Walls for $G$-Hilb via Reid's recipe

The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein 3-fold quotient singularities $\mathbb{A}^3/G$ with the representation theory of the group $G$. The first crepant resolution studied in depth was the $G$-Hilbert scheme $G$-Hilb, which is also a moduli space of $\theta$-stable representations of the McKay quiver associated to $G$. As the stability parameter $\theta$ varies, we obtain many other crepant resolutions. In this paper we focus on the case where $G$ is abelian. We compute explicit inequalities defining the chamber for $G$-Hilb inside the stability space in terms of a marking of exceptional subvarieties of $G$-Hilb 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.

outline the context and notation of [8] that we will also use, for a chamber C Ă Θ the equivalence from (BKR) induces an isomorphism ϕ C : K 0 pM C q Ñ K G pA 3 q " ReppGq. Here K 0 pM C q denotes the K-group of sheaves supported on the exceptional fibre of M C Ñ A 3 {G. Walls in Θ are cut out by hyperplanes p ř i α i¨θ pχ i q " 0q for some characters χ i P IrrpGq and integers α i P Z. The inequalities in [8] have three different forms, each coming from exceptional subvarieties. Firstly, each exceptional curve C Ă G-Hilb A 3 gives an inequality of the form θpϕ C 0 pO C qq ą 0 The characters appearing in of these inequalities are packaged in collections of monomials associated to exceptional curves that were named by Nakamura in a different context as G-igsaw pieces in [19]. Our first result is to pin down which characters lie in G-igsaw pieces. Theorem 1.1. There is a combinatorial procedure that we call the unlocking procedure for computing the characters appearing in a G-igsaw piece for an exceptional curve in G-Hilb A 3 .
To briefly illustrate how the procedure works, we consider the example of G " 1 30 p25, 2, 3q. This notation means that G is the subgroup of SL 3 pCq generated by where ε is a primitive 30th root of unity. As an abelian group quotient, the singularity A 3 {G and its crepant resolutions are toric. Crepant resolutions correspond to triangulations of the simplex at height 1the 'junior simplex' -with vertices in the lattice Z 3`Z¨p 25 30 , 2 30 , 3 30 q. The triangulation for G-Hilb is shown in Figure 1. 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˚pG-Hilb A 3 , Zq indexed by characters by Craw [7]. It was later categorified by Cautis-Craw-Logvinenko [5], who expressed the locus labelled by a character χ in terms of the support of a 'tautological line bundle' R χ on G-Hilb. We will discuss this in more detail in §2.2 and §2.4. Reid's recipe is also shown in Figure 1 for the example of G " 1 30 p2, 3, 25q.  24 We will demonstrate the unlocking procedure for the curve C shown on the left side of Figure 2 marked with the character 5; that is, the character taking g Þ Ñ ε 5 . On the right side of Figure 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 G-igsaw piece for C has characters 5, 9, 11, 14. Observe that this G-igsaw piece only picked one of the two characters 7, 14 marking a divisor containing two 5-curves. We will elaborate later on how the unlocking procedure identifies which of the two characters should be added.
Walls inside Θ are of various types denoted 0-III depending on the birational geometry of the moduli spaces near the wall following [24]. Walls of Type I correspond to flops induced by curves. By [8,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 C 0 , which is very much not true for other resolutions; see [8,Example 9.13]. These walls are the easiest to compute. We denote the set of characters appearing in a G-igsaw piece for an exceptional curve C by G-igpCq. The following result is implied by [8,Cor. 5.2 + Prop. 9.7 + Theorem 9.12] and Theorem 1.1.

Proposition 1.2 (Prop. 4.2)
. Suppose C Ă G-Hilb A 3 is an exceptional p´1,´1q-curve marked with character χ by Reid's recipe. Then, the necessary inequality corresponding to C that defines a Type I wall of C 0 is given by where G-igpCq is computed by the unlocking procedure.
Walls of Type III arise from exceptional curves corresponding to certain 'boundary' edges in the triangulation for G-Hilb. The inequalities potentially defining such walls are computed by the following result, which is a consequence of [8,Cor. 5.2] and Theorem 1.1. Proposition 1.3. Suppose C Ă G-Hilb A 3 is an exceptional boundary curve marked with character χ by Reid's recipe. Then, the inequality corresponding to C is given by where G-igpCq is computed by the unlocking procedure, and where R χ is the tautological line bundle for χ.
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 ą 0. The second kind of inequality coming from divisors is more complicated.
where C ranges over exceptional curves inside D 1 .
It was shown in [8,Prop. 3.8] that there are no Type II walls in Θ. However, it is still interesting to compute the inequalities θpϕ C pO C qq ą 0 for p1,´3q-curves that would induce contractions of this type. . Suppose C Ă G-Hilb A 3 is an exceptional p1,´3q-curve marked with character χ by Reid's recipe. Then, the inequality corresponding to C is given by where G-igpCq is computed by the unlocking procedure.
We can similarly reprove [8,Theorem 9.12] by combinatorial means. Proposition 1.7 (Prop. 4.5). Each flop in a p´1,´1q-curve in G-Hilb A 3 is induced by a wall-crossing from C 0 .
We can use these formulae to show exactly which inequalities are necessary to define C 0 . Theorem 1.8 (Theorem 4.17). Suppose G Ă SL 3 pCq is a finite abelian subgroup. The walls of the chamber C 0 for G-Hilb A 3 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, ‚ the remaining walls are of Type 0 coming from divisors as in Proposition 1.4. We discuss which of these are necessary and how to reconstruct the divisor D 1 in §4.8.
We will define the term 'generalised long side' in Definition 4.13, which is an entirely combinatorial notion.
This explicit and malleable description of the walls for C 0 has applications to studying the birational geometry of other crepant resolutions of A 3 {G. In forthcoming work [17], the author and Y. Ito use this description of C 0 to study the geometry of another Hilbert scheme-like resolution introduced in [12] called the iterated G-Hilbert scheme.
x 3 y and crepant resolutions correspond to triangulations of the cone face σ X px 1`x2`x3 " 1q 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 cone face to produce two-dimensional pictures.

