Flops and clusters in the homological minimal model programme
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Abstract
Suppose that \(f:X\rightarrow \mathrm{Spec}\,R\) is a minimal model of a complete local Gorenstein 3fold, where the fibres of f are at most one dimensional, so by Van den Bergh (Duke Math J 122(3):423–455, 2004) there is a noncommutative ring \(\Lambda \) derived equivalent to X. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of \(\Lambda \) is finite dimensional, improving a result of Donovan and Wemyss (Contractions and deformations, arXiv:1511.00406). We further show that the mutation functor of Iyama and Wemyss (Invent Math 197(3):521–586, 2014, §6) is functorially isomorphic to the inverse of the Bridgeland–Chen flop functor in the case when the factor of \(\Lambda \) is finite dimensional. These results then allow us to jump between all the minimal models of \(\mathrm{Spec}\,R\) in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling us to prove the Craw–Ishii conjecture in this setting. Second, we are able to precisely count the number of minimal models, and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity. Third, we prove a bijective correspondence between maximal modifying Rmodule generators and minimal models, and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry. This lifts the Auslander–McKay correspondence to dimension three.
1 Introduction
1.1 Setting
One of the central problems in the birational geometry of 3folds is to construct, given a suitable singular space \(\mathrm{Spec}\,R\), all its minimal models \(X_i\rightarrow \mathrm{Spec}\,R\) and to furthermore pass between them, via flops, in an effective manner.
The classical geometric method of producing minimal models is to take Proj of an appropriate graded ring. It is known that the graded ring is finitely generated, so this method produces a variety equipped with an ample line bundle. However, for many purposes this ample bundle does not tell us much information, and one of the themes of this paper, and also other homological approaches in the literature, is that we should be aiming for a much larger (ideally tilting) bundle, one containing many summands, whose determinant bundle recovers the classically obtained ample bundle. These larger bundles, and their noncommutative endomorphism rings, encode much more information about the variety than simply the ample bundle does.
At the same time, passing between minimal models in an effective way is also a rather hard problem in general. There are various approaches to this; one is to hope for some form of GIT chamber decomposition in which wandering around, crashing through appropriate walls, eventually yields all the projective minimal models. Another is just to find a curve, flop, compute all the geometry explicitly, and repeat. Neither is ideal, since both usually require a tremendous amount of calculation. For example the GIT method needs first a calculation of the chamber structure, then second a method to determine what happens when we pass through a wall. Without additional information, and without just computing both sides, standing in any given chamber it is very hard to tell which wall to crash through next in order to obtain a new minimal model.
The purpose of this paper is to demonstrate, in certain cases where we have this larger tilting bundle, that the extra information encoded in the endomorphism ring can be used to produce a very effective homological method to pass between the minimal models, both in detecting which curves are floppable, and also in producing the flop. As a consequence, this then supplies us with the map to navigate the GIT chambers, and the much finer control that this map gives means that our results imply (but are not implied by) many results in derived category approaches to GIT, braiding of flops, and faithful group actions. We outline only some in this paper, as there are a surprising number of other corollaries.
1.2 Overview of the algorithm
We work over \(\mathop {{}_{}\mathbb {C}}\nolimits \). Throughout this introduction, for simplicity of the exposition, the initial geometric input is a crepant projective birational morphism \(X\rightarrow \mathrm{Spec}\,R\), with one dimensional fibres, where R is a three dimensional normal Gorenstein complete local ring and X has only Gorenstein terminal singularities. This need not be a flopping contraction, X need not be a minimal model, and R need not have isolated singularities. We remark that many of our arguments work much more generally than this, see Sect. 1.5.
At its heart, this paper contains two key new ideas. The first is that certain factors of the algebra \(\Lambda \) encode noncommutative deformations of the curves, and thus detects which curves are floppable. The second is that when curves flop we should not view the flop as a variation of GIT, rather we should view the flop as a change in the algebra (via a universal property) whilst keeping the GIT stability constant. See 1.3. Specifically, we prove that the mutation functor of [23, §6] is functorially isomorphic to the inverse of the Bridgeland–Chen flop functor [5, 10] when the curves are floppable. It is viewing the flop via this universal property that gives us the new extra control over the process; indeed it is the mutated algebra that contains exactly the information needed to iterate, without having to explicitly calculate the geometry at each step.
The initial input is the crepant morphism \(X\rightarrow \mathrm{Spec}\,R\) above, where X has only Gorenstein terminal singularities.
Step 1: Contractions The first task is to determine which subsets of the curves contract to points without contracting a divisor, and can thus be flopped. Although this is usually obvious at the input stage (we generally understand the initial input), it becomes important after the flop if we are to continue running the programme.
Let C be the schemetheoretic fibre above the unique closed point of \(\mathrm{Spec}\,R\), so that taking the reduced scheme structure we obtain \(\bigcup _{i=1}^n C_i\) with each \(C_i\cong \mathbb {P}^1\). We pick a subset of the curves, say \(I\subseteq \{1,\ldots ,n\}\), and ask whether \(\bigcup _{i\in I}C_i\) contracts to a point without contracting a divisor. Corresponding to each curve \(C_i\) is an idempotent \(e_i\) in the algebra \(\Lambda :=\mathrm{End}_R(N)\) from (1.A), and we set \(\Lambda _I:= \Lambda /\Lambda (1\sum _{i\in I}e_i)\Lambda \). Our first main result, a refinement of [16], is the following.
Theorem 1.1
(\(=\) 3.5) \(\bigcup _{i\in I}C_i\) contracts to a point without contracting a divisor if and only if \(\dim _\mathbb {C}\Lambda _I<\infty \).
The following is our next main result. When \(C_i\) flops, we denote the flop by \(X^+\).
Theorem 1.2
 (1)
The irreducible curve \(C_i\) flops if and only if \(\upnu _i N\ne N\).
 (2)
If \(\Gamma \) denotes the natural algebra associated to the flop \(X^+\) [49], then \(\Gamma \cong \upnu _i\Lambda \).
 (3)If further \(X\rightarrow \mathrm{Spec}\,R\) is a minimal model, and \(\dim _\mathbb {C}\Lambda _i<\infty \), thenwhere \(\Psi \) are the functors in (1.A), and \(\mathsf{{Flop}}\) is the inverse of the flop functor of Bridgeland–Chen [5, 10].$$\begin{aligned} \Phi _i\cong \Psi _{X^+}\circ \mathsf{{Flop}}\circ \Psi _X^{1} \end{aligned}$$
We also remark that the proof of 1.2 does not need or refer to properties of the generic hyperplane section, so there is a good chance that in future we will be able to remove the assumption that X has only Gorenstein terminal singularities, see B.2.
However, of the results in 1.2, it is part two that is the key, since it allows us to iterate. First, 1.2(2) allows us to immediately read off the dual graph of the flop without explicitly calculating it in coordinates, since the dual graph can be read off from the mutated quiver. Second, and most importantly, combining 1.2(2) with 1.1 (applied to \(\upnu _i\Lambda \)) allows us to detect which curves are contractible after the flop by inspecting factor algebras of the form \(\upnu _i\Lambda /\upnu _i\Lambda (1e)\upnu _i\Lambda \). There is no way of seeing this information on the original algebra \(\Lambda \), which is one of the main reasons why fixing \(\Lambda \) and changing the stability there does not lend itself easily to iterations. Hence we do not change GIT stability, we instead change the algebra by plugging \(\upnu _i\Lambda \) back in as the new input, and continue the programme in an algorithmic way. This is summarised in Fig. 2.
1.3 Applications to GIT
There are various other outputs to the Homological MMP that for clarity have not been included in Fig. 2. One such output, when the curve does flop, is obtained by combining 1.2(2) with [27, 5.2.5]. This shows that it is possible to output the flop as a fixed, specified, GIT moduli space of the mutated algebra.
Corollary 1.3
 (1)
\(\mathcal {M}_{\mathsf{rk},\upphi }(\Lambda )\cong X\) for all \(\upphi \in C_+(\Lambda )\).
 (2)
\(\mathcal {M}_{\mathsf{rk},\upphi }(\upnu _i\Lambda )\cong X^+\) for all \(\upphi \in C_+(\upnu _i\Lambda )\).
This allows us to view the flop as changing the algebra but keeping the GIT chamber structure fixed, and so since mutation is easier to control than GIT wall crossing, 1.3 implies, but is not implied by, results in GIT. Mutation always induces a derived equivalence, and it turns out that it is possible to track the moduli space in 1.3(2) back across the equivalence to obtain the flop as a moduli space on the original algebra. Again, as in Step 1 in Sect. 1.2, the subtlety is that the flop of Bridgeland–Chen is constructed as a moduli with respect to the morphism g in (1.B), whereas here we want to establish the flop as a moduli with respect to global information associated to the morphism f.
The following moduli–tracking theorem allows us to do this. Later, we prove it in much greater generality, and with multiple summands.
Proposition 1.4
The technical assumptions in 1.4 hold for flopping contractions, and they also hold automatically for any noncommutative crepant resolution (\(=\)NCCR) or maximal modification algebra (\(=\)MMA) in dimension three. Thus 1.4 can be applied to situations where the fibre is two–dimensional, and we expect to be able to extend some of the techniques in this paper to cover general minimal models of general Gorenstein 3folds. We also remark that 1.4 is known in special situations; it generalises [45, 3.6, 4.20], which dealt with Kleinian singularities, and [39, 6.12], which dealt with specific examples of smooth 3folds with mutations of NCCRs given by quivers with potentials at vertices with no loops.
It is also possible to track moduli from \(\upnu _i\Lambda \) to moduli on \(\Lambda \), see 5.13(2). This leads to the following corollary.
