Hirzebruch divisors incident to the χ-chain (including those at the ends if
applicable) and include one extra vertex corresponding to u0. Let the edges be the parts of the
χ-chain between these divisors with the edge containing α split into two with one on either side
of u0. We orient an edge β � Ξχ,u0 by declaring that the tail is the boundary vertex of β closest
to u0. Note that the χ-chain is a tree and so this makes sense. We show an example of Ξχ,u0
for the group G � 1
30p25, 2, 3q with the character χ3 and the point � in Fig. 6. Note that as
an abstract quiver Ξχ,u0 depends only on the edge α or, equivalently, the curve C and so we
slightly abuse notation by subsequently denoting it ΞC .
�
Figure 6. The embedded quiver Ξχ3,� � ΞC1
.
Definition 3.1. Let C be a χ-curve. We say that a divisor D along the χ-chain is downstream
of C if it is a Hirzebruch divisor. We say that a p�1,�1q-curve E incident to such a divisor D
is downstream of C if the edge for E meets the tail of the arrow in ΞC corresponding to the
χ-curve in the same regular triangle as E.
This is illustrated schematically in Fig. 7 and concretely on the left side of Fig. 10 for the
3-curve C1 inside 1
30p25, 2, 3q-Hilb whose edge includes the point � from Fig. 6. In Fig. 7 the
bold edges indicate the boundary of a regular triangle, hence the central vertex is a Hirzebruch
12 B. Wormleighton
divisor, and only the two dotted curves are downstream according to the given orientation of
the χ-chain. In Fig. 10 the curves downstream of C1 are dotted.
χ
χ
Figure 7. Schematic of curves downstream from a p�1,�1q-curve.
Now suppose that C is a boundary curve marked with χ. By the construction of the Craw–
Reid triangulation there is a vertex ei of the junior simplex and an interior vertex v of the
junior simplex such that the edge for C is contained in the line segment rei, vs. Let vC be the
vertex furthest from ei such that this is true. At vC the χ-chain will change slope. There exists
a vertex v1C such that all curves in the χ-chain between vC and v1 are p�1,�1q-curves and hence
lie in the interior of various regular triangles. Note that vC � v1C is possible, in which case
there are no p�1,�1q-curves along the χ-chain. At v1C there are two possibilities for the χ-chain:
either the χ-chain terminates, or it continues into a line segment rv1C , ejs from v1C to a vertex ej
of the junior simplex where it then terminates. We illustrate the situation where the χ-chain
terminates at ej in Fig. 8. The dashed segment of the χ-chain represents the part between vC
and v1C , which consists of p�1,�1q-curves, and the vertices shown there correspond to Hirzebruch
divisors, which occur where the χ-chain passes between two different regular triangles.
ei
vC
v1C
ej
v1
v2
C
Figure 8. χ-chain for a boundary curve.
Similarly to the case where C was a p�1,�1q-curve, we create an embedded quiver ΞC
supported on the part of the χ-chain between the vertices v2 and v1C . The vertices are the
Hirzebruch divisors incident to this part of the χ-chain and the edges are the parts of the χ-
chain between these divisors. We orient the edges by declaring that v2 is the unique source of
the quiver and v1C is the unique sink.
We use the notation of Fig. 8 in the following definitions.
Definition 3.2. Let C be a boundary curve as depicted in Fig. 8. We say that a Hirzebruch
divisor D is downstream of C if either:
� the vertex for D lies in rv2, vCs,
� the vertex for D lies in the part of the χ-chain between vC and v1C (excluding v1C).
Walls for G-Hilb via Reid’s Recipe 13
Let E be a p�1,�1q-curve marked with a character ρ and contained in a Hirzebruch divisor D
downstream of C. We say that E is downstream of C if either:
� the vertex for D lies in the line segment rv2, vCs,
� the vertex for D lies between vC and v1C and the edge for E meets the tail of the arrow
in ΞC corresponding to the χ-curve in the same regular triangle as E,
and if either
� the ρ-chain terminates at D,
� the edges in the ρ-chain incident to D have different slopes.
We show a schematic for the downstream curves relative to C in Fig. 9. The bold arrows
represent ΞC and the additional edges correspond to sides of the various regular triangles that
the χ-chain passes through. The dotted edges are the curves downstream of C, and the dashed
edges represent two p�1,�1q-curves marked with the same character and whose edges have the
same slope, hence the dashed curves are not downstream of C.
ei
vC
v1C
ej
v1
v2
C
Figure 9. Schematic of curves downstream of a boundary curve.
Note that in Fig. 8 the divisors for v2 and vC are downstream of C but the divisor for v1 is
not. On the right of Fig. 10, when G � 1
30p25, 2, 3q the divisors D2 and D3 are all the divisors
downstream of the 15-curve C2 whereas D1 is the only divisor along the 15-chain that is not.
We have bolded the sides of regular triangles in the triangulation to make it clear that C2 is
a boundary curve, and to clarify which divisors are Hirzebruch divisors. We also show the curves
downstream of C2 dotted.
C1
C2
D3
D2
D1
Figure 10. Downstream curves and divisors in 1
30 p25, 2, 3q-Hilb.
14 B. Wormleighton
For any exceptional curve C and a Hirzebruch divisor D downstream of C, we denote the set
of curves in D downstream of C by CCpDq.
We will say that the ρ-chain for some character ρ is broken at a vertex v (or the corresponding
divisor) if either the ρ-chain terminates at v or if the edges in the ρ-chain incident to v have
different slopes as in the second part of Definition 3.2.
Lastly, we define a character χdPpC,Dq associated to a χ-curve C and a del Pezzo divisor D
along the χ-chain. Let ∆ be the regular triangle containing the vertex for D, let v be the vertex
corresponding to D, and let α be the edge of the triangulation corresponding to C.
Let tp, q,mu � t1, 2, 3u. We denote x1 � x, x2 � y, x3 � z for convenience. We assume that
∆ is a corner triangle with em as vertex and one side coming from a ray out of ep; we will treat
the meeting of champions case shortly. Here φ1, φ2 denote the characters marking the del Pezzo
divisor at v, and a, b, c, d, e, f are positive integers coming from the edges in the Craw–Reid
triangulation defining out ∆. More precisely, the two sides incident to em have the ratios xdp : xbq
and xeq : xap marking them, and the side coming from a ray out of ep has ratio xfm : xcq. We denote
by r � f the side length of the regular triangle. Each of the indices i, j, k ranges from 0, . . . , r.
Consider the local picture for p � 1, q � 2, m � 3 shown in Fig. 11 adapted from the proof of
[9, Theorem 6.1], specifically [9, Fig. 12], for eigenmonomials near v inside ∆.
Generators for Rφ1
xd�i : yb�izi
ye�j : xa�jzj
zf�k : xkyc�k
ye�jzi
ye�jzi
xa�jzf�k
xa�jzf�k
xd�iyc�k
xd�iyc�k
Generators for Rφ2
xd�i : yb�izi
ye�j : xa�jzj
zf�k : xkyc�k
xkye�j
yb�izf�k
yb�izf�k
xd�izj
xd�izj
xkye�j
Figure 11. Generators for tautological bundles near v.
Suppose χ � χ
�
xd�ip
�
; that is, if p � 1, q � 2, m � 3 then χ marks the horizontal chain
of curves in Fig. 11. Following [9] we denote by eivej the convex part of the junior simplex
enclosed by the line segments from ei to v and from ej to v. Define
χdPpC,Dq :�
#
χb χ
�
xjm
�
if α � epveq,
χb χ
�
xc�kq
�
if α � epvem.
Observe that α � eqvem is not a possibility from considering slopes. Similarly, if χ � χ
�
xe�jq
�
define
χdPpC,Dq :�
#
χb χ
�
xim
�
if α � eqvep,
χb χ
�
xkp
�
if α � eqvem,
Walls for G-Hilb via Reid’s Recipe 15
and if χ � χ
�
xf�km
�
define
χdPpC,Dq :�
#
χb χ
�
xa�jp
�
if α � emveq,
χb χ
�
xb�iq
�
if α � emvep.
When ∆ is a meeting of champions triangle with side ratios xd : yb, ye : zc, xa : zf we make
slight modifications to the above as follows. In this setting, when χ � χ
�
xd�i
�
define
χdPpC,Dq :�
#
χb χ
�
zj
�
if α � e1ve2,
χb χ
�
yk
�
if α � e1ve3.
If χ � χ
�
ye�j
�
define
χdPpC,Dq :�
#
χb χ
�
zi
�
if α � e2ve1,
χb χ
�
xa�k
�
if α � e2ve3,
and if χ � χ
�
zf�k
�
define
χdPpC,Dq :�
#
χb χ
�
xj
�
if α � e3ve2,
χb χ
�
yb�i
�
if α � e3ve1.
We remark that it follows from Case 4 of the proof of [9, Theorem 6.1] that χdPpC,Dq is one
of the characters marking D, and is moreover the unique such character φ with rχ | rφ. It also
follows from the construction that χdPpC,Dq takes the same value on χ-curves in each of the
two connected components of the χ-chain minus the vertex v.