Reid's recipe.
Let us focus on the case M C " G-Hilb A 3 . We will denote the universal G-cluster by Z and the chamber of Θ corresponding to G-Hilb by C 0 . Craw-Reid [10] 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 e 1 , e 2 , e 3 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 Figure 3: the resulting unimodal triangulation describes the resolution G-Hilb. 3. A regular triangle and its triangulation In early versions of the McKay correspondence [22] 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 [7] 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 x α y β z γ " m 1 {m 2 where x, y, z are eigencoordinates on C 3 for G. Mark the curve C with the character by which G acts on m 1 (or m 2 ). 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: ‚ v is trivalent: D -P 2 and the three exceptional curves in D are all marked with the same character ‚ 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 χ 1 . Mark D with χ b χ 1 . ‚ 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, χ 1 , χ 2 . D has two G-invariant maps to P 2 , mark D by the two characters arising from the monomials constituting these two maps. These two characters φ 1 , φ 2 satisfy For more detail see [7,]. 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 Figure 1 with G " 1 30 p25, 2, 3q, the leftmost curve marked with the character 20 has normal p´2, 25, 0q giving the G-invariant ratio y 25 {x 2 . G acts on the numerator and denominator by the character ε Þ Ñ ε 20 , 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 surfaces 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 P 2 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'.

G-igsaw pieces.
Consider the G-clusters at torus-fixed points of G-Hilb, or triangles in the Craw-Reid triangulation. The ideal defining such a cluster is a monomial ideal and one can draw a Newton polygon in the hexagonal lattice Z 3 {Z¨p1, 1, 1q to illustrate the monomial basis. An example of a torus-fixed G-cluster for the group G " 1 6 p1, 2, 3q is shown in Figure 4. Notice that there is exactly one monomial in each character space for G as desired. 4. A torus-fixed G-cluster for G " 1 6 p1, 2, 3q The monomial ideal in Crx, y, zs defining this cluster is Torus-fixed G-clusters for 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 [19] and called a G-igsaw transformation. Let χ be the character marking C. The subset of monomials in one of the two torus-invariant G-clusters partaking in the G-igsaw transformation is called a G-igsaw piece for C. There is a single monomial that divides all others in the G-igsaw piece, and this is the monomial in the G-cluster in the character space for χ. Example 2.2. We continue the example of G " 1 6 p1, 2, 3q. The Craw-Reid triangulation and Reid's recipe for this group is shown in Figure 5. The triangle labelled by ‹ is the triangle corresponding to the G-cluster from Figure 4. Passing through the 4-curve C adjacent to the triangle ‹ performs a G-igsaw transformation with Gigsaw piece centred on the monomial with character 4, which in this case is xz. The G-igsaw transformation switches xz for y 2 -since the G-invariant ratio for C is xz{y 2 -producing the new G-cluster If we pass through the 2-curve bordering ‹ then the G-igsaw piece contains the monomials y, yz and produces the G-cluster As the two 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 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.

Tautological bundles.
The sheaf R " π˚O Z is locally free with fibre H 0 pO Z q above Z P G-Hilb A 3 .

It splits into eigensheaves
and these summands are called tautological line bundles. Since G is abelian, the R χ are line bundles. [7] gives relations between these line bundles in K-theory, which translate to divisibility relations between eigenmonomials. For a triangle τ in the Craw-Reid triangulation, denote the monomial generating R χ | U τ by r χ,τ . We usually omit reference to τ so long as the context is clear. ‚ If three lines marked with the same character χ meet at a vertex marked with ψ " χ b2 then r 2 χ " r ψ ‚ If four or five or six lines consisting of two pairs marked by characters χ, χ 1 and zero or one or two extra lines marked with further characters meet at a vertex marked with ψ " χ b χ 1 then r χ¨rχ 1 " r ψ ‚ If six lines consisting of three pairs marked by characters χ, χ 1 , χ 2 meet at a vertex marked with φ, φ 1 then 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 Craw-Cautis-Logvinenko [5] categorifies Reid's recipe via the tautological bundles. Many of the constructions in [5, §3-4] resemble constructions made in §3 below, however the computations they make are for the inverse equivalence of (BKR) to that utilised in [8] and here. It would be of interest to make a more detailed comparison.
Evident from [5,8] and below, characters marking a divisor or a single curve are special. They are termed 'essential characters' and have been further examined in [9,23].