Corollary 1.5
Of course, our viewpoint is that 1.5 should be viewed as a consequence of the Homological MMP, since without the extra data the Homological MMP offers, it is hard to say which should be the next wall to crash through, and then which wall to crash through after that. The information in the next chamber needed to iterate is contained in \(\upnu _i\Lambda \), not the original \(\Lambda \). Mutation allows us to successfully track this data, and as a consequence we obtain the following corollary.
Corollary 1.6
(\(=\) 6.2(1)) There exists a connected path in the GIT chamber decomposition of \(\Lambda \) where every minimal model of \(\mathrm{Spec}\,R\) can be found, and each wall crossing in this path corresponds to the flop of a single curve.
We remark that 1.6 was verified in specific quotient singularities in [39, 1.5], and is also implicit in the setting of \(cA_n\) singularities in [24, §6], but both these papers relied on direct calculations. The Homological MMP removes the need to calculate.
The following conjecture is an extension to singular minimal models of a conjecture posed by Craw–Ishii [13], originally for quotient singularities and their NCCRs.
Conjecture 1.7
(Craw–Ishii) Suppose that S is an arbitrary complete local normal Gorenstein 3fold with rational singularities, and \(\mathrm{End}_R(N)\) is an MMA where \(R\in \mathrm{add}\,N\). Then every projective minimal model of \(\mathrm{Spec}\,R\) can be obtained as a quiver GIT moduli space of \(\mathrm{End}_R(N)\).
There are versions of the conjecture for rings R that are not complete local, but in the absence of a grading, which for example exists for quotient singularities, there are subtleties due to the failure of Krull–Schmidt. Nevertheless, a direct application of 1.6 gives the following result.
Corollary 1.8
(\(=\) 6.2(2)) The Craw–Ishii conjecture is true for all compound du Val (\(=\)cDV) singularities.
In fact we go further than 1.6 and 1.8, and describe the whole GIT chamber structure. In principle this is hard, since obtaining the numbers \(b_j\) needed in 1.4 directly on the 3fold is difficult without explicit knowledge of \(\Lambda \) or indeed without knowing the explicit equation defining R. However, the next result asserts that mutation is preserved under generic hyperplane sections, and this allows us to obtain the numbers \(b_j\) by reducing to the case of Kleinian surface singularities, about which all is known.
Lemma 1.9
(\(=\) 5.20) With the setup \(X\rightarrow \mathrm{Spec}\,R\) as above, if g is a sufficiently generic hyperplane section, then \(\Lambda /g\Lambda \cong \mathrm{End}_{R/gR}(N/gN)\), and minimal approximations are preserved under tensoring by R / gR.
Once we have obtained the \(b_j\) for all exchange sequences, which in particular depends only on the curves which appear in the partial resolution \(X_2\), we are able to use this data to do two things. First, we are able to compute the full GIT chamber structure.
Corollary 1.10
 (1)
\(C_+(\Lambda )\) is a chamber in \(\Theta \).
 (2)
For sufficiently generic \(g\in R\), the chamber structure of \(\Theta \) for \(\Lambda \) is the same as the chamber structure for \(\mathrm{End}_{R/gR}(N/gN)\). There are a finite number of chambers, and the walls are given by a finite collection of hyperplanes containing the origin. The coordinate hyperplanes \(\upvartheta _i=0\) are included in this collection.
 (3)
Tracking all the chambers \(C_+(\upnu _{i_t}\ldots \upnu _{i_1}\Lambda )\) through mutation, via knitting combinatorics, gives the full chamber structure of \(\Theta \).
We list and draw some examples in 5.26 and Sect. 7. In the course of the proof of 1.10, if \(\Pi \) denotes the preprojective algebra of an extended Dynkin diagram and e is an idempotent containing the extending vertex, then in 5.24 we describe the chamber structure of \(\Theta (e\Pi e)\) by intersecting hyperplanes with a certain subspace in a root system, a result which may be of independent interest. It may come as a surprise that the resulting chamber structures are not in general the root system of a Weyl group, even up to an appropriate change of parameters, and this has implications to the braiding of flops [17] and faithful group actions [20]. It also means, for example, that any naive extension of [46] or [7, 14] is not possible, since root systems and Weyl groups do not necessarily appear. However, this phenomenon will come as no surprise to Pinkham [41, p366].
Second, we are able to give minimal as well as maximal bounds on the number of minimal models, based only on the curves which appear in the partial resolution \(X_2\). The Homological MMP enriches the GIT chamber structure not only with the mutated quiver (allowing us to iterate), but by 1.9 it also enriches it with the information of the curves appearing after cutting by a generic hyperplane. Certainly if two minimal models X and Y cut under generic hyperplane section to two different curve configurations, then X and Y must be different minimal models. The surface curve configurations obtained via mutation can be calculated very easily using knitting combinatorics, so keeping track of this extra information (see e.g. 7.3) allows us to enhance the chamber structure, and to improve upon the results of [41] as follows.
Corollary 1.11
(\(=\) 5.28) Suppose that R is a cDV singularity, with a minimal model \(X\rightarrow \mathrm{Spec}\,R\). Set \(\Lambda :=\mathrm{End}_R(N)\) as in (1.A). By passing to a general hyperplane section g as in 1.10, the number of minimal models of \(\mathrm{Spec}\,R\) is bounded below by the number of different curve configurations obtained in the enhanced chamber structure of \(\Theta (\Lambda /g\Lambda )\).
A closer analysis (see e.g. 5.27) reveals that it is possible to obtain better lower bounds, also by tracking mutation, but we do not detail this here. See Sects. 5.4 and 7.1.
1.4 Auslander–McKay correspondence
There are also purely algebraic outputs of the Homological MMP. One such output is that we are able to lift the Auslander–McKay correspondence from dimension two [1] to 3fold compound du Val singularities. One feature is that for 3folds, unlike for surfaces, there are two correspondences. First, there is a correspondence (1.C) between maximal modifying (\(=\)MM) Rmodule generators and minimal models, and then for each such pair there is a further correspondence (in parts (1) and (2) below) along the lines of the classical Auslander–McKay Correspondence. Parts (3) and (4), the relationship between flops and mutation, describe how these two correspondences relate.
Corollary 1.12
 (1)
For any fixed MM generator, its nonfree indecomposable summands are in onetoone correspondence with the exceptional curves in the corresponding minimal model.
 (2)
For any fixed MM generator N, the quiver of \({\underline{\mathrm{End}}}_R(N)\) (for definition see 4.9) encodes the dual graph of the corresponding minimal model.
 (3)
The full mutation graph of the MM generators coincides with the full flops graph of the minimal models.
 (4)
The derived mutation groupoid of the MM generators is functorially isomorphic to the derived flops groupoid of the minimal models.
For all undefined terminology, and the detailed description of the bijection maps in (1.C), we refer the reader to Sects. 2.1, 4.2 and 6.2. We remark that the graphs in (3) are simply the framework to express the relationship between flops and mutation on a combinatorial level, and the derived groupoids in (4) are the language to express the relationship on the level of functors.
In addition to 1.12, we also establish the following. For unexplained terminology, we again refer the reader to Sect. 6.2.
Corollary 1.13
 (1)
R admits only finitely many MM generators, and any two such modules are connected by a finite sequence of mutations.
 (2)
The mutation graph of MM generators can be viewed as a subgraph of the skeleton of the GIT chamber decomposition of \(\Theta (\Lambda )\).
 (3)
The mutation graph of MM generators coincides with the skeleton of the GIT chamber decomposition. In particular, the number of basic MM generators equals the number of chambers.
Although (3) is simply a special case, the setting when R has only isolated singularities is particularly interesting since it relates maximal rigid and cluster tilting objects in certain Krull–Schmidt Homfinite 2CY triangulated categories to birational geometry.
We also remark that the above greatly generalises and simplifies [7, 14], which considered isolated \(cA_n\) singularities with smooth minimal models and observed the connection to the Weyl group \(S_n\), [39] which considered specific quotient singularities, again with smooth minimal models, and [24] which considered general \(cA_n\) singularities. All these previous works relied heavily on direct calculation, manipulating explicit forms.
Based on the above results, we offer the following conjecture.
Conjecture 1.14
Let R be a Gorenstein 3fold with only rational singularities. Then R admits only a finite number of basic MM generators if and only if the minimal models of \(\mathrm{Spec}\,R\) have onedimensional fibres (equivalently, R is cDV).
The direction \((\Leftarrow )\) is true by 1.13. Although we cannot yet prove \((\Rightarrow )\), by strengthening some results in [2] to cover nonisolated singularities, we do show the following as a corollary of a more general ddimensional result.
Proposition 1.15
(\(=\) 6.12) Suppose that R is a complete local 3dimensional normal Gorenstein ring, and suppose that R admits an NCCR (which by [50] implies that the minimal models of \(\mathrm{Spec}\,R\) are smooth). If R admits only finitely many basic MM generators up to isomorphism, then R is a hypersurface singularity.
1.5 Generalities
In this paper we work over an affine base, restrict to complete local rings, work over onedimensional fibres and sometimes restrict to minimal models. Often these assumptions are not necessary, and are mainly made just for technical simplification of the notation and exposition. In “Appendix B” we outline questions and conjectures for when R is not Gorenstein, including flips and other aspects of the MMP.