3.2 Unlocking procedure
In this subsection we outline the algorithm that we use to compute G-igpCq. We will spend the
remainder of this section proving its validity.
Algorithm 3.3 (unlocking procedure). Input: An exceptional curve C � G-HilbA3 marked
with a character χ by Reid’s recipe.
Ch Let S � tχu.
dP For each del Pezzo divisor D along the χ-chain, add χdPpC,Dq to S.
H1 For each Hirzebruch divisor along the χ-chain, add the character marking it to S.
Re For each Hirzebruch divisor D downstream of C and for each E P CCpDq, compute G-igpEq
by running the unlocking procedure with E as input.
H2 For each Hirzebruch divisor D downstream of C, add the characters in
�
EPCCpDqG-igpEq
to S.
Output: G-igpCq � S.
We call this the unlocking procedure as passing through a Hirzebruch divisor “unlocks” simpler
G-igsaw puzzles for the curves E downstream of C that one recursively solves in the step Re
and then feeds into the total G-igsaw piece for C. It can be visualised as a flow through the
triangulation emanating from the curve C with preferred paths defining its tributaries. We note
that the convoluted definition of χdPpC,Dq is only important for explicit calculations and not
for qualitative discussion; the step dP states that G-igpCq contains exactly one of the characters
16 B. Wormleighton
marking each del Pezzo divisor along the χ-chain. We will often refer to curves downstream of C
as being “unlocked” by C.
As an example use case, if G-Hilb has a meeting of champions of side length 0 with the three
champions marked with a character χ then for any curve C along the χ-chain the characters
in the G-igsaw piece are given by the unlocking procedure applied to the branch of the χ-chain
that C lies on, combined with all the characters from (Hirzebruch) divisors along the other two
branches of the χ-chain. We will see an example of this in Section 3.8.
3.3 Monomials for divisors
We will begin by relating the characters marking divisors along the χ-chain to G-igsaw pieces
for χ-curves.
Lemma 3.4. Suppose C is a χ-curve. Then G-igpCq includes exactly one character from each
divisor that is along the χ-chain. Moreover, if D is a del Pezzo divisor along the χ-chain then
χdPpC,Dq is the character marking D that appears in G-igpCq.
That is, G-igpCq contains the characters marking each Hirzebruch divisor along the χ-chain
and precisely one of the two characters marking each del Pezzo surface along the χ-chain.
Proof. Let D be a Hirzebruch divisor along the χ-chain marked with a character ψ. Cases 2–3
of the proof of [9, Theorem 6.1] give that Rψ has degree 1 on a given χ-curve, and hence it
follows that rχ | rψ. It follows that any G-igsaw piece featuring rχ – such as a G-igsaw piece
for C – will also feature each rψ and so ψ P G-igpCq. Now let D be a del Pezzo divisor along the
χ-chain marked with characters φ1, φ2. It follows from Case 4 of the proof of [9, Theorem 6.1]
that exactly one of Rφ1 , Rφ2 has degree 1 on a given χ-curve and so rχ | rφ1 or rχ | rφ2 but
not both. It follows that exactly one of φ1, φ2 lie in G-igpCq. Let φ � χdPpC,Dq P tφ1, φ2u.
As noted above, it follows from Case 4 of the proof of [9, Theorem 6.1] that rχ | rφ and so rφ is
the unique character marking D that appears in G-igpCq. �
Lemma 3.4 gives an effective way of finding the characters in G-igpCq coming from divisors.
However, this does not usually supply all characters in G-igpCqztχu.
3.4 Counting characters
Our method for showing that Algorithm 3.3 is valid for a curve C is to source many characters
from divisors (as discussed in the previous subsection) and from curves (coming next) that
feature in G-igpCq and to then count how many characters are actually in G-igpCq to verify that
all characters in the total G-igsaw piece have been located. To move towards this second aim
we cite a lemma of Craw–Ishii.
Lemma 3.5 ([10, Lemma 9.1]). A character χ marks a torus-invariant compact divisor D �
G-HilbA3 iff rχ is in the socle of every G-cluster corresponding to a torus-fixed point in D.
Select a p�1,�1q-curve C marked with χ. This lies in two del Pezzo divisors from the
endpoints of the corresponding line segment. From Lemma 3.4 we see that rχ divides exactly two
of the monomials in the character spaces labelling these two divisors. Suppose τ is a χ-triangle
neighbouring C. By the shape of the ratios in Fig. 11 we can assume that rχ is not a power
of a single variable. The Unique Valley Lemma [22, Lemma 3.3] of Nakamura implies that rχ
divides exactly two elements of the socle of the torus-invariant G-cluster Zτ corresponding to τ .
Lemma 3.5 implies that the elements in the socle of Zτ that rχ divides correspond exactly to
these two characters labelling the neighbouring del Pezzo divisors. These are the outermost
Walls for G-Hilb via Reid’s Recipe 17
monomials in the G-igsaw piece for C on τ , so that knowing them will allow us to count how
many characters appear in G-igpCq.
Using this observation we will first prove the validity of the unlocking procedure for curves
inside regular triangles (i.e. those able to define flops, or p�1,�1q-curves) before justifying the
procedure for the other exceptional curves.
3.5 p�1,�1q-curves
We consider four cases covering all p�1,�1q-curves in G-Hilb based on the different ratios
labelling edges in Fig. 11. For this subsection denote x1 � x, x2 � y, x3 � z. We will use indices
tp, q, ru � t1, 2, 3u to symmetrise the discussion. A pp, qq-triangle is an ep-corner triangle with
one edge coming from a straight line out of eq.
� Type Ix: curves in the interior of a pr, pq-triangle with ratios xd�ip : xb�iq xir.
� Type Iy: curves in the interior of an pr, pq-triangle with ratios xe�jq : xa�jp xjr.
� Type Iz: curves in the interior of an pr, pq-corner triangle with ratios xf�kr : xkpx
c�k
q .
� Type Ic: curves in the interior of the meeting of champions triangle (if existent).
We will treat each of these types of p�1,�1q-curves but will specialise to the case p � 1,
q � 2, m � 3, which suffices to cover all possibilities by symmetry. Fix a p3, 1q-triangle ∆.
3.5.1 Type Iy curves
We consider the edges in the interior of ∆ marked with ratios of the form ye�j : xa�jzj ; that
is, of Type Iy. The analysis from Section 3.1 gives a precise description of the socle of the
G-clusters corresponding to nearby torus-invariant points as depicted in Fig. 12 and hence we
identify the total G-igsaw pieces for such χ-curves. The only additional calculation required is
of the monomials rφ0 and rφm for the characters at the endpoints. Consider rφm . The ratio
marking the side of ∆ containing the vertex marked with φm is zf : yc. It follows that yc | rφm
on ∆ but then it cannot be the case that rχ divides rφm since rχ � xa�jzj for some basic
triangles in ∆. It follows from Section 2.2 that rφm is given by xa�jyc in the basic triangle
where it is displayed in Fig. 12. A similar argument applies to compute rφ0 .
φ0
φ1
φ2
φ3
φ4
. . . φm�2
φm�1
φm
rφ0 � ybzj
rφ1 � xa�jzj�1
rφ2 � xd�1zj
rφ3 � xa�jzj�2
rφ4 � xd�2zj
rφ5 � xa�jzj�3
rφm�3 � xd�pf�j�2qzj
rφm�2 � xa�jzf�1
rφm�1 � xd�pf�j�1qzj
rφm � xa�jyc
Figure 12. Generators of eigenspaces along a χpxa�jzjq-chain inside a regular triangle.
Lemma 3.6. A total G-igsaw piece for a χ-curve of Type Iy on a χ-triangle chosen so that
in the coordinates used above rχ � xa�jzj is
rχ xrχ . . . xf�i�j�1rχ
zrχ
. .
.
zirχ
18 B. Wormleighton
where the curve corresponds to the edge whose endpoints are the intersection of the χ-chain with
the lines marked with xd�i : yb�izi and xd�i�1 : yb�i�1zi�1. Moreover, the χ-chain does not
continue outside of this regular triangle. In particular, Hirzpχq � H.
Proof. The calculation of the total G-igsaw piece follows immediately from the description of
the eigenmonomials in Fig. 12. As noted the χ-chain cannot continue outside of this regular
triangle since neither rφ0 nor rφm are divisible by rχ and so Theorem 2.4 implies that there
cannot be two edges marked with χ incident to either boundary vertex. �
Notice that this means that there are f � j � 1 characters to account for, excluding χ. But
this is exactly the number of del Pezzo surfaces along the χ-chain, each of which contributes one
character.
Corollary 3.7. For an exceptional curve C of Type Iy G-igpCq consists exactly of χ and the
characters χdPpC,Dq for each del Pezzo divisor D along the χ-chain.
Observe that this is a situation in which there is no recursion necessary since Hirzpχq � H.
This is one of the base cases that we will reduce to.
3.5.2 Type Ix curves
Suppose now that C is a χ-curve inside ∆ that is marked with the ratio xd�i : yb�izi; that is, C
is of Type Ix. [9, Theorem 6.1] yields the identities in Fig. 13 for eigenmonomials on triangles
neighbouring the χ-chain, which allow us to completely describe G-igsaw pieces inside regular
triangles. In the following we continue the notation of Fig. 11 and let κ � r � pi� 1q.