C G-
Our motivating question for this section is the following: let C be a χ-curve, what are the characters that appear in a G-igsaw piece for C? As we shall see, the answer depends somewhat on how C sits inside G-Hilb, though it is still completely combinatorial.
3.1. Monomials for divisors. We will begin by proving results for p´1,´1q-curves (or those lying in the interior of a regular triangle), starting with the following results relating the characters marking divisors along the χ-chain to G-igsaw pieces for χ-curves.
Then G-igpCq includes the character for each Hirzebruch divisor along the χ-chain.
Proof. Theorem 2.3 implies that r χ | r ψ for each ψ P Hirzpχq on every triangle. Hence, any G-igsaw piece featuring r χ -such as a G-igsaw piece for C -will also feature each r ψ and so ψ P G-igpCq.

Lemma 3.2.
Suppose C is a p´1,´1q-curve inside a regular triangle ∆ marked with χ. Then the G-igsaw piece for C includes exactly one of each pair of characters marking a del Pezzo divisor inside ∆ that is along the χ-chain.
Proof. Consider the local picture deduced from the proof of [7] Theorem 6.1 shown in Figure 6 for eigenmonomials near a vertex v inside ∆. We assume that ∆ is a corner triangle with e 3 as vertex and one side coming from a ray out of e 1 ; the meeting of champions case is similar. Here φ 1 , φ 2 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 e 3 have the ratios x d : y b and y e : x c marking them, and the side coming from a ray out of e 1 has ratio z f : y c . r " f is the side length of the regular triangle. Suppose C is a curve marked with x d´i : y b`i z i -not necessarily incident to v -and hence that χ " χpx d´i q. We will consider only this case as the same analysis goes through for each of the curves marked with the other two ratios. The computation of eigenmonomials in Figure 6 implies the divisibility relations shown in Figure 7. The χ-chain is dashed for emphasis.
Indeed, some of the divisibility relations are clear; for example in the lower half of the diagrams when r χ " x d´i and r φ 1 " x d´i y c`k or r φ 2 " x d´i z j . The remaining claims are: respectively. Observe that the monomial x d´i doesn't appear in any G-graphs for triangles above the χ-chain in the diagram but x a`j z f´k does. Hence x d´i ∤ x a`j z f´k by the convexity of monomial bases. Similarly one sees that x d´i ∤ x k y e´j . From [10] with the˘depending on whether the χ-triangle we are using to compute a G-igsaw piece for C is 'up' or 'down' (see [10] §3.2). Notice also that since v is in the interior of a regular triangle, each of i, j, k ě 1.
f´k " i`j¯1 ě i and so y b`i z i | y b`i z f´k . It follows that r χ divides r φ 1 and does not divide r φ 2 for every del Pezzo divisor 'to the right' of C in the orientation of Figure 7, and that r χ divides r φ 2 but not r φ 1 for every del Pezzo divisor 'to the left' of C, which establishes the lemma.
This also expands on how the position of a curve determines which of the characters marking a del Pezzo surface along the χ-chain makes it into the G-igsaw piece, as alluded to while calculating walls for the example G " 1 30 p25, 2, 3q in §1.

Lemma 3.3.
Suppose C is a p´1,´1q-curve marked with χ. Then G-igpCq contains exactly one of each pair of characters marking a del Pezzo divisor along the χ-chain.
Proof. Note that the only generalisation of Lemma 3.2 in this claim is that its conclusion also holds for del Pezzo divisors along the χ-chain but in a different regular triangle to C. This follows since at least one of the monomials in the ratio marking C (and the part of the χ-chain inside ∆) still marks the χ-chain after it passes into a new regular triangle.
Lemmas 3.1-3.3 give an effective way of finding characters in G-igpCq. However, this will turn out to not supply all characters in G-igpCq. We will transition into discussion of the recursive procedure for filling in the remaining characters, and of the methods we will use to verify that all characters have been located. We start with a lemma of Craw-Ishii.
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 3.2 r χ 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 Figure 6 we can assume that r χ is not a power of a single variable. The Unique Valley Lemma [19, Lemma 3.3] of Nakamura implies that r χ divides exactly two elements of the socle of the torus-invariant G-cluster Z τ corresponding to τ. Lemma 3.4 implies that the elements in the socle of Z τ that r χ divides correspond exactly to these two characters labelling the neighbouring del Pezzo divisors. These are the outermost 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.