1.6 Notation and conventions
Everything in this paper takes place over the complex numbers \(\mathbb {C}\), or any algebraically closed field of characteristic zero. All complete local rings appearing are the completions of finitely generated \(\mathbb {C}\)algebras at some maximal ideal. Throughout modules will be left modules, and for a ring A, \(\hbox {mod}\,A\) denotes the category of finitely generated left Amodules, and \(\mathrm{fdmod}\,A\) denotes the category of finite length left Amodules. For \(M\in \hbox {mod}\,A\) we denote by \(\mathrm{add}\,M\) the full subcategory consisting of summands of finite direct sums of copies of M. We say that M is a generator if \(R\in \mathrm{add}\,M\), and we denote by \(\mathrm{proj}\,A:=\mathrm{add}\,A\) the category of finitely generated projective Amodules. Throughout we use the letters R and S to denote commutative noetherian rings, whereas Greek letters \(\Lambda \) and \(\Gamma \) will denote noncommutative noetherian rings.
We use the convention that when composing maps fg, or \(f{\cdot }g\), will mean f then g, and similarly for quivers ab will mean a then b. Note that with this convention \(\mathrm{Hom}_R(M,X)\) is a \(\mathrm{End}_R(M)\)module and \(\mathrm{Hom}_R(X,M)\) is a \(\mathrm{End}_R(M)^\mathrm{op}\)module. Functors will use the opposite convention, but this will always be notated by the composition symbol \(\circ \), so throughout \(F\circ G\) will mean G then F.
2 General preliminaries
We begin by outlining the necessary preliminaries on aspects of the MMP, MM modules, MMAs, perverse sheaves, and mutation. With the exception of 2.15, 2.21, 2.22, 2.25 and 2.26 nothing in this section is original to this paper, and so the confident reader can skip to Sect. 3.
2.1 General background
Singular dCY algebras are a convenient language that unify the commutative Gorenstein algebras and the mildly noncommutative algebras under consideration.
Definition 2.1
When \(\Lambda =R\), it is known [21, 3.10] that R is dsCY if and only if R is Gorenstein and equicodimensional with \(\dim R=d\). One noncommutative source of dsCY algebras are maximal modification algebras, introduced in [23] as the notion of a noncommutative minimal model.
Definition 2.2
The notion of a smooth noncommutative minimal model, called a noncommutative crepant resolution, is due to Van den Bergh [50].
Definition 2.3
Suppose that R is a normal dsCY algebra. By a noncommutative crepant resolution (NCCR) of R we mean \(\Lambda :=\mathrm{End}_R(N)\) where \(N\in \mathrm{ref}\,R\) is such that \(\Lambda \in \mathrm{CM}\,R\) and \(\mathrm{gl.dim}\,\Lambda =d\).
In the setting of the definition, provided that N is nonzero, it is equivalent to ask for \(\Lambda \in \mathrm{CM}\,R\) and \(\mathrm{gl.dim}\,\Lambda <\infty \) [50, 4.2]. Note that any modifying module N gives rise to a dsCY algebra \(\mathrm{End}_R(N)\) by [23, 2.22(2)], and \(\mathrm{End}_R(N)\) is dCY if and only if \(\mathrm{End}_R(N)\) is an NCCR [23, 2.23]. Further, an NCCR is precisely an MMA with finite global dimension, that is, a smooth noncommutative minimal model. On the base R, those NCCRs where \(N\in \mathrm{CM}\,R\) can be characterised in terms of CT modules [23, 5.9(1)].
Definition 2.4
Throughout this paper we will freely use the language of terminal, canonical and compound Du Val (\(=\)cDV) singularities in the MMP, for which we refer the reader to [11, 35, 43] for a general overview. Recall that a normal scheme X is defined to be \(\mathbb {Q}\)factorial if for every Weil divisor D, there exists \(n\in \mathbb {N}\) for which nD is Cartier. Also, if X and \(X_{\mathrm {con}}\) are normal, then recall that a projective birational morphism \(f:X\rightarrow X_{\mathrm {con}}\) is called crepant if \(f^*\omega _{X_{\mathrm {con}}}=\omega _X\). A \(\mathbb {Q}\)factorial terminalisation, or minimal model, of \(X_{\mathrm {con}}\) is a crepant projective birational morphism \(f:X\rightarrow X_{\mathrm {con}}\) such that X has only \(\mathbb {Q}\)factorial terminal singularities. When X is furthermore smooth, we call f a crepant resolution.
The following theorem, linking commutative and noncommutative minimal models, will be used implicitly throughout.
Theorem 2.5
 (1)
\(X\rightarrow \mathrm{Spec}\,R\) is a minimal model.
 (2)
\(\Lambda \) is an MMA of R.
The result is also true when R is complete local, see [24, 4.19].
Recall that a \(\mathbb {Q}\)Cartier divisor D is called gnef if \(D\cdot C\ge 0\) for all curves contracted by g, and D is called gample if \(D\cdot C> 0\) for all curves contracted by g. There are many (equivalent) definitions of flops in the literature, see e.g. [34]. We will use the following.
Definition 2.6
Suppose that \(f:X\rightarrow \mathrm{Spec}\,R\) is a crepant projective birational morphism, where R is complete local, with at most onedimensional fibres. Choose \(\bigcup _{i\in I}C_i\) in X, contract them to give \(g:X\rightarrow X_{\mathrm {con}}\), and suppose that g is an isomorphism away from \(\bigcup _{i\in I}C_i\). Then we say that \(g^+:X^+\rightarrow X_{\mathrm {con}}\) is the flop of g if for every line bundle \(\mathcal {L}=\mathcal {O}_X(D)\) on X such that \(D\) is gnef, then the proper transform of D is \(\mathbb {Q}\)Cartier, and \(g^+\)nef.
The following is obvious, and will be used later.
Lemma 2.7
With the setup in 2.6, suppose that \(D_i\) is a Cartier divisor on X such that \(D_i\cdot C_j=\delta _{ij}\) for all \(i,j\in I\) (such a \(D_i\) exists since R is complete local), let \(D'_i\) denote the proper transform of \(D_i\) to \(X^+\). Then if \(D_i'\) is Cartier and there is an ordering of the exceptional curves \(C_i^+\) of \(g^+\) such that \(D'_i \cdot C^+_j =\delta _{ij}\), then \(g^+:X^+\rightarrow X_{\mathrm {con}}\) is the flop of g.
2.2 Perverse sheaves and tilting
Some of the arguments in this paper are not specific to dimension three, and are not specific to crepant morphisms. Consequently, at times we will refer to the following setup.
Setup 2.8
(General Setup) Suppose that \(f:X\rightarrow \mathrm{Spec}\,R\) is a projective birational morphism, where R is complete local, X and R are noetherian and normal, such that \(\mathbf{R}{f}_*\mathcal {O}_X=\mathcal {O}_R\) and the fibres of f have dimension at most one.
However, some parts will require the following restriction.
Setup 2.9
 (1)
If \(d=2\) we allow X to have canonical Gorenstein singularities, so \(X\rightarrow \mathrm{Spec}\,R\) is a partial crepant resolution of a Kleinian singularity.
 (2)
If \(d=3\) we further assume that X has only Gorenstein terminal singularities.
Notation 2.10
 (1)
Set \(\mathcal {N}_i:=\mathcal {M}_i^*\), and \(\mathcal {V}_X:=\mathcal {O}_{X}\oplus \bigoplus _{i=1}^n\mathcal {N}_i\).
 (2)
Set \(N_i:=H^0(\mathcal {N}_i)\) and \(N:=H^0(\mathcal {V}_X)\).
By [49, 3.5.5], \(\mathcal {V}_X\) is a basic progenerator of \({}^{0}\mathfrak {Per}(X,R)\), and furthermore is a tilting bundle on X. Note that \(\mathrm{rank}_{R} N_i\) is equal to the schemetheoretic multiplicity of the curve \(C_i\) [49, 3.5.4].
Remark 2.11
Under the derived equivalence \(\Psi _X\) in (2.A), the coherent sheaves \(\mathcal {O}_{C_i}(1)\) belong to \({}^{0}\mathfrak {Per}(X,R)\) and correspond to simple left \(\mathrm{End}_X(\mathcal {V}_X)\)modules \(S_i\).
Unfortunately, at this level of generality \(\mathrm{End}_X(\mathcal {V}_X)\ncong \mathrm{End}_R(N)\) (see e.g. [16, §2]). However, in the crepant setup of 2.9, this does hold, which later will allow us to reduce many problems to the base \(\mathrm{Spec}\,R\).
Lemma 2.12
([49, 3.2.10]) In the setup of 2.9, \(\mathrm{End}_X(\mathcal {V}_X)\cong \mathrm{End}_R(N)\).
Notation 2.13
 (1)
\(\mathcal {N}_I:=\bigoplus _{i\in I}\mathcal {N}_i\) and \(\mathcal {N}_{I^c}:=\mathcal {O}_X\oplus \bigoplus _{j\notin I}\mathcal {N}_j\), so that \(\mathcal {V}_X=\mathcal {N}_I\oplus \mathcal {N}_{I^c}\).
 (2)
\(N_I:=\bigoplus _{i\in I}N_i\) and \(N_{I^c}:=R\oplus \bigoplus _{j\notin I}N_j\), so that \(N=N_I\oplus N_{I^c}\).
The following result is implicit in the literature.
Proposition 2.14
 (1)
\(\mathcal {V}_{X_{\mathrm {con}}}\cong g_*\mathcal {N}_{I^c}\cong \mathbf{R}{g}_*\mathcal {N}_{I^c}\), and \(\mathrm{End}_X(\mathcal {N}_{I^c})\cong \mathrm{End}_{X_{\mathrm {con}}}(\mathcal {V}_{X_{\mathrm {con}}})\).
 (2)The following diagram commutes
Proof
The following is an easy extension of [52, 3.2], and will be needed later to read off the dual graph after the flop.