φ0
φ11
φ21
φ12
φ22
. . . φ1κ
φ2κ
φm
rφ0
� xazi
rφ1
1
� xd�iyc�κ rφ2
1
� xd�iz
rφ1
2
� xd�iyc�κ�1
rφ2
2
� xd�iz2
rφ5
� xd�iyc�κ�2
rφ2
κ�1
� xd�izκ�2
rφ1
κ
� xd�iyc�1
rφ2
κ
� xd�izκ�1
rφm � xd�iyc
Figure 13. Generators of eigenspaces along a χ-chain inside a regular triangle.
Lemma 3.8. The G-igsaw piece for a χ-curve C of Type Ix on a χ-triangle chosen so that
in the coordinates used above rχ � xd�i is
yc�k�1rχ
. . .
yrχ
rχ
zrχ
. .
.
zjrχ
where C corresponds to the edge whose endpoints are the intersection of the χ-chain with the
lines marked with ye�j : xa�jzj and ye�j�1 : xa�j�1zj�1, and where i � j � k � r. Moreover,
the χ-chain continues to the right and does not continue to the left of Fig. 13.
Walls for G-Hilb via Reid’s Recipe 19
Proof. The same argument as for Lemma 3.6 applies, except that rχ does divide rφm and so
by Theorem 2.4 the χ-chain must continue past the rightmost vertex. �
Notice that the only characters in any such G-igsaw piece that are unaccounted for by divisors
along the χ-chain in the same regular triangle are those for the monomials
yrχ, . . . , y
crχ
though ycrχ � rφm , which we have seen corresponds to a Hirzebruch divisor appearing along
the χ-chain.
Lemma 3.9. Suppose C is a χ-curve of Type Ix such that the χ-chain continues into a boundary
edge of a corner triangle. Then G-igpCq consists of χ, one character from every del Pezzo divisor
along the χ-chain, and the characters marking Hirzebruch divisors along the χ-chain.
This follows since the corner triangle has side length c and so there are exactly c Hirzebruch
divisors along the boundary part of the χ-chain that contribute the remaining c characters to
the G-igsaw piece. We say that the curves from Lemma 3.9 are of Type Ixb. This is the other
base case to which the unlocking procedure reduces. Note that the character in G-igpCq from
a del Pezzo divisor D along the χ-chain is by definition χdPpC,Dq.
We consider the remaining possibilities where the χ-chain merges into the interior of an e2-
corner triangle or the interior of an e1-corner triangle.
Lemma 3.10. Suppose C is a χ-curve of Type Ix and suppose that the χ-chain continues into
the interior of an e2-corner triangle ∆1. Then G-igpCq consists of χ, one character from each
del Pezzo surface along the χ-chain, the character marking the Hirzebruch divisor D between
the two regular triangles, and the characters from the total G-igsaw piece of the Iy curve also
incident to D inside ∆1.
As above, the character in G-igpCq from a del Pezzo divisor D along the χ-chain is χdPpC,Dq.
Proof. Let C 1 be the Type Iy curve incident to D in ∆1. Denote its character by χ1. From
Lemma 3.6 the characters in the total G-igsaw piece for C 1 are χ1 and one character from each
del Pezzo divisor along the χ1-chain inside ∆1. Let the ratios marking sides of ∆1 be
xd
1
: zb
1
, ze
1
: xa
1
, zf : yc,
where a1 d1 and where zf : yc marks the common side with ∆. Let the part of the χ-chain
in ∆1 be marked by the ratio xd
1�i1 : zb
1�i1yi
1
and so d1 � i1 � d� i. It follows that the χ1-chain
is marked by ze
1�j1 : xa
1�j1yj
1
where i1 � j1 � f . Using the relation d1 � a1 � c we see that
a1 � j1 � d� i.
Examining the situation explicitly, we see that on the lower χ-triangle neighbouring C one
has rχ � xd�i and rχ1 � xa
1�j1yj
1
so that rχ | rχ1 near C. Moreover, one can see that the zone
where rχ1 divides one character from each del Pezzo divisor along the χ1-chain includes this χ-
triangle containing C and so these divisibility relations remain. Hence, the G-igsaw piece for C 1
is contained in the G-igsaw piece for C. The divisibility relations are depicted in Fig. 14.
From Lemma 3.8 the total G-igsaw piece for C is missing c characters after counting the
characters in ∆. There are c�i1 new characters along the χ-chain corresponding to the del Pezzo
divisors along the χ-chain and the boundary Hirzebruch divisorD. There are c�pc�i1q�1 � i1�1
divisors along the χ1-chain, making a contribution of i1 characters in total including χ1 itself.
Thus these account for all of the c missing characters. �
20 B. Wormleighton
χ χ1
rχ � xd�i
rχ1 � xd�iyj
1
rχ | rχ1 | rφ1
i
φ11, φ
2
1
. . .
φ1i , φ
2
i
Figure 14. Unlocking for a Type Ix curve merging into a p2, 1q-triangle.
Note that this vindicates the unlocking procedure for such curves, where only one recursion
was required to unlock the single Type Iy curve downstream of C. The final case to consider is
when the χ-chain merges into an e1-corner triangle.
Suppose the χ-chain passes through n e1-corner triangles before entering an e2-corner tri-
angle ∆1.
Let the ratio xdm : ybm mark the edge opposite e1 for the mth e1-corner triangle ∆m from the
left and so ∆m has side length dm. Suppose the χ-chain enters ∆m at height im. This means
that the χ-chain picks up dm�im divisors from del Pezzo divisors and a single Hirzebruch divisor
inside ∆m. From analysing local divisibility relations as above, it is clear that rχ divides all
of the monomials in the G-igsaw pieces for the Type Ix curve incident to the χ-chain and the
leftmost Hirzebruch divisor inside each of these regular triangles. See Fig. 15 for a schematic.
We denote Dm :�
°m
q�1 dq and BDm :�
°m
q�1pbq � dqq.
C
χpxb1�i�1zf�1q
χpzf�1q
χpzf�d1�1q
χpxb1�i�1zf�d1�1q
χpzf�Dn�1�1q
χpxbn�1�i�1zf�Dn�1�1q
zf : yc zf�d1 : yc�pb1�d1q zf�Dn : yc�BDn
Figure 15. Unlocking for a Type Ix curve in a series of e1-corner triangles.
By computing the characters on the nearby del Pezzo divisor, one can tell that these Type
Ix curves each have bm � im characters in their G-igsaw pieces, making the total number of
characters they contribute to the G-igsaw piece of C
ņ
q�1
pdq � iq � bq � iqq �
ņ
q�1
pbq � dqq.
Walls for G-Hilb via Reid’s Recipe 21
From [9, Section 2] the ratios marking the edges from e1 for the e1-corner triangles are of the
form
zf�
°m
q�1 dq : yc�
°m
q�1pbq�dqq for m � 0, . . . , n
with the last edge marked by zf�
°n
q�1 dq : yc�
°n
q�1pbq�dqq. In particular, this means that the ∆1
has side length c�
°n
q�1pbq � dqq. Assume the χ-chain continues into a chain of Type Ix curves
in ∆1. By the same reasoning as for Lemma 3.10 this produces c�
°n
q�1pbq�dqq new characters
in G-igpCq coming from ∆. But then we have
ņ
q�1
pbq � dqq � c�
ņ
q�1
pbq � dqq � c
characters in total so far, which exhausts all characters in G-igpCq by Lemma 3.8. Hence in this
case ∆1 is the rightmost regular triangle containing χ-curves. Note that the χ-chain cannot merge
into a chain of Type Iy curves in ∆1 as such curves cannot escape a single regular triangle. Also,
it is clear from convex geometric considerations that the χ-chain cannot continue into a chain
of Type Iz curves. The only remaining option is that the χ-chain continues into a chain of
boundary curves, thus again producing c �
°n
q�1pbq � dqq new characters from the Hirzebruch
divisors along the side of ∆1. In either case the number of characters coming from ∆1 is exactly
the number of characters in G-igpCq not accounted for by del Pezzo divisors in ∆ by Lemma 3.8.
This completes the proof of validity of the unlocking procedure for curves of Type Ix.
3.5.3 Type Iz curves
The third type of curve occurring inside regular triangles is Type Iz: the curves marked by
ratios of the form zf�k : xkyc�k in the coordinates we have been using for an e3-corner triangle.
We repeat the G-igsaw analysis for these curves, represented in Fig. 16 with the χ � χ
�
zf�k
�
-
chain dashed.
rχ - rφ1
rχ - rφ1
rχ | rφ1
rχ | rφ1
rχ | rφ1
rχ | rφ1
rχ | rφ2
rχ | rφ2
rχ | rφ2
rχ - rφ2
rχ - rφ2
rχ | rφ2
Figure 16. Divisibility relations near v.
As in all previous cases, exactly one character marking each incident del Pezzo surface has
a monomial divisible by rχ and so we can pin down the socle and hence the G-igsaw piece for
such a curve.