3.2.
Recursive procedure: 'unlocking'. We will describe the recursive procedure to compute G-igsaw pieces using only the data of Reid's recipe.
Input: an exceptional p´1,´1q-curve C marked with a character χ. Let S " tχu. dP for each del Pezzo surface along the χ-chain, add one of the two characters marking it to S H1 for each Hirzebruch divisor along the χ-chain, add the character marking it to S For each Hirzebruch divisor D along the χ-chain, define a set of curves C χ pDq by: ‚ if D is a boundary vertex on either side of which the χ-chain consists of boundary edges, then C χ pDq " H, ‚ if D is a boundary vertex of a regular triangle ∆ that is not in the previous case, then C χ pDq consists of all curves strictly inside ∆ incident to D that are not marked with χ.
H2 for each Hirzebruch divisor D along the χ-chain, add the characters from the G-igsaw pieces for curves in C χ pDq to S.
Observe that to compute the characters on the G-igsaw piece for these curves combinatorially, one may need to iteratively apply the procedure until reaching a curve where the characters in the G-igsaw piece can be read off immediately (see below for a description of such curves).
We call this the unlocking procedure as passing through a Hirzebruch divisor 'unlocks' the simpler Gigsaw puzzles for the relevant curves incident to it that one can recursively solve and then feed into the 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 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.
3.3. Type Iy curves. In order to study G-igsaw pieces for curves inside regular triangles our treatment of the three kinds of curve marked with different ratios as in Figure 6 will now diverge. We consider the first case, when the ratio marking the curve is of the formWe now consider the edges marked with ratios of the form y e´j : x a`j z j . We say that such curves are of Type Iy. The analysis from Lemma 3.2 gives a precise description of the socle of the nearby torus-invariant G-clusters -depicted in Figure 8 -and hence the G-igsaw pieces for χ-curves. 8. Generators of eigenspaces along a χpx a`j z j q-chain inside a regular triangle The G-igsaw piece for a χ-curve of Type Iy on a χ-triangle chosen so that in the coordinates used above r χ " x a`j z j is where the curve corresponds to the ith line segment from the bottom edge of the triangle. Moreover, the χ-chain does not continue outside of this regular triangle. In particular, Hirzpχq " H.
Proof. The calculation of the G-igsaw piece follows immediately from the description of the eigenmonomials in Figure 9. The χ-chain cannot continue outside of this regular triangle since neither r φ 0 nor r φ m are divisible by r χ and so Theorem 2.3 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 that depends on how far along the chain the curve is. 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.
3.4. Type Ix curves. Suppose now that C is a χ-curve inside a corner triangle with e 3 as a vertex that is marked with the ratio x d´i : y b`i z i . We say that such curves are of Type Ix. [7,Lemma 3.4] yields the identities in Figure 9 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 Figure 6 and let κ " r´pi`1q. 9. Generators of eigenspaces along a χ-chain inside a regular triangle r φm " x d´i y c Lemma 3.7. The G-igsaw piece for a χ-curve C of Type Ix on a χ-triangle chosen so that in the coordinates used above r χ " x d´i is y c`k´1 r χ . . .
where C corresponds to the pj`1qth line segment from the left edge of the triangle, and i`j`k " r. Moreover, the χ-chain continues to the right and does not continue to the left of Figure 9.
Proof. The same argument as for Lemma 3.5 applies, except that r χ does divide r φ m and so by Theorem 2.3 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 yr χ , . . . , y c r χ though y c r χ " r φ m , which we have seen corresponds to a Hirzebruch divisor appearing along the χ-chain. This follows since the e 2 -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 3.8 are of Type Ixb. This is the other base case to which the unlocking procedure reduces. Lemma 3.9. Suppose C is a χ-curve of Type Ix inside an e 2 -corner triangle and suppose that the χ-chain continues into an e 3 -corner triangle (not necessarily the boundary). 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 G-igsaw piece of the Iy curve also incident to D inside the e 3 -corner triangle.
Proof. Let C 1 be the Type Iy curve incident to D in the e 3 -corner triangle. Denote its character by χ 1 . From Lemma 3.5 the characters in the G-igsaw piece for C 1 are χ 1 and one character from each del Pezzo divisor along the χ 1 -chain inside this regular triangle. From examining the situation explicitly, on the lower χ-triangle neighbouring C one has r χ " x d´i and r χ 1 " x d´i z i so that r χ | r χ 1 near C. Also, one can see that the zone where r χ 1 divides one character from each del Pezzo divisor includes this basic 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 Figure 10.
Suppose the ratio marking the common edge of the two corner triangles is z f : y c . From Lemma 3.7 noting the change in notation coming from using an e 2 -instead of an e 3 -corner triangle, the G-igsaw piece for C is missing f characters from the χ-chain to the right. Continuing the adapted notation, we let the χ-chain enter the e 3 -corner triangle at height d´i so that there are f´i new characters along the χ-chain corresponding to the del Pezzo divisors along the χ-chain and the boundary Hirzebruch divisor D. There are f´p f´iq´1 " i´1 divisors along the χ 1 -chain, making a contribution of i characters in total including χ 1 itself. Thus these account for all of the f missing characters.
Note that this vindicates the unlocking procedure for such curves, where only one recursion was required to unlock the single Type Iy curve. The final case to consider is when the χ-chain merges into an e 1 -corner triangle, whence it becomes a different kind of curve that we will treat separately after the present case. We are still able to analyse it however.
Suppose the χ-chain passes through n e 1 -corner triangles before entering an e 2 -corner triangle. Note that the curves in this last triangle are either boundary or Iy curves and so this is the final triangle the   Figure 11 for a schematic. We denote D m :" ř m q"1 d q and BD m :" ř m q"1 pb q`dq q.
Unlocking for a Type Ix curve in a series of e 1 -corner triangles By computing the characters on the nearby del Pezzo divisor, one can tell that these Type Ix curves each have b m`im characters in their G-igsaw pieces, making the total number of characters they contribute to the G-igsaw piece of C n ÿ q"1 pd q´iq`bq`iq q " n ÿ m"1 pb q`dq q From the equations (3.2) the ratios marking the edges from e 1 for the e 1 -corner triangles are of the form z f`ř m q"1 d q : y c´ř m q"1 pb q`dq q for m " 0, . . . , n with the last edge marked by z f`ř n q"1 d q : y c´ř n q"1 pb q`dq q . In particular, this means that the e 2 -corner triangle has side length c´ř n q"1 pb q`dq q and so the final part of the χ-chain contributes c´ř n q"1 pb q`dq q characters to G-igpCq. Thus, in total we have n ÿ q"1 pb q`dq q`c´n ÿ q"1 pb q`dq q " c characters, which is exactly the number that are not accounted for by del Pezzo divisors in the e 3 -corner triangle that C inhabits by Lemma 3.7. This completes the proof of validity of the unlocking procedure for curves of Type Ix.