Theorem 2.15
Number of arrows  If setup 2.9, \(d=3\) and X is smooth  

\(\star \rightarrow \star \)  \(\dim _\mathbb {C}\mathrm{Ext}_X^1(\omega _C,\omega _C)\).  
\(i\rightarrow \star \)  \(\dim _\mathbb {C}\mathrm{Hom}_X(\mathcal {O}_{C_i}(1),\omega _C)\)  
\(\star \rightarrow i\)  \(\dim _\mathbb {C}\mathrm{Ext}_X^2(\omega _C,\mathcal {O}_{C_i}(1))\)  
\(i\rightarrow i\)  \(\dim _\mathbb {C}\mathrm{Ext}_X^1(\mathcal {O}_{C_i},\mathcal {O}_{C_i})\)  \(=\left\{ \begin{array}{rl} 0&{} \hbox {if }(1,1)\hbox {curve}\\ 1 &{}\hbox {if }(2,0)\hbox {curve}\\ 2&{}\hbox {if }(3,1)\hbox {curve} \end{array}\right. \) 
\(i\rightarrow j\)  \(\left\{ \begin{array}{rl} 1&{} \hbox {if }C_i\cap C_j=\{\mathrm {pt}\}\\ 0 &{}\hbox {else} \end{array}\right. \) 
2.3 Mutation
Throughout this subsection R denotes a normal dsCY complete local commutative algebra, with \(d\ge 2\), and \(M\in \mathrm{ref}\,R\) denotes a basic modifying module M. We summarise and extend the theory of mutation from [23, §6] and [15, §5].
Setup 2.16
 (1)Denote \(M_{I^c}\) to be the complement of \(M_I\), so that$$\begin{aligned} M=M_I\oplus M_{I^c}. \end{aligned}$$
 (2)
We define \([M_{I^c}]\) to be the twosided ideal of \(\Lambda \) consisting of morphisms \(M\rightarrow M\) which factor through a member of \(\mathrm{add}\,M_{I^c}\). We define \(\Lambda _I:=\Lambda /[M_{I^c}]\). Equivalently, if \(e_{I}\) denotes the idempotent of \(\Lambda =\mathrm{End}_R(M)\) corresponding to the summand \(M_{I}\) of M, then \(\Lambda _I=\Lambda /\Lambda (1e_{I})\Lambda \).
Given our choice of summand \(M_I\), we then mutate. In the theory of mutation, the complement submodule \(M_{I^c}\) is fixed, and the summand \(M_I\) changes in a universal way. Recall from Sect. 2.1 that \(()^*:=\mathrm{Hom}_R(,R)\).
Setup 2.17
 (1)
\(V_i\in \mathrm{add}\,M_{I^c}\) and \((\cdot a_i):\mathrm{Hom}_R(M_{I^c},V_i)\rightarrow \mathrm{Hom}_R(M_{I^c},M_i)\) is surjective,
 (2)
If \(g\in \mathrm{End}_R(V_i)\) satisfies \(a_i=ga_i\), then g is an automorphism.
Definition 2.18
 (1)We define the right mutation of M at \(M_I\) asthat is we remove the summand \(M_I\) and replace it with \(K_I\).$$\begin{aligned} \upmu _{I}M:=M_{I^c}\oplus K_I, \end{aligned}$$
 (2)We define the left mutation of M at \(M_I\) as$$\begin{aligned} \upnu _{I}M:=M_{I^c}\oplus (J_I)^*. \end{aligned}$$
In this level of generality, \(\upnu _IM\) is not necessarily isomorphic to \(\upmu _IM\).
Remark 2.19
Lemma 2.20
 (1)
\(T_I\) is a tilting \(\Lambda \)module with \(\mathrm {pd}_\Lambda T_I=1\).
 (2)\(T_I\) is a tilting \(\Gamma ^{\mathrm{op}}\cong \mathrm{End}_R((\upnu _IM)^*)\)module, with \(T_I\cong \mathrm{Hom}_R((\upnu _IM)^*, M_{I^c}^*)\oplus D_I\) where \(D_I\) arises from the exact sequenceof \(\Gamma ^{\mathrm{op}}\)modules. Thus \(\mathrm {pd}_{\Gamma ^{\mathrm{op}}}T_I=1\) and \(\mathrm{End}_{\Gamma ^{\mathrm{op}}}(T_I)\cong \Lambda ^{\mathrm{op}}\).$$\begin{aligned} 0\rightarrow \mathrm{Hom}_R((\upnu _IM)^*,J_I)\xrightarrow {\cdot d}\mathrm{Hom}_R((\upnu _IM)^*,U_I^*)\rightarrow D_I\rightarrow 0 \end{aligned}$$
Proof
For our purposes later we will require the finer information encoded in the following two key technical results. They are both an extension of [23, §6] and [15, §4], and are proved using similar techniques, so we postpone the proofs until “Appendix A”.
Theorem 2.21
 (1)
\(T_I=\Lambda (1e_I)\Lambda \) and \(\Gamma :=\mathrm{End}_\Lambda (T_I)\cong \Lambda \).
 (2)
\(\Omega _\Lambda \Lambda _I=T_I\), thus \(\mathrm {pd}_\Lambda \Lambda _I=2\) and \(\mathrm{Ext}^1_\Lambda (T_I,)\cong \mathrm{Ext}^2_\Lambda (\Lambda _I,)\).
Theorem 2.22
 (1)
\(T_I\cong \mathrm{Hom}_R(M,\upnu _IM)\).
 (2)
\(\Omega _\Lambda ^2\Lambda _I=T_I\), thus \(\mathrm {pd}_\Lambda \Lambda _I=3\) and \(\mathrm{Ext}^1_\Lambda (T_I,)\cong \mathrm{Ext}^3_\Lambda (\Lambda _I,)\).
The following, one of the main results in [23], will allow us to establish properties nonexplicitly when we restrict to minimal models and mutate at single curves.
Theorem 2.23
 (1)
We have \(\upmu _{j}(M)\cong \upnu _{j}(M)\).
 (2)
Always \(\upnu _j\upnu _j(M)\cong M\).
 (3)
\(\upnu _j(M)\ncong M\) if and only if \(\dim _\mathbb {C}\Lambda _j<\infty \).
 (4)
\(\upnu _j(M)\cong M\) if and only if \(\dim _\mathbb {C}\Lambda _j=\infty \).
Proof
(1) and (2) are special cases of [23, 6.25].
(3)(\(\Rightarrow \)) is [23, 6.25(2)], and (4)(\(\Rightarrow \)) is [23, 6.25(1)]. (3)(\(\Leftarrow \)) is the contrapositive of (4)(\(\Rightarrow \)), and (4)(\(\Leftarrow \)) is the contrapositive of (3)(\(\Rightarrow \)). \(\square \)
Remark 2.24
Theorem 2.23(3)(4) shows that there is a dichotomy in the theory of mutation depending on whether the dimension of \(\Lambda _j\) is finite or not. In the flops setting, this dichotomy will correspond to the fact that in a 3fold, an irreducible curve may or may not be floppable. In either case we will obtain a derived equivalence from mutation, and the results 2.21 and 2.22 will allow us to control it.
The above 2.23 will allow us to easily relate flops and mutations in the case when \(d=3\) and the singularities of X are \(\mathbb {Q}\)factorial. When we want to drop the \(\mathbb {Q}\)factorial assumption, or consider \(d=2\) with canonical singularities, we will need the following.
Proposition 2.25
With the crepant setup of 2.9, and notation from 2.13, choose a subset \(\bigcup _{i\in I}C_i\) of curves above the origin and contract them to obtain \(X\rightarrow X_{\mathrm {con}}\rightarrow \mathrm{Spec}\,R\). If \(X_{\mathrm {con}}\) has only isolated hypersurface singularities, then \(\upnu _I\upnu _IN\cong N\) in such a way that \(N_i\) mutates to \(J_i^*\) mutates to \(N_i\).
Proof
In the study of terminal (and even smooth) 3folds, canonical surfaces appear naturally via hyperplane sections, and in this setting \(\mathrm {pd}_\Lambda \Lambda _i\) can be infinite, which is very different to 2.21 and 2.22. The next result will allow us to bypass this problem.
Proposition 2.26
Suppose that R is a normal complete local 2sCY commutative algebra, and \(M\in \mathrm{CM}\,R\) is basic. Choose a summand \(M_I\), set \(\Lambda :=\mathrm{End}_R(M)\) and denote the simple \(\Lambda \)modules by \(S_j\). Assume that \(\upnu _I\upnu _IM\cong M\). If \(x\in \mathrm{fdmod}\,\Lambda \) with \(\mathrm{Hom}_\Lambda (x,S_i)=0\) for all \(i\in I\), then \(\mathrm{Ext}^1_\Lambda (T_I,x)=0\).
Proof
3 Contractions and deformation theory
The purpose of this section is use noncommutative deformations to detect whether a divisor has been contracted to a curve, in such a manner that is useful for iterations, improving [16]. This part of the Homological MMP does not need any restriction on singularities, so throughout this section we adopt the general setup of 2.8.
3.1 Background on noncommutative deformations
Noncommutative deformations add two new features to this classical picture. First, the test objects are enlarged from commutative artinian rings to allow certain (basic) noncommutative artinian \(\mathbb {C}\)algebras. This thickens the universal sheaf. Second, they allow us to deform a finite collection \(\{E_i\mid i\in I\}\) of objects whilst remembering Ext information between them.
For the purposes of this paper, we will not deform coherent sheaves, but rather their images under the derived equivalence in Sect. 2.2. Deforming on either side of the derived equivalence turns out to give the same answer [16], but the noncommutative side is slightly easier to formulate. Thus we input a finite collection \(\{S_i\mid i\in I\}\) of simple \(\Lambda \)modules, and define the associated noncommutative deformation functor as follows.