22 B. Wormleighton
Lemma 3.11. The G-igsaw piece for a p�1,�1q-curve marked with χ on a χ-triangle chosen
so that in the coordinates used above rχ � zf�k is
yb�i�1rχ
. . .
yrχ
rχ xrχ . . . xd�i�krχ
where the curve corresponds to the edge whose endpoints are the intersection of the χ-chain with
the lines marked with xd�i : yb�izi and xd�i�1 : yb�i�1zi�1.
This means that there are b � d � k characters in the G-igsaw piece for such a Iz curve.
We shift notation to match the setup of the final case for Type Ix curves shown in Fig. 15.
In particular, we assume our Type Iz curve C lies in an e1-corner triangle. Suppose it lies in the
mth triangle from the left. From considering local divisibility relations near Hirzebruch divisors
along the χ-chain this implies that C unlocks m�1 Type Iy curves to the left and n�m Type Ix
curves to the right. From the calculations for Type Ix curves, the n �m Type Ix curves each
feature bq� iq characters in their G-igsaw pieces. From a similar calculation, one can verify that
the Type Iy curves contain iq characters in their G-igsaw pieces. These unlocked curves thus
contribute
m�1̧
q�1
iq �
ņ
q�m�1
pbq � iqq �
ņ
q�m�1
bq �
ņ
q�1
iq � im
characters to G-igpCq. The part of the χ-chain in the e3-corner triangle studied in the pre-
vious case contributes f � i0 characters, and the part in the e2-corner triangle contributes
c�
°n
q�1pbq � dqq. If i0 �� 0 then we unlock another Iy curve with i0 characters appearing in its
G-igsaw piece. If i0 � 0 then the χ-chain continues along the boundary of an e3-corner triangle,
contributing f characters. In either case there are f characters coming from the e3-corner tri-
angle. Lastly, there are
°n
q�1pdq � iqq del Pezzo and Hirzebruch divisors along the part of the
χ-chain inside e1-corner triangles, giving in total
floomoon
e3-corner
�
ņ
q�m�1
bq �
ņ
q�1
iq � imlooooooooooooomooooooooooooon
unlocked curves
�
ņ
q�1
pdq � iqqlooooomooooon
e1-corner
� c�
ņ
q�1
pbq � dqqloooooooomoooooooon
e2-corner
� f � c�
m̧
q�1
bq � im
characters. Compare to the quantity b� d� k in Lemma 3.11, which in these coordinates is
c�
m̧
q�1
pbq � dqq � f �
m̧
q�1
dq � im � f � c�
m̧
q�1
bq � im
showing that every character in G-igpCq is accounted for.
3.5.4 Type Ic curves
As in [9] the case of curves whose chains meet the interior of a meeting of champions triangle only
requires minor notational changes for the arguments above to carry over verbatim. For brevity
we omit it.
Walls for G-Hilb via Reid’s Recipe 23
3.6 Boundary curves
Suppose now that C is a curve lying on the boundary of a regular triangle. We will see that the
unlocking procedure computes G-igpCq by a similar argument to the case of p�1,�1q-curves.
We will again use neighbouring divisors to compute the socle and hence the total G-igsaw piece
for C, and then assess local divisibility relations to evidence that all these characters come from
the subvarieties in the unlocking procedure. We will sketch the novel elements of the proof
below.
Choose coordinates so that C lies along a straight line from e1. Assume for the moment that
the edge for C is actually incident to e1.
Suppose that D is a Hirzebruch divisor along the χ-chain. If D is at the boundary of two e1-
corner triangles or an e1-corner triangle and a meeting of champions – as shown in Fig. 17 – then
one can check that rχ divides the monomials in the G-igsaw pieces for the Type Iy curves C3
and C4.
D
χ
χ
C1 C2
C3 C4
Figure 17. D bordering two e1-corner triangles or meeting of champions.
Suppose now that D borders an e2- and an e3-corner triangle, or an e1-corner triangle and
an e3-corner triangle. We illustrate this situation in Fig. 18, along with some of the ratios
marking curves.
D
χ
χ
C1 C2
C3 C4
D
C3 C4
zf : yc
zf : yc
xd�i : yb�izi xd�i : zh�i
1
yi
1
Figure 18. D bordering an e1- or an e2-corner triangle and an e3-corner triangle.
The same argument as in the previous case gives that rχ divides the G-igsaw pieces for C3
and C4.
To treat the remaining two curves C1 and C2 in each case, we use a generalised form of [12,
Section 3.3.2]: an edge ` continues in a straight line past a boundary edge `0 if and only if the
ratio marking ` features any common variables x, y, z raised to a strictly lower exponent than
in the ratio marking `0. One can verify this by a case-by-case analysis using as its base the
24 B. Wormleighton
original result from [12]. This implies that rχ divides the G-igsaw pieces for “broken edges” that
do not continue in a straight line past the χ-chain and that it does not divide any monomials
in the G-igsaw pieces for “straight edges” that do continue past the χ-chain. This is captured
exactly in the notion of downstream curves relative to boundary curves in Definition 3.2, and
hence in the unlocking procedure.
For the case when C is not incident to e1, consider two boundary curves C and C 1 shown
in Fig. 19, where C is closer to e1. One can verify using local divisibility relations that the
only difference between the G-igsaw piece for C and for C 1 is that the latter loses the characters
in the G-igsaw pieces for the dashed curves in broken chains; that is, exactly the curves that
are downstream from C but not from C 1. By retracing back to the edge incident to e1 the first
calculation performed above suffices to compute the G-igsaw piece for an arbitrary boundary
curve.
C
C 1
Figure 19. Two boundary curves.
Variations of the arguments above work just as well for the cases not depicted when some
of the edges incident to D are also boundary edges of regular triangles. Counting up all these
monomials and comparing them with a socle calculation shows that these are all the characters
in the G-igsaw piece for C, which validates the unlocking procedure for boundary curves and
hence for all exceptional curves in G-Hilb.
3.7 Example: G � 1
30
p25, 2, 3q
We will illustrate the unlocking procedure for G-Hilb in the case that G � 1
30p25, 2, 3q. In the
figure below, dashed lines are edges within a regular triangle and undashed lines are the result
of the first stage of the Craw–Reid triangulation.
We will demonstrate the unlocking procedure for a few curves in G-Hilb. Consider the 15-
curve C15 shown in Fig. 21. This curve is of Type Ixb since it is the only p�1,�1q-curve marked
with 15 and feeds to the right into boundary edges only. This gives
G-igpC15q � t15, 17, 19, 21u.
Consider the 5-curve C5 shown in Fig. 22. It passes into the right side of the junior simplex,
unlocking the 9-curve of Type Iy and giving
G-igpC5q � t5, 7, 9, 11u.
Consider the 2-curve C2 shown in Fig. 23. This is a curve of Type Iz. We first get the
character 17 marking the divisor on the 2-chain, unlocking the 27-chain. The 27-chain contains
a del Pezzo divisor contributing the character 22 in this case. Hence
G-igpC2q � t2, 17, 22, 27u.
Walls for G-Hilb via Reid’s Recipe 25
1
26
21
16
11
6
6
6
6
6
6
23
13
3
3
3
25
28
15
18
5
8
19
17
29
22
14
720
20
10
10
15
15
15
2
2
4
4
4
4
5 5
9
12
12
10
20
20
25
27
27
24
Figure 20. Reid’s recipe for G � 1
30 p25, 2, 3q.
15
21
19
17
15
15
15
15
Figure 21. Unlocking for a 15-curve.
Lastly, we will consider the boundary 15-curve C 1
15 shown in Fig. 24.
At the first step we include the 15-chain and the curves of Type Ix and Iy unlocked by
it. These curves are marked with characters 10, 24, 18. The 18-curve and the 24-curve are
of Type Iy and only contribute their own character to the G-igsaw piece. The 10-curve is of
Type Ixb and so we add the Hirzebruch divisors along the 10-chain. As a result
G-igpC 1
15q � t10, 13, 15, 16, 17, 18, 19, 21, 24u.
3.8 Example: G � 1
35
p1, 3, 31q
We will use the example of G � 1
35p1, 3, 31q to illustrate a phenomenon implicit, but less clear
in the long side picture. The triangulation for G-Hilb and Reid’s recipe are found in Fig. 25.
Consider the 3-curve C3 incident to e1. The unlocking procedure for this curve is shown
in Fig. 26 giving
G-igpC3q � t1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29,
30, 32, 33, 34u.
26 B. Wormleighton
5
11
75 5 5
9
Figure 22. Unlocking for a 5-curve.
2
17
2
2
27
17
22
2
2
27
27
Figure 23. Unlocking for a 2-curve.
Notice that every chain meeting the 3-chain in a vertex is broken there. Repeating for the next
3-curve along the chain produces the same unlocking sequence except that the topmost part
including the 1-chain and the 12-chain are not included, capturing that the monomials in the
corresponding character spaces are no longer divisible by r3 there.
4 Walls of C0
In this section we will compute explicit inequalities carving out C0, and will determine which of
these inequalities are necessary and hence define walls of C0.