3.5.
Type Iz curves. The final type of curve occurring inside regular triangles is Type Iz: the curves marked by ratios of the form z f´k : x k y c`k in the coordinates we have been using for an e 3 -corner triangle. We repeat the G-igsaw analysis for these curves, represented in Figure 12 with the χ " χpz f´k q-chain dashed.
As in all previous cases, exactly one character marking each incident del Pezzo surface has a monomial divisible by r χ and so we can pin down the socle and hence the G-igsaw piece for such a curve. Lemma 3.10. 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 r χ " z f´k is y b`i´1 r χ . . . 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 Figure 11. In particular, we assume our Type Iz curve C lies in an e 1 -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 b q`iq characters in their G-igsaw pieces. From a similar calculation, one can verify that the Type Iy curves contain i q characters in their G-igsaw pieces. These unlocked curves thus contribute m´1 ÿ q"1 i q`n ÿ q"m`1 pb q`iq q " n ÿ q"m`1 b q`n ÿ q"1 i q´im characters to G-igpCq. The part of the χ-chain in the e 3 -corner triangle studied in the previous case contributes f´i 0 characters, and the part in the e 2 -corner triangle contributes c´ř n q"1 pb q`dq q. If i 0 " 0 then we unlock another Iy curve with i 0 characters appearing in its G-igsaw piece. If i 0 " 0 then the χ-chain continues along the boundary of an e 3 -corner triangle, contributing f characters. In either case there are f characters coming from the e 3 -corner triangle. Lastly, there are ř n q"1 pd q´iq q del Pezzo and Hirzebruch divisors along the part of the χ-chain inside e 1 -corner triangles, giving in total f lo omo on characters. Compare to the quantity b`d´k in Lemma 3.10, which in these coordinates is showing that every character in G-igpCq is accounted for.
3.6. G-igsaw pieces for other curves. The procedure described above also works to compute G-igsaw pieces for curves not found in the interior of regular corner triangles. Firstly, direct computations of divisibility relations show that unlocking procedure as described above carries over verbatim to curves inside or whose chains pass through a meeting of champions triangle. There is a more significant expansion required for curves corresponding to boundary edges of regular triangles.