Given \(\Gamma \in \mathsf {art}_n\), the morphism i produces n idempotents \(e_1,\ldots ,e_n\in \Gamma \), and we denote \(\Gamma _{ij}:=e_i\Gamma e_j\).
Definition 3.1
 (1)For \(\Gamma \in \mathsf {art}_{I}\), we say that \(M\in \mathrm{Mod}\,\Lambda \otimes _{\mathbb {C}}\Gamma ^{\mathrm{op}}\) (i.e. a \(\Lambda \)\(\Gamma \) bimodule) is \(\Gamma \)matricfree ifas \(\Gamma ^{\mathrm{op}}\)modules, where the righthand side is the matrix built by varying \(i,j\in \{1,\ldots ,n\}\), which has an obvious \(\Gamma ^{\mathrm{op}}\)module structure.$$\begin{aligned} M \cong (S_i\otimes _{\mathbb {C}} \Gamma _{ij}) \end{aligned}$$
 (2)The noncommutative deformation functoris defined by sending$$\begin{aligned} \mathcal {D}ef_{\mathcal {S}}:\mathsf {art}_n\rightarrow \mathsf {Sets}\end{aligned}$$where \((M,\delta )\sim (M',\delta ')\) if there exists an isomorphism \(\tau :M\rightarrow M'\) of bimodules such that commutes, for all i.$$\begin{aligned} (\Gamma ,\mathfrak {n})\mapsto \left. \left\{ (M,\delta ) \left \begin{array}{l}M\in \mathrm{Mod}\,\Lambda \otimes _{\mathbb {C}}\Gamma ^{\mathrm{op}}\\ M \text { is }\Gamma \text {matricfree}\\ \delta =(\delta _i)\hbox { with }\delta _i:M\otimes _\Gamma (\Gamma /\mathfrak {n})e_i\xrightarrow {\sim } S_i \end{array}\right. \right\} \bigg / \sim \right. \end{aligned}$$
 (3)
The commutative deformation functor is defined to be the restriction of \(\mathcal {D}ef_{\mathcal {S}}\) to \(\mathsf {cart}_{I}\), and is denoted \(c\mathcal {D}ef_{\mathcal {S}}\).
Theorem 3.2
(Contraction theorem) With the general setup in 2.8, if \(d=3\) then f contracts \(\bigcup _{i\in I}C_i\) to a point without contracting a divisor if and only if \(\mathcal {D}ef_{\mathcal {S}}\) is representable.
3.2 Global and local contraction algebras
We maintain the notation from the general setup of the previous subsection. In this subsection we detect whether g contracts a curve without contracting a divisor by using the algebra \(\Lambda =\mathrm{End}_X(\mathcal {V}_X)\), constructed in Sect. 2.2 using the morphism f. This will allow us to iterate.
Definition 3.3
For \(\Lambda =\mathrm{End}_X(\mathcal {V}_X)\), with notation from 2.13 define \([\mathcal {N}_{I^c}]\) to be the 2sided ideal of \(\Lambda \) consisting of morphisms that factor through \(\mathrm{add}\,\mathcal {N}_{I^c}\), and set \(\Lambda _I:=\Lambda /[\mathcal {N}_{I^c}]\).
Example 3.4
The following is the main result of this section.
Theorem 3.5
 (1)
\(\mathcal {D}ef_{\mathcal {S}}\cong \mathrm{Hom}_{\mathsf {Alg}_{I}}(\Lambda _I,)\).
 (2)
\(\Lambda _I\cong \mathrm {A}_{\mathrm {con}}^I\).
 (3)
\(\bigcup _{i\in I}C_i\) contracts to point without contracting a divisor \(\Leftrightarrow \dim _{\mathbb {C}}\Lambda _I<\infty \).
Proof
(3) This follows by combining (1) and 3.2. \(\square \)
4 Mutation, flops and twists
4.1 Flops and mutation
We now consider the crepant setup of 2.9 with \(d=3\), namely \(f:X\rightarrow \mathrm{Spec}\,R\) is a crepant projective birational morphism, with one dimensional fibres, where R is a complete local Gorenstein algebra, and X has at worst Gorenstein terminal singularities. As in Sect. 2.2, we consider \(\mathcal {V}_X\), the basic progenerator of \({}^{0}\mathfrak {Per}(X,R)\), set \(N:=H^0(\mathcal {V}_X)\) and by 2.12 denote \(\Lambda :=\mathrm{End}_X(\mathcal {V}_X)\cong \mathrm{End}_R(N)\).
Remark 4.1
It also follows from 2.12 that \(\mathrm{End}_X(\mathcal {V}_X)/[\mathcal {N}_{I^c}]\cong \mathrm{End}_R(N)/[N_{I^c}]\) and so the \(\Lambda _I\) defined in 3.3 and 2.16 coincide. This allows us to link the previous contraction section to mutation.
We aim to prove the following theorem.
Theorem 4.2
 (1)
\(\upnu _IN\cong H^0(\mathcal {V}_{X^+})\), where \(\mathcal {V}_{X^+}\) is the basic progenerator of \({}^{0}\mathfrak {Per}(X^+,R)\).
 (2)
The proof of 4.2 will be split into two stages. Stage one, proved in this subsection, is to establish 4.2(1) in the case where X is a minimal model and \(I=\{i \}\). Stage two is then to prove 4.2 in full generality, lifting the \(\mathbb {Q}\)factorial and \(I=1\) assumption. The second stage uses the Auslander–McKay correspondence in Sect. 4.2, together with the Bongartz completion to pass to the minimal model, before we then contract back down. Thus, the full proof of 4.2 will not finally appear until Sect. 4.3.
To establish functoriality, the following results will be useful later.
Proposition 4.3
 (1)
\(\mathbf{R}{g}'_*\circ \Theta \cong \mathbf{R}{g}_*\)
 (2)
\(\Theta (\mathcal {O}_X)\cong \mathcal {O}_{X'}\)
 (3)
\(\Theta (\mathcal {O}_{C_i}(1))\cong \mathcal {O}_{C'_i}(1)\) for all \(i\in I\),
Proof
This is identical to [15, §7.6], which itself is based on [46]. As in [15, 7.17], from properties (1) and (3) it follows that \(\Theta \) takes \({}^{0}\mathfrak {Per}(X,X_{\mathrm {con}})\) to \({}^{0}\mathfrak {Per}(X',X_{\mathrm {con}})\). The argument is then wordforword identical to the proof of [15, 7.17, 7.18], since although there it was assumed that X was projective, this is not needed in the proof. \(\square \)
Corollary 4.4
Suppose that \(f:X\rightarrow \mathrm{Spec}\,R\) and \(f':X'\rightarrow \mathrm{Spec}\,R\) both satisfy the crepant setup 2.9. If \(H^0(\mathcal {V}_X)\cong H^0(\mathcal {V}_{X'})\), then there is an isomorphism \(X\cong X'\) compatible with f and \(f'\).
Proof
The following lemma, an easy consequence of Riedtmann–Schofield, proves 4.2(1) with restricted hypotheses.
Lemma 4.5
With the crepant setup of 2.9, suppose further that \(d=3\) and X is \(\mathbb {Q}\)factorial, that is \(f:X\rightarrow \mathrm{Spec}\,R\) is a minimal model. Choose a single curve \(C_i\) above the origin, suppose that \(\dim _{\mathbb {C}}\Lambda _i<\infty \) (equivalently, by 3.5, \(C_i\) flops), and let \(X^+\) denote the flop of \(C_i\). Then \(\upnu _iN\cong H^0(\mathcal {V}_{X^+})\).
Proof
Denote the base of the contraction of \(C_i\) by \(X_{\mathrm {con}}\), set \(M:=H^0(\mathcal {V}_{X^+})\) and let \(M_i\) denote the indecomposable summand of M corresponding to \(C_i^+\). It is clear that \(M\ncong N\). Applying 2.14 to both sides of the contraction, \(\frac{M}{M_i}\cong H^0(\mathcal {V}_{X_{\mathrm {con}}})\cong \frac{N}{N_i}\), so the module M differs from N only at the summand \(N_i\). Similarly, by 2.23(3) \(\upnu _iN\ncong N\), and by definition of mutation, \(\upnu _iN\) differs from N only at the summand \(N_i\). Consequently, as Rmodules, \(\upnu _iN\) and M share all summands except one, and neither is isomorphic to N.
But by 2.5 both \(\mathrm{End}_R(M)\) and \(\mathrm{End}_R(N)\) are MMAs, and since \(\mathrm{End}_R(\upnu _iN)\) is also derived equivalent to these, it too is an MMA [23, 4.16]. Further, \(\mathrm{Hom}_R(N,\upnu _iN)\) and \(\mathrm{Hom}_R(N,M)\) are tilting \(\mathrm{End}_R(N)\)modules by [23, 4.17(1)], and by above as \(\mathrm{End}_R(N)\)modules they share all summands except one. Hence as in [23, 6.22], a Riedtmann–Schofield type theorem implies that \(\upnu _iN\cong M\). \(\square \)
Corollary 4.6
The above allows us to verify Fig. 2 under restricted hypotheses.
Corollary 4.7
We can run the Homological MMP in Fig. 2 when \(d=3\) and X has only \(\mathbb {Q}\)factorial Gorenstein terminal singularities, and we choose only irreducible curves.
Later in Sect. 4.3 we will drop the \(\mathbb {Q}\)factorial assumption, and also drop the restriction to single curves.