4.1 Type I walls
We know from [10, Theorem 9.12] that all flops in a single p�1,�1q-curve C are achieved
by a wall-crossing from C0. Moreover, we have degpRρ|Cq � 1 for all ρ P G-igpCq from
[10, Corollary 6.3]. The unlocking procedure hence gives a combinatorial way of writing down
the equations of these walls.
Proposition 4.1. Suppose C � G-HilbA3 is an exceptional p�1,�1q-curve marked with cha-
racter χ by Reid’s recipe. Then, the Type I wall corresponding to C is given by
θpϕC0pOCqq �
¸
χPG-igpCq
θpχq ¡ 0,
where G-igpCq is computed by the unlocking procedure.
Walls for G-Hilb via Reid’s Recipe 27
15
21
19
17
15
18
15
15
15
10
24
21
16
13
19
17
15
18
10
10
15
15
15
10
24
Figure 24. Unlocking for a boundary 15-curve.
34
30
26
22
18
14
10
6
4
5
32
29
20
25
17
8
13
28
16
31
27 23 19 15 11 7 3
1
2
1
1
31
27
27
27
15
15
15
3
3
3
3
3
3
3
3
3
3
3
31
33
2
1
23
21
19
19
2
2
24
24
1
9
11 12
12
2
2
7
7
Figure 25. Reid’s recipe for G � 1
35 p1, 3, 31q.
4.2 No Type II walls
Proposition 4.2. Suppose C � G-HilbA3 is an exceptional p1,�3q-curve marked with charac-
ter χ by Reid’s recipe. Then, the inequality corresponding to C is given by
θ
�
ϕC0pOCq
�
� 2 � θ
�
χb2
�
�
¸
χPG-igpCqztχb2u
θpχq ¡ 0,
where G-igpCq is computed by the unlocking procedure.
Proof. Notice that such a curve C lies inside the exceptional P2 in the meeting of champions
case when the meeting of champions triangle has side length 0. Thus the P2 is marked with χb2
and lies in the socle of any torus-invariant G-cluster. From Theorem 2.4 rχb2 � r2χ and so r2χ is
the furthest character from rχ in the G-igsaw piece in some direction. Note that
degpRρ|Cq � min
k : rkχ | rρ
(
28 B. Wormleighton
6
4
5
3
1
2
3
3
3
3
1
9
12
2
34
30
26
22
18
14
10
6
4
5
32
29
20
17
8
13
28
16
3
1
2
1
1
3
3
3
3
3
3
3
3
3
3
3
33
2
1
21
2
2
24
1
9
12
12
2
2
34
30
26
22
18
14
10
6
4
5
32
29
20
25
17
8
13
28
16
3
1
2
1
1
3
3
3
3
3
3
3
3
3
3
3
33
2
1
21
2
2
24
24
1
9
12
12
2
2
Figure 26. Unlocking for a 3-curve.
Walls for G-Hilb via Reid’s Recipe 29
and so all the characters in G-igpCq appear with multiplicity 1 except for r2χ, which appears
with multiplicity 2. This gives the required formula. �
As a result we can immediately deduce the conclusion of [10, Proposition 3.8] for C0.
Corollary 4.3. C0 has no Type II walls.
Proof. Suppose C is an exceptional p1,�3q-curve marked with χ. From the unlocking procedure
a totalG-igsaw piece for C consists of χ, χ2, and the characters marking the (Hirzebruch) divisors
along the χ-chain. Let D1 be the exceptional P2 containing C. Consider the inequality for rigid
quotients parameterised by D1: from Proposition 2.6 the characters appearing in this inequality
are exactly the characters in the G-igsaw pieces of all three χ-curves converging at D1. These are
χ, χb2
(
YHirzpχq,
which are exactly the characters appearing in the inequality for C. However, the inequality for
rigid quotients parameterised by D1 has multiplicities all equal to 1. When combined with the
inequality θ
�
χb2
�
¡ 0 coming from rigid subsheaves parameterised by D1 – note that we can
use both inequalities from subsheaves and from quotients since D1 is irreducible – this implies
that the inequality ϕC0pOCq ¡ 0 is redundant. �
4.3 All flops in p�1,�1q-curves
Using Proposition 4.1 and the unlocking procedure one can show directly that every p�1,�1q-
curve produces a necessary inequality, recovering [10, Theorem 9.12] by purely combinatorial
means.
In order to test redundancy of inequalities we say that an inequality
°
i αiθpχiq ¡ 0 with
nonnegative coefficients is a summand of another inequality
°
j βjθpρjq ¡ 0 with nonnegative
coefficients if the difference
°
i αiθpχiq �
°
j βjθpρjq also has nonnegative coefficients in the
basis IrrG. If an inequality coming from curves or divisors decomposes into other inequalities
as summands, then it is redundant and does not define a wall of C0.
Proposition 4.4. Suppose C is an exceptional p�1,�1q curve inside G-HilbA3. Then the
inequality θpϕC0pOCqq ¡ 0 is necessary and so defines a wall of C0.
Proof. Suppose C is marked with χ. From the unlocking procedure we can write the inequality
corresponding to C in the form
θpχq �
¸
i
θpψiq � θpρ1q �
¸
i
θ
�
ψ1
i
�
� � � � � θpρmq �
¸
i
θ
�
ψmi
�
¡ 0, (4.1)
where ρj are the characters marking curves Cj unlocked by C and ψji are the characters in the
G-igsaw piece for Cj . Note that curves unlocked by C cannot continue on both sides of the
χ-chain, since they meet the χ-chain at a Hirzebruch divisor found at the intersection of the
χ-chain and an edge of a regular triangle, where only two chains can continue. The inequality
for the p�1,�1q-curve Cj is
θ
�
ϕC0pOCj q
�
� θpρjq �
¸
i
θ
�
ψji
�
¡ 0.
In order to express (4.1) in terms of other inequalities, we must have an inequality featuring the
character χ. These can only arise from other χ-curves or divisors parameterising rigid quotients
not featuring χ. Other χ-curves will feature at least one different character in their G-igsaw piece
compared to G-igpCq: indeed, other curves in the same regular triangle will feature a different
30 B. Wormleighton
collection of del Pezzo divisors, curves in other regular triangles will either feature different del
Pezzo divisors or unlock different curves, and χ-curves along a boundary edge will have different
unlocking behaviour. In particular, the inequalities from these curves will not be summands of
the inequality (4.1). Inequalities from rigid quotients not containing χ will also not be summands
of (4.1) since the unlocking procedure implies that there are no divisors Dρ along the χ-chain
for which all characters marking curves incident to Dρ are represented in G-igpCq. It follows
that (4.1) is necessary. �
Proposition 4.4 has an analog for Type III walls in Lemma 4.15 where we classify the
p0,�2q-curves producing those walls in terms of explicit combinatorics.
4.4 Irredundant inequalities – examples
The aim of these final sections is to precisely describe all of the walls of C0, which primarily means
identifying which curves produce redundant inequalities and subsequently classifying the walls
of Type III. We start with an example.
Example 4.5. Consider G � 1
6p1, 2, 3q. G-Hilb and Reid’s recipe are shown in Fig. 27.
5
32
2
1
3
4
Figure 27. G-Hilb and Reid’s recipe for 1
6 p1, 2, 3q.
We compute the inequalities coming from curves and divisors that define C0 via the unlocking
procedure:
θpχ1q ¡ 0, (A1)
θpχ2q � θpχ5q ¡ 0 (A2)
θpχ2q � θpχ3q � 2θpχ4q � 2θpχ5q ¡ 0, (B2)
θpχ3q � θpχ5q ¡ 0, (A3)
θpχ3q � θpχ4q � θpχ5q ¡ 0, (B3)
θpχ4q ¡ 0, (A4)
θpχ5q ¡ 0, (A5)
θpχ2q � θpχ3q � θpχ4q � θpχ5q ¡ 0. (B5)
(A1) is from the curve marked with the essential character 1. Similarly for (A4). We then
have two inequalities (A2) and (B2) coming from the two 2-curves, and two (A3) and (B3) from
the two 3-curves. The 5-divisor gives two inequalities (A5) and (B5) for rigid subsheaves and
quotients it parameterises.
We can see that (B2) is redundant by expressing it as a combination of (A2), (A3) and (A4).