Input:
An exceptional curve C corresponding to a boundary edge of a regular triangle. Let S " tχu.
H1 for each Hirzebruch divisor along the χ-chain, add the character marking it to S. For each Hirzebruch divisor D along the χ-chain, define a set of curves C χ pDq to consist of the curves contained in D whose chain terminates at D or whose corresponding edges are along 'broken chains' at D. A broken chain at D is a ρ-chain for some character ρ such that D is contained in the interior of the chain and the two ρ-curves incident to D are marked with different ratios. Pictorally, this means that the edges corresponding to these curves have different slopes. We say that a chain passing through D is 'straight' at D if the two curves incident to D in the chain are marked with the same ratio; that is, the corresponding edges have the same slope. These situations are shown in Figure 13. For some examples in the case G " 1 30 p25, 2, 3q depicted in Figures 1 and 18, the 15-chain is broken at the divisor D 21 marked with 21 but straight at the divisors D 19 and D 17 marked with 19 and 17 respectively, and the 5-chain is straight at all the divisors it contains. It follows that, in this example, C 6 pD 11 q " tC 8 , C 9 u and C 6 pD 21 q " tC 1 15 , C 2 15 , C 1 8u where C ρ is the curve incident to the relevant divisor marked with ρ, and C 1 15 , C 2 15 are the two 15-curves incident to D 21 .
We say that a divisor D is 'ahead' of a boundary curve C if the edge corresponding to C lies between the vertex for D and the vertex that the straight line containing C emanates from. For example, when G " 1 30 p25, 2, 3q, D 19 and D 21 are ahead of the 15-curve C contained in D 17 and D 19 , whereas D 17 is not ahead of C.
H2 For each Hirzebruch divisor D ahead of C along the χ-chain, add the characters from the G-igsaw pieces for curves in C χ pDq to S.
Output: G-igpCq " S. One proves that this procedure is valid in an analogous way to the procedure for curves of Types Ix-Iz using neighbouring divisors to compute the socle and hence the G-igsaw piece for C and then testing local divisibility relations to evidence that all these characters come from the subvarieties in the procedure. We will sketch some new elements of the proof below.
Proof. Choose coordinates so that C lies along the boundary of an e 1 -corner triangle. Consider the two boundary curves C and C 1 shown in Figure 14. 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 curves on one side of the χ-chain. It hence suffices to just compute the G-igsaw piece for the curve in the χ-chain incident to e 1 .
Suppose that D is a Hirzebruch divisor along the χ-chain. If D is at the boundary of two e 1 -corner triangles or an e 1 -corner triangle and a meeting of champions -as shown in Figure 15 -then one can check that r χ divides the G-igsaw pieces for the Type Iy curves C 3 and C 4 .
Suppose now that D borders an e 2 -and an e 3 -corner triangle, or an e 1 -corner triangle and an e 3 -corner triangle. We illustrate this situation in Figure 16, along with some of the ratios marking curves.
The same argument as in the previous case gives that r χ divides the G-igsaw pieces for C 3 and C 4 . To treat the remaining two curves C 1 and C 2 in each case, we use a generalised form of [10, §3.3.2]: an edge ℓ continues in a straight line past a boundary edge ℓ 0 if and only if the ratio marking ℓ features any common variables x, y, z raised to a strictly lower exponent than in the ratio marking ℓ 0 . One can verify this by a case-by-case analysis using as its base the original result from [10]. This implies that r χ 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.
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.
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 §3.8.
Observe that the unlocking procedure for these curves directly generalises the unlocking procedure for p´1,´1q-curves in the sense that the construction of C χ pDq in §3.2 agrees with the construction via broken chains here.
3.7. Example: G " 1 30 p25, 2, 3q. We will illustrate the unlocking procedure for G-Hilb in the case that G  We will demonstrate the unlocking procedure for a few curves in G-Hilb. Consider the 15-curve C 15 shown in Figure 19. This curve is of Type Ixb since after the p´1,´1q-curve on the left side it feeds into a boundary edge of a regular triangle. This gives G-igpC 15 q " t15, 17, 19, 21u Consider the 5-curve C 5 shown in Figure 20. It passes into the right side of the junior simplex, unlocking the 9-curve of Type Iy and giving G-igpC 5 q " t5, 9, 11, 14u Consider the 2-curve C 2 shown in Figure 21. 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-igpC 2 q " t2, 17, 22, 27u Lastly, we will consider the boundary 15-curve C 1 15 shown in Figure 22.  3.8. Example: G " 1 35 p1, 3, 31q. We will use the example of G " 1 35 p1, 3, 31q to illustrate a phenomenon implicit, but less clear in the long side picture. The triangulation for G-Hilb is shown in Figure 23. Reid's recipe is found in Figure 24.
Consider the 3-curve C 3 incident to e 1 . The unlocking procedure for this curve is shown in Figure 25 giving G-igpC 3 q " 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 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 r 3 there. for all exceptional curves C Ă G-Hilb. If this inequality is necessary to define C 0 , the geometry of C determines the type of the wall as follows: ‚ If C is a p´1,´1q-curve -that is, it corresponds to an interior edge inside a regular triangle -then pθpϕ C 0 pO C qq " 0q X C 0 is a Type I wall.
‚ If C is a p1,´3q-curve -that is, it corresponds to one of the edges incident to a trivalent vertexthen pθpϕ C 0 pO C qq " 0q X C 0 is a Type II wall.
‚ If C is contained in a Hirzebruch divisor but it is not in either of the previous cases, then As discussed, there are no Type II walls and so the inequalities from the second case cannot be necessary. One can express the inequality θpϕ C pO C qq ą 0 abstractly via [8, Corollary 5.2], a consequence of which is Any character ρ not in G-igpCq has R ρ | C " O C and so it doesn't appear in the sum above. It follows that which empowers the unlocking procedure to compute and interpret these inequalities.