4.2 Auslander–McKay correspondence
Throughout this subsection we keep the crepant setup of 2.9, and as in the previous subsection assume further that \(d=3\) and X is \(\mathbb {Q}\)factorial. The R admitting such a setup are of course wellknown to be precisely the cDV singularities [43].
Definition 4.8
 (1)
We define the full mutation graph of the MM generators to have as vertices the basic MM generators (up to isomorphism of Rmodules), where each vertex N has an edge to \(\upnu _IN\) provided that \(\dim _{\mathbb {C}}\Lambda _I<\infty \), for I running through all possible summands \(N_I\) of N that are not generators. The simple mutation graph is defined in a similar way, but we only allow mutation at indecomposable summands.
 (2)
We define the full flops graph of the minimal models of \(\mathrm{Spec}\,R\) to have as vertices the minimal models of \(\mathrm{Spec}\,R\) (up to isomorphism of Rschemes), and we connect two vertices if the corresponding minimal models are connected by a flop at some curve. The simple flops graph is defined in a similar way, but we only connect two vertices if the corresponding minimal models differ by a flop at an irreducible curve.
The following is standard.
Definition 4.9
For \(N\in \hbox {mod}\,R\), the stable endomorphism ring \({\underline{\mathrm{End}}}_R(N)\) is defined to be the quotient of \(\mathrm{End}_R(N)\) by the two sided ideal consisting of those morphisms \(N\rightarrow N\) which factor through \(\mathrm{add}\,R\).
Recall from the introduction Sect. 1.3 that there is a specified region \(C_+\) of the GIT chamber decomposition of \(\Theta (\mathrm{End}_R(N))\), and \(\mathcal {M}_{\mathsf{rk},\upphi }(\Lambda )\) denotes the moduli space of \(\upphi \)semistable \(\Lambda \)modules of dimension vector \(\mathsf{rk}\) (see Sect. 5.1 for more details).
Theorem 4.10
 (1)
For any fixed MM generator, its nonfree indecomposable summands are in onetoone correspondence with the exceptional curves in the corresponding minimal model.
 (2)
For any fixed MM generator N, the quiver of \({\underline{\mathrm{End}}}_R(N)\) encodes the dual graph of the corresponding minimal model.
 (3)
The simple mutation graph of the MM generators coincides with the simple flops graph of the minimal models.
Proof
 (1)
Let N be an MM generator, then since by the above \(N\cong H^0(\mathcal {V}_Y)\) for some minimal model \(Y\rightarrow \mathrm{Spec}\,R\), the statement follows from the construction of the bundle \(\mathcal {V}_Y\) in Sect. 2.2.
 (2)
Since the quiver of \({\underline{\mathrm{End}}}_R(N)\) is the quiver of \(\mathrm{End}_R(N)\cong \mathrm{End}_Y(\mathcal {V}_Y)\) with the vertex \(\star \) (corresponding to the summand R of N) removed, it follows from 2.15 that the quiver of \({\underline{\mathrm{End}}}_R(N)\) is the double of the dual graph, together with some loops.
 (3)
This follows from 4.6 and the above argument. \(\square \)
We extend the correspondence later in Sects. 6.2–6.4.
Remark 4.11
\(\mathrm{Spec}\,R\) may be its own minimal model, in which case the correspondence in 4.10 reduces to the statement that R is the only basic modifying generator.
Remark 4.12
The proof of 4.10 uses the fact that there are only a finite number of minimal models, and that they are connected by a finite sequence of simple flops. We use these results only to simplify the exposition; it is possible to instead use the moduli tracking results of Sect. 5, specifically 5.25, to give a purely homological proof of 4.10.
Remark 4.13
Sometimes the minimal models of \(\mathrm{Spec}\,R\) can be smooth. Recall from 2.4 the definition of a CT module.
Corollary 4.14
Proof
If one of (equivalently, all of) the minimal models is smooth, then \(\mathrm{End}_X(\mathcal {V}_X)\cong \mathrm{End}_R(N)\) has finite global dimension and hence is an NCCR. Thus N is a CT module. Since \(\mathrm{CM}\,R\) has a CT module, by [23, 5.11(2)] CT modules are precisely the MM generators. \(\square \)
4.3 Flops and mutation revisited
In this subsection we use the Auslander–McKay Correspondence to finally prove 4.2 in full generality, then run the Homological MMP in Fig. 2 when X has only Gorenstein terminal singularities.
We first track certain objects under the mutation functor \(\Phi _I\) in (2.L). As notation, suppose that M is a basic modifying Rmodule, where R is complete local dsCY. Then for each indecomposable summand \(M_j\) of M, denote the corresponding simple and projective \(\mathrm{End}_R(M)\)modules by \(S_j\) and \(P_j\) respectively. For an indecomposable summand \(X_j\) of \(\upnu _IM\), we order the indecomposable summands of \(\upnu _IM\) so that either \(X_j\cong M_j\) when \(j\notin I\), or \(X_i\cong J_i^*\) when \(i\in I\). We denote the corresponding simple and projective \(\mathrm{End}_R(\upnu _IM)\)modules by \(S'_i\) and \(P'_i\) respectively.
Lemma 4.15
 (1)
\(\Phi _I(P_j)=P'_j\) for all \(j\notin I\).
 (2)
\(\Phi _I(S_i)=S'_i[1]\) for all \(i\in I\).
Proof
We also require the following, which does not need the crepant assumption.
Lemma 4.16
In the general setup of 2.8, \(\Psi _X^{1}\Lambda _I\) is a sheaf in degree zero, which we denote by \(\mathcal {E}_I\).
Proof
Clearly \(\Lambda _I\) is a finitely generated \(\Lambda \)module, so \(\Psi _X^{1}\Lambda _I\in {}^{0}\mathfrak {Per}(X,R)\). Further, by 2.14 \(\mathbf{R}{f}_*(\Psi _X^{1}\Lambda _I)=0\) since \(e\Lambda _I=0\), so since the spectral sequence collapses it follows that \(H^{1}(\Psi _X^{1}\Lambda _I)\in \mathcal {C}_f\) and \(H^{0}(\Psi _X^{1}\Lambda _I)\in \mathcal {C}_f\). But since \(\Psi _X^{1}\Lambda _I\in {}^{0}\mathfrak {Per}(X,R)\), by definition \(\mathrm{Hom}(\mathcal {C}_f,H^{1}(\Psi _X^{1}\Lambda _I))=0\). Thus \(\mathrm{Hom}(H^{1}(\Psi _X^{1}\Lambda _I),H^{1}(\Psi _X^{1}\Lambda _I))=0\) and so \(H^{1}(\Psi _X^{1}\Lambda _I)=0\). Thus \(\Psi _X^{1}\Lambda _I\) is a sheaf in degree zero. \(\square \)
The following is one of the main results, from which 4.2 will follow easily.
Proposition 4.17
 (1)
\(X^+\) is the flop of X at the curves \(\bigcup _{i\in I} C_i\).
 (2)The following diagram of equivalences is naturally commutative
Proof
On the other hand, since \(\mathrm{End}_R(\upnu _IN)\) is derived equivalent to \(X^+\) via \(H^0(\mathcal {V}_{X^+})\cong \upnu _IN\), the summands of \(\upnu _IN\) correspond to exceptional curves. If we contract all the curves corresponding to the summand \(J_I^*\), we obtain \(X^+_{\mathrm {con}}\) say. But again by 2.14 \(\mathrm{End}_R(\frac{\upnu _IN}{J_I^*})= \mathrm{End}_R(N_{I^c})\) is derived equivalent to \(X^+_{\mathrm {con}}\) with \(H^0(\mathcal {V}_{X^+_{\mathrm {con}}})\cong N_{I^c}\), so by 4.4 \(X_{\mathrm {con}}\cong X^+_{\mathrm {con}}\) and we can suppose that we are in the situation of the diagram above. As notation, we denote \(C_i^+\) to be the curve in \(X^+\) corresponding to the summand \(J_i^*\) of \(\upnu _IN\).
We next claim that \(g^+:X^+\rightarrow X_{\mathrm {con}}\) is the flop of g, and to do this we use 2.7. First, \(g^+:X^+\rightarrow X_{\mathrm {con}}\) does not contract a divisor to a curve, since \(\Lambda _I\cong (\upnu _I\Lambda )_I\) and so \(\dim _{\mathbb {C}}(\upnu _I\Lambda )_I<\infty \) by [23, 6.20]. Now with the notation as in Sect. 2.2, we let \(D_i\) in X be the Cartier divisor cutting exactly the curve \(C_i\), and \(D^+_i\) be the Cartier divisor in \(X^+\) cutting exactly the curve \(C_i^+\). Since \((D_i)\) is gample, we let \(D_i'\) denote the proper transform of \(D_i\). In what follows, we will use the notation \([]_X\) to denote something viewed in the class group of X. Since g and \(g^+\) give reflexive equivalences, we will also abuse notation and for example refer to the divisor \(D_i\) on \(X_{\mathrm {con}}\), and on \(X^+\).
Denote \(\mathcal {G}_i:=\mathrm{Cok}\,\upvarepsilon _i\). Since \(\mathbf {R}^1 g^+_*\mathcal {U}_i=0\) as in the proof of 2.14, it follows that \(\mathbf {R}^1 g^+_*\mathcal {G}_i=0\). Then since \(\mathbf{R}{g}^+_*\mathcal {E}=0\), it then follows that \(\mathbf {R}^1 g^+_*\mathcal {N}_i^+=0\). Thus \(\mathbf{R}{g}^+_*\mathcal {N}_i^+=g^+_*\mathcal {N}_i^+\), and again as in the proof of 2.14, \(\mathbf{R}{g}^+_*\mathcal {W}_i=g^+_*\mathcal {W}_i\).