Similarly, (B3) can be expressed in terms of (A3) and (A4). No further reductions are possible,
and so the walls of C0 (with their types) in this example are
θpχ1q � 0, (I)
θpχ2q � θpχ5q � 0, (III)
Walls for G-Hilb via Reid’s Recipe 31
θpχ3q � θpχ5q � 0, (I)
θpχ4q � 0, (I)
θpχ5q � 0, (0)
θpχ2q � θpχ3q � θpχ4q � θpχ5q � 0. (0)
Example 4.6. We continue with a more detailed example for G � 1
30p25, 2, 3q. Continuing the
calculations in Section 3.7, we find that the inequalities from curves in G-Hilb are
θ2 � θ27 � θ22 � θ17 ¡ 0, (A2)
θ2 � θ5 � θ8 � θ11 � θ14 ¡ 0, (B2)
θ3 � θ13 � θ18 � θ23 � θ28 ¡ 0, (A3)
θ3 � θ5 � θ7 � θ9 � θ11 � θ13 � θ23 � θ28 ¡ 0, (B3)
θ3 � θ5 � θ7 � θ9 � θ11 � θ13 � θ15 � θ17 � θ19 � θ21 ¡ 0, (C3)
θ4 � θ29 � θ24 � θ19 � θ14 ¡ 0, (A4)
θ4 � θ7 � θ29 � θ24 � θ19 ¡ 0, (B4)
θ4 � θ7 � θ10 � θ13 � θ16 � θ19 � θ29 ¡ 0, (C4)
θ4 � θ7 � θ10 � θ13 � θ16 � θ19 � θ22 ¡ 0, (D4)
θ5 � θ7 � θ9 � θ11 ¡ 0, (A5)
θ5 � θ7 � θ8 � θ11 ¡ 0, (B5)
θ5 � θ8 � θ11 � θ14 ¡ 0, (C5)
θ6 � θ8 � θ9 � θ10 � θ11 � θ13 � 2θ12 � 2θ14 � 2θ16 � 2θ15 � 2θ17 � 2θ19 � 3θ18
� 3θ20� 3θ22� 3θ21� 3θ23� 3θ25� 4θ24� 4θ26� 4θ28� 4θ27� 4θ29� 4θ1 ¡ 0, (A6)
θ6 � θ8 � θ10 � 2θ12 � 2θ14 � 2θ16 � 3θ18 � θ9 � θ11 � θ13 � 2θ15 � 2θ17
� 2θ19 � 3θ21 � θ1 � 4θ26 � 2θ16 � 3θ23 � 3θ20 � 3θ22 � 4θ24 � 4θ26 ¡ 0, (B6)
θ6 � θ8 � θ10 � 2θ12 � 2θ14 � 2θ16 � 3θ18 � θ9 � θ11 � θ13
� 2θ15 � 2θ17 � 2θ19 � 3θ21 � θ1 � θ26 � θ21 � θ16 ¡ 0, (C6)
θ6 � θ8 � θ10 � 2θ12 � 2θ14 � 2θ16 � θ1 � θ26 � θ21 � θ16 � θ9 � θ11 � θ13 ¡ 0, (D6)
θ6 � θ1 � θ26 � θ21 � θ16 � θ11 � θ8 � θ9 ¡ 0, (E6)
θ6 � θ1 � θ26 � θ21 � θ16 � θ11 ¡ 0, (F6)
θ8 ¡ 0, (A8)
θ9 ¡ 0, (A9)
θ10 � θ13 � θ16 ¡ 0, (A10)
θ10 � θ12 � θ14 � θ16 � θ18 � θ5 � θ7 � θ9 � θ11 � θ13 ¡ 0, (B10)
θ10 � θ13 � θ12 � θ14 � θ16 ¡ 0, (C10)
θ12 � θ7 ¡ 0, (A12)
θ12 � θ14 ¡ 0, (B12)
θ15 � θ17 � θ19 � θ21 ¡ 0, (A15)
θ15 � θ17 � θ19 � θ18 � θ21 ¡ 0, (B15)
θ15 � θ17 � θ18 � θ21 � θ24 � θ10 � θ13 � θ16 � θ19 ¡ 0, (C15)
θ15 � θ18 � θ21 � θ24 � θ27 � θ10 � θ13 � θ16 � θ19 � θ22 � θ5 � θ8
� θ11 � θ14 � θ17 ¡ 0, (D15)
θ18 ¡ 0, (A18)
θ20 � θ23 � θ26 � θ29 ¡ 0, (A20)
32 B. Wormleighton
θ20 � θ22 � θ23 � θ26 ¡ 0, (B20)
θ20 � θ23 � θ22 � θ24 � θ26 ¡ 0, (C20)
θ20 � θ15 � θ17 � θ19 � θ21 � θ22 � θ24 � θ26 � θ28 ¡ 0, (D20)
θ24 ¡ 0, (A24)
θ25 � θ27 � θ29 � θ1 ¡ 0, (A25)
θ25 � θ28 � θ1 ¡ 0, (B25)
θ27 � θ22 ¡ 0, (A27)
θ27 � θ29 ¡ 0, (B27)
θ28 ¡ 0. (A28)
The bolded inequalities correspond to curves C with NC not of type p�1,�1q. We know by
[10, Theorem 9.12] that the other inequalities are necessary and define Type I walls of C0. The
inequalities from divisors parameterising rigid subsheaves are
θ1 ¡ 0, (A1)
θ7 ¡ 0, (A7)
θ11 ¡ 0, (A11)
θ13 ¡ 0, (A13)
θ14 ¡ 0, (A14)
θ16 ¡ 0, (A16)
θ17 ¡ 0, (A17)
θ19 ¡ 0, (A19)
θ21 ¡ 0, (A21)
θ22 ¡ 0, (A22)
θ23 ¡ 0, (A23)
θ26 ¡ 0, (A26)
θ29 ¡ 0. (A29)
We record the redundancies for the bold (or potentially redundant) inequalities:
(F6)� (A8)� (A9)� (A10)� (B12)� (A15)� (A18)� (A20)� (A24)
� (A25)� (B27)� (A28) ùñ (A6),
(F6)� (A8)� (A9)� (A10)� (B12)� (A15)� (A18)� (A20)� (A24) ùñ (B6),
(F6)� (A8)� (A9)� (A10)� (B12)� (A15)� (A18) ùñ (C6),
(F6)� (A8)� (A9)� (A10)� (B12) ùñ (D6),
(F6)� (A8)� (A9) ùñ (E6),
(A5)� (B12)� (A18) ùñ (B10),
(A10)� (B12) ùñ (C10),
(A15)� (A18) ùñ (B15),
(A15)� (A18)� (A10)� (A24) ùñ (C15),
(A15)� (A18)� (A10)� (A24)� (A27)� (C5) ùñ (D15),
(B20)� (A24) ùñ (C20),
(A15)� (B20)� (A24) ùñ (D20).
We have killed off the inequalities from all curves except for the p�1,�1q-curves and one
curve (F6) from the long side.
Walls for G-Hilb via Reid’s Recipe 33
4.5 Redundant inequalities from curves
Observe that the vast majority of inequalities in Examples 4.5–4.6 define walls of Type I.
We should be unsurprised by the cancellation of all except one bolded inequality in Example 4.6
due to the following result from [10].
Lemma 4.7 ([10, Corollaries 6.3 and 6.5]). Suppose w � p
°
αiθi � 0q is a Type I or III wall
of C0. Then all αi P t0, 1u.
Chambers other than C0 can have coefficients αi � �1, however since the trivial representa-
tion does not appear in G-igpCq for any curve C we can exclude this possibility.
Corollary 4.8. Suppose G-HilbA3 has a meeting of champions of side length 0. Then the
inequality for any curve along one of the three champions is redundant.
Proof. Suppose χ is the character marking each of the champions. Then, by Theorem 2.4,
r2χ � rχ2 globally on G-Hilb and so degpRχ2 |Cq � 2 for all χ-curves C. It follows from Lemma 4.7
that none of these inequalities can be strict. �
We can also show this directly via unlocking. This reproves Corollary 4.3.
Lemma 4.9. Suppose C is a χ-curve. If the unlocking procedure for C doesn’t unlock a curve
or divisor marked with χ2 then all the coefficients in the inequality θpϕC0pOCqq ¡ 0 are equal
to 0 or 1.
Proof. This is because if some ρ has degpRρ|Cq ¥ 2 then r2χ | rρ and so r2χ must feature in the
G-igsaw piece for C and is hence equal to rχ2 near C. �
Lemma 4.10. Suppose a curve C0 unlocks a curve C1 of character ρ. Let ψ P G-igpC1q. If C
is a curve that unlocks C0, then degpRψ|Cq ¥ degpRρ|Cq.
Proof. As used previously, degpRρ|Cq � max
k P Z¥0 : rkχ | rρ
(
. From this formulation, clearly
if rρ | rψ then degpRψ|Cq ¥ degpRρ|Cq, but this is the case by definition of G-igsaw piece. �
Lemma 4.11. Suppose C is a curve on the boundary of a regular triangle marked with a char-
acter χ. Suppose the χ-chain contains a p�1,�1q-curve. Then the inequality θpϕC0pOCqq ¡ 0 is
redundant.
Proof. Suppose C is marked with character χ. Let C0 be the first p�1,�1q-curve in the χ-chain
moving inwards from C. Then the G-igsaw piece for C consists of exactly the characters in the
G-igsaw piece for C0 along with the characters in the G-igsaw pieces for any curves C1, . . . , Cn
unlocked by C at Hirzebruch divisors before C0. Let the character marking Ci be χi. The
inequality for C decomposes as
θ
�
ϕC0pOCq
�
�
¸
ρPG-igpC0q
αρθpρq �
ņ
i�1
¸
ρPG-igpCiq
βiρθpρq, (4.2)
where αρ and βiρ are nonnegative multiplicities given by the appropriate calculation of degpRρ|?q,
possibly computing the degree of Rρ on multiple curves. Note that αχ � 1. One can thus write
θ
�
ϕC0pOCq
�
� θ
�
ϕC0pOC0q
�
�
¸
ρPG-igpC0q
pαρ � 1qθpρq �
m̧
i�1
�
βiχiθ
�
ϕC0pOCiq
�
�
¸
ρPG-igpCiq
pβiρ � βiχiqθpρq
�
.