Inequalities from divisors.
To complete the classification, the walls of Type 0 are obtained from divisors. Suppose F is a G-equivariant coherent sheaf on C 3 with H 0 pFq " CrGs as G-modules; in the language of [8] F is a G-constellation, which generalises the notion of G-cluster. Suppose w is a Type 0 wall of C 0 with unstable locus D. Let θ P w and suppose that S Ă F is a nontrivial subsheaf such that θpSq " 0. That is, F is θ-destabilised by S. [8,Cor. 4.6 + Remark 4.7] says that either S or Q " F{S is rigid and so defines a constant family on D. It follows that D is exactly the divisor parameterising such rigid sub-or quotient sheaves. From [8, Cor. 5.6 + Theorem 9.5] the equation of a type 0 wall w is one of the following: ‚ if w comes from a divisor D parameterising rigid subsheaves, then D " D ψ is irreducible and the inequality defining the wall is θpϕ C 0 pR´1 ψ | D q " θpψq ą 0.
‚ if w comes from a divisor D 1 parameterising rigid quotient sheaves, then D 1 is connected but potentially reducible and the inequality defining the wall is θpω D 1 q " θpQq ă 0 where Q is the representation defined by a quotient sheaf in D 1 .
We can be more precise in the second case. For the representation Q to be constant across D, it means that R ρ | D 1 is trivial for all ρ Ă Q. Equivalently, all torus-invariant G-clusters in D 1 share the same eigenmonomial r ρ for each ρ Ă Q or, also equivalently, ρ R G-igpCq for any C Ă D 1 .

Type I walls.
We know from [8,Theorem 9.12] that all flops in a single p´1,´1q-curve C are achieved by a wall-crossing from C 0 . Moreover, we have degpR ρ | C q " 1 for all ρ P G-igpCq from [8, Corollary 6.3].
The unlocking procedure hence gives a combinatorial way of writing down the equations of these walls.

Proposition 4.2.
Suppose C Ă G-Hilb A 3 is an exceptional p´1,´1q-curve marked with character χ by Reid's recipe. Then, the Type I wall corresponding to C is given by where G-igpCq is computed by the unlocking procedure.

No Type II walls.
Proposition 4.3. Suppose C Ă G-Hilb A 3 is an exceptional p1,´3q-curve marked with character χ by Reid's recipe. Then, the inequality corresponding to C is given by where G-igpCq is computed by the unlocking procedure.
Proof. Notice that such a curve C lies inside the exceptional P 2 in the meeting of champions case when the meeting of champions triangle has side length 0. Thus the P 2 is marked with χ b2 and lies in the socle of any torus-invariant G-cluster. From Theorem 2.3 r χ b2 " r 2 χ and so r 2 χ is the furthest character from r χ in the G-igsaw piece in some direction. Note that degpR ρ | C q " mintk : r k χ | r ρ u and so all the characters in G-igpCq appear with multiplicity 1 except for r 2 χ , which appears with multiplicity 2. This gives the required formula.
As a result we can immediately deduce the conclusion of [8,Prop. 3.8] for C 0 . Proof. Suppose C is an exceptional p1,´3q-curve marked with χ. From Prop. 4.3 a G-igsaw piece for C consists of χ, χ 2 , and the characters marking the (Hirzebruch) divisors along the χ-chain. Let D 1 be the exceptional P 2 containing C. Consider the inequality for rigid quotients parameterised by D 1 : from Prop. 4.1 the characters appearing in this inequality are exactly the characters in the G-igsaw pieces of all three χ-curves converging at D 1 . These are tχ, χ b2 u Y Hirzpχq which are exactly the characters appearing in the inequality for C. However, the inequality for rigid quotients parameterised by D 1 has multiplicities all equal to 1. When combined with the inequality θpχ b2 q ą 0 coming from rigid subsheaves parameterised by D 1 this implies that the inequality ϕ C 0 pO C q ą 0 is redundant. 4.5. All flops in p´1,´1q-curves. Using Proposition 4.2 and the unlocking procedure one can show directly that every p´1,´1q-curve produces a necessary inequality, recovering [8,Theorem 9.12] by purely combinatorial means. Proposition 4.5. Suppose C is an exceptional p´1,´1q curve inside G-Hilb A 3 . Then the inequality θpϕ C 0 pO C qq ą 0 is necessary and so defines a wall of C 0 .
Proof. Suppose C is marked with χ. From the unlocking procedure we can write the inequality corresponding to C in the form where ρ j are the characters marking curves C j unlocked by C and ψ j i are the characters in the G-igsaw piece for C j . 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 C j is In order to express (:) 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 collection of del Pezzo divisors, curves in other regular triangles will either feature different del Pezzo divisors or unlock different curves, and χcurves along a boundary edge will have different unlocking behaviour. In particular, the inequalities from these curves will not be summands of the inequality (:). Inequalities from rigid quotients not containing χ will also not be summands of (:) since the unlocking procedure implies that there are no divisors D ρ along the χ-chain for which all characters marking curves incident to D ρ are represented in G-igpCq. It follows that (:) is necessary.