 (1)
\(\mathbf{R}{g}^+_*\circ F\cong \mathbf{R}{g}_*\),
 (2)
\(F(\mathcal {O}_X)\cong \mathcal {O}_{X^+}\),
 (3)
\(F(\mathcal {O}_{C_i}(1))\cong \mathcal {O}_{C^+_i}(1)[1]\) for all \(i\in I\),
Thus to prove 4.2, by 4.17 we just need to establish that \(\upnu _IN\cong H^0(\mathcal {V}_{X^+})\) for some \(X^+\rightarrow \mathrm{Spec}\,R\). The trick in 4.5 in the minimal model case with \(I=\{i\}\) was to use Riedtmann–Schofield. To work in full generality requires another standard technique from representation theory, namely the Bongartz completion.
Proof of 4.2
Since R admits an MM module, by Bongartz completion we may find \(F\in \mathrm{ref}\,R\) such that \(\mathrm{End}_R(\upnu _IN\oplus F)\) is an MMA [23, 4.18]. Since \(\upnu _IN\) is a generator, necessarily \(F\in \mathrm{CM}\,R\). By the Auslander–McKay Correspondence 4.10 \(\upnu _IN\oplus F\) is one of the finite number of MM generators, and further \(\upnu _IN\oplus F=H^0(\mathcal {V}_{Y})\) for some minimal model Y, where the nonfree summands of \(\upnu _IN\oplus F\) correspond to the exceptional curves for \(Y\rightarrow \mathrm{Spec}\,R\). Contracting all the curves in Y that correspond to the summand F, as in (1.B) we factorise \(Y\rightarrow \mathrm{Spec}\,R\) as \(Y\rightarrow X^+\rightarrow \mathrm{Spec}\,R\) for some \(X^+\). By 2.14 \(\upnu _IN\cong H^0(\mathcal {V}_{X^+})\), so parts (1) and (2) both follow from 4.17. \(\square \)
Corollary 4.18
We can run the Homological MMP in Fig. 2 when X has only Gorenstein terminal singularities, for arbitrary subsets of curves.
Corollary 4.19
 (1)
\(\mathcal {M}_{\mathsf{rk},\upvartheta }(\Lambda )\cong X\) for all \(\upvartheta \in C_+(\Lambda )\).
 (2)
\(\mathcal {M}_{\mathsf{rk},\upvartheta }(\upnu _I\Lambda )\cong X^+\) for all \(\upvartheta \in C_+(\upnu _I\Lambda )\).
Proof
Part (1) follows immediately from [27, 5.2.5] applied to X. By 4.2, part (2) follows from [27, 5.2.5] applied to \(X^+\). \(\square \)
The following extends 4.6 by dropping the \(\mathbb {Q}\)factorial assumption and considering multiple curves, but now the statement is a little more subtle.
Corollary 4.20
 (1)
If \(\bigcup _{i\in I}C_i\) flops (equivalently, by 3.5, \(\dim _{\mathbb {C}}\Lambda _I<\infty \)), then \(\upnu _IN\cong H^0(\mathcal {V}_{X^+})\).
 (2)
If \(I=\{i\}\), \(\mathrm {pd}_\Lambda \Lambda _i<\infty \) and \(C_i\) does not flop (equivalently, \(\dim _{\mathbb {C}}\Lambda _i=\infty \)), then \(\upnu _IN\cong N\).
Proof
Part (1) is just 4.2. For part (2), since \(\Lambda _i\) is local and has finite projective dimension, by [42, 2.15] \(\mathrm{depth}_R\Lambda _i=\dim _R\Lambda _i=\mathrm{inj.dim}_{\Lambda _i}\Lambda _i\). The result follows using the argument of [23, 6.23(1)]. \(\square \)
Remark 4.21
The statement in 4.20(2) is not true for multiple curves, indeed the hypothesis in 4.20(2) cannot be weakened. First, if \(I\ne \{i\}\) then \(\Lambda _i\) is not local and there are examples that satisfy \(\upnu _IN\ncong N\) even when \(\mathrm {pd}_\Lambda \Lambda _I<\infty \) and \(\dim _{\mathbb {C}}\Lambda _I=\infty \). Second, if \(I=\{i\}\) and \(\mathrm {pd}_\Lambda \Lambda _I=\infty \), there are examples that satisfy \(\dim _{\mathbb {C}}\Lambda _i=\infty \) but \(\upnu _IN\ncong N\).
There are two separate problems here, namely in general \(\Lambda _I\) need not be perfect, and it need not be Cohen–Macaulay. Both cause independent technical difficulties, and this will also be evident in Sect. 5. See also B.1.
One further corollary of this section is that both commutative and noncommutative deformations of curves are preserved under flop.
Corollary 4.22
 (1)
The noncommutative deformation functor of \(\bigcup _{i\in I}C_i\) is represented by the same ring as the noncommutative deformation functor of \(\bigcup _{i\in I}C^+_i\).
 (2)
The statement in (1) also holds for commutative deformations.
Proof
By 3.5, since \(\bigcup _{i\in I}C_i\) flops, \(\dim _{\mathbb {C}}\Lambda _I<\infty \) and the noncommutative deformations of \(\bigcup _{i\in I}C_i\) are represented by \(\Lambda _I\). By 3.5 and 4.2, the noncommutative deformations of \(\bigcup _{i\in I}C^+_i\) are represented by \((\upnu _I\Lambda )_I\). By [23, 6.20] \(\Lambda _I\cong (\upnu _I\Lambda )_I\).
(2) This follows by taking the abelianization of (1). \(\square \)
4.4 Auslander–McKay Revisited
Now that 4.2 has been established in full generality, we can extend the Auslander–McKay Correspondence in 4.10.
Definition 4.23
 (1)
The derived mutation groupoid is defined by the following generating set. It has vertices \(\mathrm{{D}^b}(\hbox {mod}\,\mathrm{End}_R(N))\), running over all isomorphism classes of basic MM generators N, and as arrows each vertex \(\mathrm{{D}^b}(\hbox {mod}\,\mathrm{End}_R(N))\) has the mutation functors \(\Phi _I\) emerging, as I runs through all possible summands satisfying \(\dim _{\mathbb {C}}\Lambda _I<\infty \).
 (2)
The derived flops groupoid is defined by the following generating set. It has vertices \(\mathrm{{D}^b}(\mathrm{coh}\,X)\), running over all minimal models X, and as arrows we connect vertices by the inverse of the Bridgeland–Chen flop functors, running through all possible combinations of flopping curves.
Theorem 4.24
 (3)\('\)

The full mutation graph of the MM generators coincides with the full flops graph of the minimal models.
 (4)

The derived groupoid of the MM modules is functorially isomorphic to the derived flops groupoid of the minimal models.
Proof
(3)\('\) By definition, the full mutation graph and derived mutation groupoid only considers \(\upnu _I\) provided that \(\dim _{\mathbb {C}}\Lambda _I<\infty \), which by 3.5 is equivalent to the condition that \(\bigcup _{i\in I}C_i\) flops. Hence the result follows by combining the bijection in 4.10 with 4.2(1).
(4) This follows by combining the bijection with 4.2(2).
5 Stability and mutation
In this section we relate stability and mutation, then use this together with the Homological MMP (proved in 4.18) to give results in GIT, specifically regarding chamber decompositions and later in Sect. 6.1 the Craw–Ishii conjecture.
After first proving general moduli–tracking results in Sect. 5.2, running Fig. 2 over all possibilities and tracking all the moduli back then computes the full GIT chamber decomposition. We further prove in Sect. 5.3 that mutation is preserved under generic hyperplane section, which in effect means (in Sect. 5.4) that the chamber decomposition reduces to knitting on ADE surface singularities, which is very easy to calculate. Amongst other things, this observation can be used to prove the braiding of flops in dimension three [17].
5.1 GIT background
There are two GIT approaches to moduli that could be used in this paper. The first is quiver GIT, which relies on presenting \(\Lambda ^{\mathrm{op}}\) as (the completion of) a quiver with relations, and the second is the more abstract approach given in [50, §6.2]. For most purposes either is sufficient, so for ease of exposition we use quiver GIT.
Definition 5.1
[32] Given \(\upvartheta \in \Theta \), \(x\in \mathrm{fdmod}\,\Lambda =\mathrm{Rep}(Q,I)\) is called \(\upvartheta \)semistable if \(\upvartheta \cdot {\underline{\mathrm{dim}}\,}x=0\) and every subobject \(x'\subseteq x\) satisfies \(\upvartheta \cdot {\underline{\mathrm{dim}}\,}x'\ge 0\). Such an object x is called \(\upvartheta \)stable if the only subobjects \(x'\) with \(\upvartheta \cdot {\underline{\mathrm{dim}}\,}x'=0\) are x and 0. Two \(\upvartheta \)semistable modules are called Sequivalent if they have filtrations by \(\upvartheta \)stable modules which give isomorphic associated graded modules. Further, for a given \(\upbeta \), we say that \(\upvartheta \) is generic if every \(\upvartheta \)semistable module of dimension vector \(\upbeta \) is \(\upvartheta \)stable.
Notation 5.2
 (1)
Denote by \(\mathcal {M}_{\upbeta ,\upvartheta }(\Lambda )\) the moduli space of \(\upvartheta \)semistable \(\Lambda \)modules of dimension vector \(\upbeta \).