34 B. Wormleighton
From Lemma 4.10, αρ � 1 and βρ � βχ1 are both nonnegative. If all the remaining ρ in these
sums with nonzero coefficients after this reduction are characters marking divisors then one can
express each term γρθpρq � γρθ
�
ϕC0
�
R�1
ψ |D
��
for some divisor D, thus evidencing that (4.2)
is redundant. Suppose instead that some ρ � ρ1 marks a curve unlocked by C0 or some Ci.
We assume the latter; the former is treated identically. Denote this new curve by Ci,1. Then¸
ρPG-igpCiq
�
βiρ � βiχi
�
θpρq �
�
βiρ1 � βiχi
�
θ
�
ϕC0pOCi,1q
�
�
¸
ρPG-igpCi,1q
�
βiρ � βiρ1
�
θpρq �
¸
ρRG-igpCi,1q
�
βiρ � βiχi
�
θpρq,
where again each coefficient is nonnegative by Lemma 4.10 applied to Ci,1. Observe that there
are strictly fewer nonzero coefficients in this expression than before, since at the least we removed
the term for ρ1. Continuing in this way for each character appearing that marks a curve, we
can reduce to the situation where the only characters with nonzero coefficients in the error term
are those that mark divisors. At that point we have already seen how to express the error term
in terms of inequalities coming from divisors, and so we have shown that (4.2) is redundant. �
4.6 Classifying Type III walls
We provide a combinatorial classification of the Type III walls for C0. We start with the
following definition.
Definition 4.12. Let χ be a character marking a curve in G-Hilb. We say that the χ-chain is
a generalised long side if it starts and ends on the boundary of the junior simplex, and all the
edges along the χ-chain are boundary edges of regular triangles. We exclude the lines meeting
at a trivalent vertex if there is a meeting of champions of side length 0 from this definition.
For example, any long side is a generalised long side. The 15-chain for 1
35p1, 3, 31q is a ge-
neralised long side as can be seen in Fig. 25.
Example 4.13. We compute the inequalities for curves along the 15-chain in G-Hilb for G �
1
35p1, 3, 31q. From the unlocking procedure or computing G-igsaw pieces directly, the inequalities
for the 15-curves starting from e1 and moving downwards are
θ15 � θ18 � θ21 � θ24 � θ7 � θ10 � θ13 � θ16 � θ11 � θ14 � θ17 � θ20 ¡ 0, (A15)
θ15 � θ18 � θ21 � θ16 � θ11 � θ14 � θ17 ¡ 0, (B15)
θ15 � θ16 � θ17 � θ18 ¡ 0, (C15)
θ15 � θ16 � θ17 � θ18 ¡ 0. (D15)
Clearly (C15) and (D15) depend on each other; the inequality is the same since they are fibres of
the P1-bundle structure on the Hirzebruch surface marked with 18, and so contracting one must
contract the other. We consider some of the additional inequalities coming from p�1,�1q-curves:
θ7 � θ10 � θ13 ¡ 0, (A7)
θ11 � θ14 ¡ 0, (A11)
θ21 ¡ 0, (A21)
θ24 � θ20 ¡ 0. (A24)
We can deduce
(C15)� (A7)� (A11)� (A21)� (A24) ùñ (A15),
(C15)� (A11)� (A21) ùñ (B15),
so that (A15) and (B15) are redundant.
Walls for G-Hilb via Reid’s Recipe 35
Definition 4.14. Consider a generalised long side marked with character χ. Recall that each
χ-chain consists of potentially several straight line segments. We call a curve in the χ-chain final
if it is the furthest curve along the χ-chain away from a vertex along such a line segment.
For example, for G � 1
35p1, 3, 31q, the bolded curves in Fig. 28 are final.
Figure 28. Final curves for G � 1
35 p1, 3, 31q.
Final curves not along a long side are also those contained in an exceptional Hirzebruch
surface (with no blowups) or, equivalently, those corresponding to edges incident to a 4-valent
vertex. There can be at most two final curves for each generalised long side, with exactly one
when the generalised long side is actually a long side.
Lemma 4.15. Suppose χ is a character marking a curve and that the χ-chain is a generalised
long side. Then, the inequality for each non-final curve C in the χ-chain is redundant. The final
curves all produce the same inequality:
θpχq �
¸
ψPHirzpχq
θpψq ¡ 0,
which is a necessary inequality defining a Type III wall of C0.
Proof. First, the inequality for a final χ-curve C features only the Hirzebruch divisors along the
χ-chain by the unlocking procedure. It has all nonzero coefficients equal to 1 for the following
reason. χ2 cannot mark a Hirzebruch divisor along the χ-chain because to do so one would
require another chain, say with character ρ, to cross the χ-chain and have χ b ρ � χ2. Of
course, this would mean that ρ � χ, but chains do not self-intersect. Hence, χ2 does not appear
in the G-igsaw piece for C and so all multiplicities must be equal to 1 by Lemma 4.9. This is
clearly a necessary inequality, as χ is the only character in the inequality coming from a curve
and there is no divisor that contains only χ-curves – in contrast to the case of a trivalent vertex.
To see that the other inequalities coming from curves along a generalised long side are re-
dundant, we will decompose these inequalities similarly to before. Let C be such a curve and
write
θ
�
ϕC0pOCq
�
� θpχq �
¸
ψPHirzpCq
αψθpψq �
ņ
i�1
¸
ρPG-igpCiq
βiρθpρq,
where C1, . . . , Cn are the curves unlocked by C. By exactly the same methods as in the proof of
Lemma 4.11, one can express the final term as a sum of inequalities from curves and divisors.
36 B. Wormleighton
The first two terms are equal to
θpχq �
¸
ψPHirzpCq
αψθpψq � θ
�
ϕC0pOC1q
�
�
¸
ψPHirzpχq
pαψ � 1qθ
�
ϕC0
�
R�1
ψ |Dψ
��
,
where C 1 is a final χ-curve and Dψ is the divisor marked with ψ. Of course αψ ¥ 1 and so we
have shown that the inequality from C is redundant. �
We consider the example G � 1
25p1, 3, 21q, which has a meeting of champions of side length 2.
Example 4.16. We show the triangulation for G-Hilb and Reid’s recipe for G � 1
25p1, 3, 21q
in Fig. 29. Observe that of the three champions, the 3-chain and 9-chain are generalised long
sides whilst the 1-chain contains a p�1,�1q-curve. We hence obtain two Type III walls from
the champions and another for the 21-chain that is also a generalised long side, with inequalities
θ3 � θ4 � θ8 � θ12 � θ16 � θ20 � θ24 ¡ 0, (F3)
θ9 � θ10 � θ11 � θ12 ¡ 0, (C9)
θ21 � θ22 � θ23 � θ24 ¡ 0. (C21)
24
20
16
12
8
4
7
11
10
23
14
19
22
9
9
9
1
1
1
21
21
21
3
3
3
3
3
3
2
1
21
17
13
9
5
6
3
13
13
1
2
2
18
18
15
17
6
2
5
Figure 29. Reid’s recipe for G � 1
25 p1, 3, 21q.
4.7 Summary
We compile the main results – Corollary 4.3, Proposition 4.4, Lemmas 4.11 and 4.15 – of this
section.
Theorem 4.17. Suppose G � SL3pCq is a finite abelian subgroup. The walls of the chamber C0
for G-HilbA3 and their types are as follows:
� a Type I wall for each exceptional p�1,�1q-curve,
Walls for G-Hilb via Reid’s Recipe 37
� a Type III wall for each generalised long side,
� a Type 0 wall for each irreducible exceptional divisor,
� each remaining wall is of Type 0 and comes from a divisor parameterising a rigid quotient.
Moreover, for every contraction of Type I or III for G-HilbA3 there is a wall of the correspon-
ding type that induces the contraction by VGIT.
Proposition 2.6 describes how to recover the unstable locus or the corresponding reducible
divisor D1 for each wall of Type 0 from a rigid quotient. Let w be a wall of C0. Denote
by Epwq the set of edges in the Craw–Reid triangulation corresponding to curves C for which
all characters in G-igpCq appear in the equation of the wall. The desired divisor D1 inducing w is
then the union of the divisors corresponding to vertices for which all incident edges are in Epwq.
We observe that the unlocking procedure allows the check of which walls from rigid quotients
are necessary to be performed combinatorially.
5 Future directions
There are several natural avenues of further study opened up by the results of this paper, three
of which are
� attuning the results here with the derived interpretation of Reid’s recipe [6],
� exploring any relations between analogs to Reid’s recipe in other settings and walls in sta-
bility (for instance, dimer models [3, 15]),
� reverse-engineering a partial Reid’s recipe for other resolutions MC from an explicit des-
cription of the walls of C, and examining whether this has any categorical content.