Irredundant inequalities -examples.
The aim of these final sections is to precisely describe all the walls of C 0 . We start with an accessible example. inequalities coming from curves and divisors that define C 0 via the unlocking procedure.
We can also show this directly via unlocking. This reproves Corollary 4.4. Lemma 4.10. Suppose C is a χ-curve. If the unlocking procedure for C doesn't unlock a curve or divisor marked with χ 2 then all the coefficients in the inequality θpϕ C 0 pO C qq ą 0 are equal to 0 or 1.
Proof. This is because if some ρ has degpR ρ | C q ě 2 then r 2 χ | r ρ and so r 2 χ must feature in the G-igsaw piece for C and is hence equal to r χ 2 near C. Lemma 4.11. Suppose a curve C 0 unlocks a curve C 1 of character ρ. Let ψ P G-igpC 1 q. If C is a curve that unlocks C 0 , then degpR ψ | C q ě degpR ρ | C q.
Proof. As used previously, degpR ρ | C q " maxtk P Z ě0 : r k χ | r ρ u. From this formulation, clearly if r ρ | r ψ then degpR ψ | C q ě degpR ρ | C q, but this is the case by definition of G-igsaw piece.
We say that an inequality ř i α i θpχ i q ą 0 with nonnegative coefficients is a summand of another inequality ř j β j θpρ j q ą 0 with nonnegative coefficients if the difference ř i α i θpχ i q´ř j β j θpρ j q also has nonnegative coefficients in the basis Irr G. 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 C 0 . Lemma 4.12. Suppose C is a curve on the boundary of a regular triangle marked with a character χ. Suppose the χ-chain contains a p´1,´1q-curve. Then the inequality θpϕ C 0 pO C qq ą 0 is redundant.
Proof. Suppose C is marked with character χ. Let C 0 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 C 0 along with the characters in the G-igsaw pieces for any curves C 1 , . . . , C n unlocked by C at Hirzebruch divisors before C 0 . Let the character marking C i be χ i . The inequality for C decomposes as where α ρ and β i ρ are nonnegative multiplicities given by the appropriate calculation of degpR ρ | ? q. Note that α χ " 1. One can thus write From Lemma 4.11, α ρ´1 and β ρ´βχ 1 are both nonnegative. If all the remaining ρ in these sums with nonzero coefficients are characters marking divisors then one can express each term γ ρ θpρq " γ ρ θpϕ C 0 pR´1 ψ | D qq for some divisor D, thus evidencing that (♠) is redundant. Suppose instead that some ρ " ρ 1 marks a curve unlocked by C 0 or some C i . We assume the latter; the former is treated identically. Denote this new curve by C i,1 . Then where again each coefficient is nonnegative by Lemma 4.11 applied to C i,1 . Observe that there are strictly fewer nonzero coefficients in this expression than before, since at the least we removed the term for ρ 1 .
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 (♠) is redundant.
Definition 4.13. 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 35 p1, 3, 31q is a generalised long side as can be seen in Figure 24.
Example 4.14. We compute the inequalities for curves along the 15-chain in G-Hilb for G " 1 35 p1, 3, 31q. From the unlocking procedure or computing G-igsaw pieces directly, the inequalities for the 15-curves starting from e 1 and moving downwards are θ 15`θ18`θ21`θ24`θ7`θ10`θ13`θ16`θ11`θ14`θ17`θ20 ą 0 (A 15 ) θ 15`θ18`θ21`θ16`θ11`θ14`θ17 ą 0 (B 15 ) θ 15`θ16`θ17`θ18 ą 0 (C 15 ) θ 15`θ16`θ17`θ18 ą 0 (D 15 ) Clearly (C 15 ) and (D 15 ) depend on each other; the inequality is the same since they are fibres of the P 1 -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: 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 35 p1, 3, 31q, the bolded curves in Figure 27  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. 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. χ 2 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 " χ 2 . Of course, this would mean that ρ " χ, but chains do not self-intersect. Hence, χ 2 does not appear in the G-igsaw piece for C and so all multiplicities must be equal to 1 by Lemma 4.10. 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 redundant, we will decompose these inequalities similarly to before. Let C be such a curve and write θpϕ C 0 pO C qq " θpχq`ÿ ψPHirzpCq α ψ θpψq`n ÿ i"1 ÿ ρPG-igpC i q β i ρ θpρq where C 1 , . . . , C n are the curves unlocked by C. By exactly the same methods as in the proof of Lemma 4.12, one can express the final term as a sum of inequalities from curves and divisors. The first two terms are equal to θpχq`ÿ ψPHirzpCq α ψ θpψq " θpϕ C 0 pO C 1 qq`ÿ ψPHirzpχq pα ψ´1 qθpϕ C 0 pR´1 ψ | D ψ qq where C 1 is a final χ-curve and D ψ is the divisor marked with ψ. Of course α ψ ě 1 and so we have shown that the inequality from C is redundant.
We consider the example G " 1 25 p1, 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 25 p1, 3, 21q in Figures  28-29. 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, with inequalities θ 3`θ8`θ12`θ16`θ20`θ24 ą 0 (F 3 ) θ 9`θ10`θ11`θ12 ą 0 (C 9 ) θ 21`θ22`θ23`θ24 ą 0 (C 21 )  ‚ 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, ‚ each remaining wall is of Type 0 and comes from a divisor parameterising a rigid quotient.
Proposition 4.1 describes how to recover the unstable locus or the corresponding reducible divisor D 1 for each wall of Type 0 from a rigid quotient. Let w be a wall of C 0 . 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 D 1 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.

F
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 [5]. ‚ exploring any relations between analogs to Reid's recipe in other settings and walls in stability (for instance, dimer models [3,13]). ‚ reverse-engineering a partial Reid's recipe for other resolutions M C from an explicit description of the walls of C, and examining whether this has any categorical content.