Acknowledgements
The author would like to thank Yukari Ito and Nagoya University for hosting him as this research
began. He would also like to thank Alastair Craw, Álvaro Nolla de Celis, and David Nadler for
many fruitful and enjoyable conversations about this project, as well as the referees for their
thoughtful suggestions on how to improve its exposition.
References
[1] Artin M., Verdier J.L., Reflexive modules over rational double points, Math. Ann. 270 (1985), 79–82.
[2] Batyrev V.V., Dais D.I., Strong McKay correspondence, string-theoretic Hodge numbers and mirror sym-
metry, Topology 35 (1996), 901–929, arXiv:alg-geom/9410001.
[3] Bocklandt R., Craw A., Quintero Vélez A., Geometric Reid’s recipe for dimer models, Math. Ann. 361
(2015), 689–723, arXiv:1305.0156.
[4] Bridgeland T., Flops and derived categories, Invent. Math. 147 (2002), 613–632, arXiv:math.AG/0009053.
[5] Bridgeland T., King A., Reid M., The McKay correspondence as an equivalence of derived categories,
J. Amer. Math. Soc. 14 (2001), 535–554, arXiv:math.AG/9908027.
[6] Cautis S., Craw A., Logvinenko T., Derived Reid’s recipe for abelian subgroups of SL3pCq, J. Reine Angew.
Math. 727 (2017), 1–48, arXiv:1205.3110.
[7] Cautis S., Logvinenko T., A derived approach to geometric McKay correspondence in dimension three,
J. Reine Angew. Math. 636 (2009), 193–236, arXiv:0803.2990.
[8] Craw A., The McKay correspondence and representations of the McKay quiver, Ph.D. Thesis, Warwick
University, 2001.
https://doi.org/10.1007/BF01455531
https://doi.org/10.1016/0040-9383(95)00051-8
https://arxiv.org/abs/alg-geom/9410001
https://doi.org/10.1007/s00208-014-1085-8
https://arxiv.org/abs/1305.0156
https://doi.org/10.1007/s002220100185
https://arxiv.org/abs/math.AG/0009053
https://doi.org/10.1090/S0894-0347-01-00368-X
https://arxiv.org/abs/math.AG/9908027
https://doi.org/10.1515/crelle-2014-0086
https://doi.org/10.1515/crelle-2014-0086
https://arxiv.org/abs/1205.3110
https://doi.org/10.1515/CRELLE.2009.086
https://arxiv.org/abs/0803.2990
38 B. Wormleighton
[9] Craw A., An explicit construction of the McKay correspondence for A-Hilb C3, J. Algebra 285 (2005),
682–705, arXiv:math.AG/0010053.
[10] Craw A., Ishii A., Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, Duke
Math. J. 124 (2004), 259–307, arXiv:math.AG/0211360.
[11] Craw A., Ito Y., Karmazyn J., Multigraded linear series and recollement, Math. Z. 289 (2018), 535–565,
arXiv:1701.01679.
[12] Craw A., Reid M., How to calculate A-Hilb C3, in Geometry of Toric Varieties, Sémin. Congr., Vol. 6, Soc.
Math. France, Paris, 2002, 129–154, arXiv:math.AG/9909085.
[13] Denef J., Loeser F., Motivic integration, quotient singularities and the McKay correspondence, Compositio
Math. 131 (2002), 267–290, arXiv:math.AG/9903187.
[14] Ishii A., Ito Y., Nolla de Celis A., On G{N -Hilb of N -Hilb, Kyoto J. Math. 53 (2013), 91–130,
arXiv:1108.2310.
[15] Ishii A., Ueda K., Dimer models and the special McKay correspondence, Geom. Topol. 19 (2015), 3405–3466,
arXiv:0905.0059.
[16] Ito Y., Nakajima H., McKay correspondence and Hilbert schemes in dimension three, Topology 39 (2000),
1155–1191, arXiv:math.AG/9803120.
[17] Ito Y., Nakamura I., McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72
(1996), 135–138.
[18] Ito Y., Wormleighton B., Wall-crossing for iterated G-Hilbert schemes, in preparation.
[19] King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. 45 (1994), 515–530.
[20] Logvinenko T., Derived McKay correspondence via pure-sheaf transforms, Math. Ann. 341 (2008), 137–167,
arXiv:math.AG/0606791.
[21] McKay J., Cartan matrices, finite groups of quaternions, and Kleinian singularities, Proc. Amer. Math. Soc.
81 (1981), 153–154.
[22] Nakamura I., Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001), 757–779.
[23] Reid M., La correspondance de McKay, arXiv:math.AG/9911165.
[24] Takahashi K., On essential representations in the McKay correpondence for SL3pCq, Master’s Thesis, Nagoya
University, 2011.
[25] Wilson P.M.H., The Kähler cone on Calabi–Yau threefolds, Invent. Math. 107 (1992), 561–583.
https://doi.org/10.1016/j.jalgebra.2004.10.001
https://arxiv.org/abs/math.AG/0010053
https://doi.org/10.1215/S0012-7094-04-12422-4
https://doi.org/10.1215/S0012-7094-04-12422-4
https://arxiv.org/abs/math.AG/0211360
https://doi.org/10.1007/s00209-017-1965-1
https://arxiv.org/abs/1701.01679
https://arxiv.org/abs/math.AG/9909085
https://doi.org/10.1023/A:1015565912485
https://doi.org/10.1023/A:1015565912485
https://arxiv.org/abs/math.AG/9903187
https://doi.org/10.1215/21562261-1966080
https://arxiv.org/abs/1108.2310
https://doi.org/10.2140/gt.2015.19.3405
https://arxiv.org/abs/0905.0059
https://doi.org/10.1016/S0040-9383(99)00003-8
https://arxiv.org/abs/math.AG/9803120
https://doi.org/10.3792/pjaa.72.135
https://doi.org/10.1093/qmath/45.4.515
https://doi.org/10.1007/s00208-007-0186-z
https://arxiv.org/abs/math.AG/0606791
https://doi.org/10.2307/2044010
https://arxiv.org/abs/math.AG/9911165
https://doi.org/10.1007/BF01231902
1 Introduction
2 Resolutions of A3/G
2.1 Setup
2.2 Reid's recipe
2.3 G-igsaw pieces
2.4 Tautological bundles
2.5 Abstract inequalities for C0
3 Computing characters in total G-igsaw pieces
3.1 Combinatorial definitions
3.2 Unlocking procedure
3.3 Monomials for divisors
3.4 Counting characters
3.5 (-1,-1)-curves
3.5.1 Type Iy curves
3.5.2 Type Ix curves
3.5.3 Type Iz curves
3.5.4 Type Ic curves
3.6 Boundary curves
3.7 Example: G=1/30(25,2,3)
3.8 Example: G=1/35(1,3,31)
4 Walls of C0
4.1 Type I walls
4.2 No Type II walls
4.3 All flops in (-1,-1)-curves
4.4 Irredundant inequalities – examples
4.5 Redundant inequalities from curves
4.6 Classifying Type III walls
4.7 Summary
5 Future directions
References
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| id | nasplib_isofts_kiev_ua-123456789-211014 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T20:34:07Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wormleighton, Ben 2025-12-22T09:28:46Z 2020 Walls for -Hilb via Reids Recipe. Ben Wormleighton. SIGMA 16 (2020), 106, 38 pages 1815-0659 2020 Mathematics Subject Classification: 14E16; 14M25; 16G20 arXiv:1908.05748 https://nasplib.isofts.kiev.ua/handle/123456789/211014 https://doi.org/10.3842/SIGMA.2020.106 The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein 3-fold quotient singularities ³/ with the representation theory of the group . The first crepant resolution studied in depth was the -Hilbert scheme -HilbA3, which is also a moduli space of θ-stable representations of the McKay quiver associated to . As the stability parameter θ varies, we obtain many other crepant resolutions. In this paper, we focus on the case where is abelian, and compute explicit inequalities for the chamber of the stability space defining -Hilb ³ in terms of a marking of exceptional subvarieties of -Hilb ³ called Reid's recipe. We further show which of these inequalities define walls. This procedure depends only on the combinatorics of the exceptional fibre and has applications to the birational geometry of other crepant resolutions. The author would like to thank Yukari Ito and Nagoya University for hosting him as this research began. He would also like to thank Alastair Craw, Alvaro Nolla de Celis, and David Nadler for many fruitful and enjoyable conversations about this project, as well as the referees for their thoughtful suggestions on how to improve its exposition. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Walls for -Hilb via Reids Recipe Article published earlier |
| spellingShingle | Walls for -Hilb via Reids Recipe Wormleighton, Ben |
| title | Walls for -Hilb via Reids Recipe |
| title_full | Walls for -Hilb via Reids Recipe |
| title_fullStr | Walls for -Hilb via Reids Recipe |
| title_full_unstemmed | Walls for -Hilb via Reids Recipe |
| title_short | Walls for -Hilb via Reids Recipe |
| title_sort | walls for -hilb via reids recipe |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211014 |
| work_keys_str_mv | AT wormleightonben wallsforhilbviareidsrecipe |