 (2)
Denote by \(\mathcal {S}_{\upvartheta }(\Lambda )\) the full subcategory of \(\mathrm{fdmod}\,\Lambda \) which has as objects the \(\upvartheta \)semistable objects, and denote by \(\mathcal {S}_{\upbeta ,\upvartheta }(\Lambda )\) the full subcategory of \(\mathcal {S}_{\upvartheta }(\Lambda )\) consisting of those elements with dimension vector \(\upbeta \).
By King [32] (see also [50, 6.2.1]) \(\mathcal {M}_{\upbeta ,\upvartheta }(\Lambda )\) is a coarse moduli space that parameterises Sequivalence classes of \(\upvartheta \)semistable modules of dimension vector \(\upbeta \). If further \(\upbeta \) is an indivisible vector and \(\upvartheta \) is generic, then \(\mathcal {M}_{\upbeta ,\upvartheta }(\Lambda )\) is a fine moduli space, and Sequivalence classes coincide with isomorphism classes.
5.2 Tracking stability through mutation
In this subsection we track stability conditions through mutation, extending [39, 45] to work in a much greater level of generality. Throughout, we will make use of the following setup.
Setup 5.3
Suppose that R is a normal complete local dsCY commutative algebra with \(d\ge 2\), M is a basic modifying module and \(M_I\) is a summand of M. Set \(\Lambda :=\mathrm{End}_R(M)\) and \(\Gamma :=\upnu _I\Lambda \). We denote the projective \(\Lambda \)modules by \(P_j\), the simple \(\Lambda \)modules by \(S_j\), and the simple \(\Gamma \)modules by \(S'_j\).
Definition 5.4
Thus given the data of \(\mathbf {b_I}=(b_{i,j})\), we thus view \(\upnu _{\mathbf {b_I}}\) as an operation on dimension vectors, and as a (different) operation on stability parameters.
Remark 5.5
We remark that the b’s are defined with respect to the mutation \(\Lambda \mapsto \upnu _I\Lambda \). When we iterate and consider another mutation \(\upnu _I\Lambda \mapsto \upnu _J\upnu _I\Lambda \), the b’s may change for this second mutation. This change may occur even in the situation \(\upnu _I\upnu _I\Lambda \cong \Lambda \), and we are considering the mutation back \(\upnu _I\Lambda \mapsto \upnu _I\upnu _I\Lambda \cong \Lambda \). The papers [39, 45] involve a global rule for \(\upnu _{\mathbf {b_I}}\upvartheta \) (in their notation, \(s_i\upvartheta \)), and this is the reason why their combinatorial rule, and proofs, only work in a very restricted setting.
The following two lemmas are elementary.
Lemma 5.6
 (1)
\(\upnu _{\mathbf {b_I}}\upbeta \cdot \upnu _{\mathbf {b_I}}\upvartheta =\upbeta \cdot \upvartheta \).
 (2)
\(\upnu _{\mathbf {b_I}}\upnu _{\mathbf {b_I}}\upbeta =\upbeta \).
 (3)
\(\upvartheta \cdot \upnu _{\mathbf {b_I}}\upbeta =\upnu _{\mathbf {b_I}}\upvartheta \cdot \upbeta \).
Proof
This is easily verified by direct calculation. \(\square \)
Lemma 5.7
 (1)
If \(\mathrm{Ext}^1_\Lambda (T_I,x)=0\), then \({\underline{\mathrm{dim}}\,}\mathrm{Hom}_\Lambda (T_I,x)=\upnu _{\mathbf {b_I}}{\underline{\mathrm{dim}}\,}x\).
 (2)
If \(\mathrm{Tor}_1^{\Gamma }(T_I,y)=0\), then \({\underline{\mathrm{dim}}\,}(T_I\otimes _{\Gamma } y)=\upnu _{\mathbf {b_I}}{\underline{\mathrm{dim}}\,}y\).
Proof
When tracking stability under mutation, as in 4.21 the fact that \(\Lambda _I\) need not be CohenMacaulay and need not be perfect causes problems. The following two technical results allows us to overcome the first. To avoid cases in the statement and proof, as a convention \(\frac{M}{(a_1,\ldots ,a_t)M}:=M\) when \(t=0\).
Lemma 5.8
 (1)
\(\mathrm {pd}_\Lambda M<\infty \),
 (2)
\(N\in \mathrm{fdmod}\,\Lambda \),
 (3)
\(\mathrm{Hom}_\Lambda \left( N,\frac{M}{(a_1,\ldots ,a_t)M}\right) =0\),
Proof
Corollary 5.9
 (a)
\(\upnu _IM\cong M\), or
 (b)
\(d= 3\), \(\upnu _I\upnu _IM\cong M\) and \(\dim _{\mathbb {C}}\Lambda _I<\infty \).
Proof
By either 2.21(2) or 2.22(2), \(\mathrm {pd}_\Lambda \Lambda _I<\infty \). Thus by 5.8 applied with \(M=\Lambda _I\) and \(N=x\), we only need to verify that \(\mathrm{Hom}_\Lambda (x,\tfrac{\Lambda _I}{(a_1,\ldots ,a_t)\Lambda _I})\) is zero. Consider an element f, then since x is finite dimensional, so is \(\mathrm{Im}\, f\). Thus being a submodule of a factor of \(\Lambda _I\), \(\mathrm{Im}\, f\) must have a finite filtration with factors from the set \(\{ S_i\mid i\in I\}\). Since \(\mathrm{Hom}_\Lambda (x,S_i)=0\) for all \(i\in I\), inducting along the finite filtration gives \(\mathrm{Hom}_\Lambda (x,\mathrm{Im}\, f)=0\), and hence \(\mathrm{Hom}_\Lambda (x,\tfrac{\Lambda _I}{(a_1,\ldots ,a_t)\Lambda _I})=0\).\(\square \)
The following, which is a consequence of 2.26 and 5.9, will be needed in 5.12.
Corollary 5.10
 (a)
\(\upnu _IM\cong M\), or
 (b)
\(\upnu _I\upnu _IM\cong M\) and \(\dim _{\mathbb {C}}\Lambda _I<\infty \).
 (1)
\(\mathrm{Ext}^{1}_\Lambda (T_I,x)=0\) provided \(\mathrm{Hom}_\Lambda (x,S_i)=0\) for all \(i\in I\).
 (2)
\(\mathrm{Tor}_{1}^{\Gamma }(T_I,y)=0\) provided \(\mathrm{Hom}_\Gamma (S'_i,y)=0\) for all \(i\in I\).
Proof
Denote \(t=\mathrm{depth}\,\Lambda _I\).
(1) In situation (a), by 2.21(2) \(\mathrm {pd}_\Lambda \Lambda _I=2\) and \(\mathrm{Ext}_{\Lambda }^1(T_I,x)\cong \mathrm{Ext}_\Lambda ^{2}(\Lambda _I,x)\), which is \(\mathrm{Ext}_\Lambda ^{dt}(\Lambda _I,x)\) by Auslander–Buchsbaum. This is zero by 5.9. In situation (b), by A.7(4) the assumptions in fact force \(d\le 3\). If \(d=2\) then the result is precisely 2.26, so we can assume that \(d= 3\). In this case, by 2.22(2) \(\mathrm {pd}_\Lambda \Lambda _I=3\) and \(\mathrm{Ext}_{\Lambda }^1(T_I,x)\cong \mathrm{Ext}_\Lambda ^{3}(\Lambda _I,x)=\mathrm{Ext}_\Lambda ^{dt}(\Lambda _I,x)\), which again is zero by 5.9.
The following lemma is elementary.
Lemma 5.11
 (1)
If \(\upvartheta _i>0\) for all \(i\in I\), then \(\mathrm{Hom}_\Lambda (x,S_i)=0\) for all \(i\in I\).
 (2)
If \(\upphi _i<0\) for all \(i\in I\), then \(\mathrm{Hom}_\Gamma (S'_i,y)=0\) for all \(i\in I\).
Proof
(2) Any nonzero morphism \(S'_i\rightarrow y\) is necessarily injective, so \(\upphi _i=\upphi \cdot {\underline{\mathrm{dim}}\,}S'_i\ge 0\) since \(y\in \mathcal {S}_{\upphi }(\Gamma )\). Since \(\upphi _i<0\), the morphism must be zero. \(\square \)
Given the technical preparation above, the following is now very similar to [45, 3.5].
Theorem 5.12
 (a)
\(\upnu _IM\cong M\), or
 (b)
\(\upnu _I\upnu _IM\cong M\) and \(\dim _{\mathbb {C}}\Lambda _I<\infty \).
 (1)
\(\mathrm{Hom}_{\Lambda }(T_I,):\mathcal {S}_{\upvartheta }(\Lambda )\rightarrow \mathcal {S}_{\upnu _{\mathbf {b_I}}\upvartheta }(\Gamma )\) is an exact functor.
 (2)
\(T_I\otimes _\Gamma :\mathcal {S}_{\upnu _{\mathbf {b_I}}\upvartheta }(\Gamma )\rightarrow \mathcal {S}_{\upvartheta }(\Lambda )\) is an exact functor.
 (3)There is a categorical equivalence(5.C)
 (4)
\(\upvartheta \) is generic if and only if \(\upnu _{\mathbf {b_I}}\upvartheta \) is generic.
Proof
(1) By 5.10(1) and 5.11(1), \(\mathrm{Hom}_\Lambda (T_I,)\) is exact out of \(\mathcal {S}_{\upvartheta }(\Lambda )\). To see that \(\mathrm{Hom}_{\Lambda }(T_I,)\) maps \(\mathcal {S}_{\upvartheta }(\Lambda )\) to \(\mathcal {S}_{\upnu _{\mathbf {b_I}}\upvartheta }(\Gamma )\), suppose that \(x\in \mathcal {S}_{\upvartheta }(\Lambda )\), let \(y:=\mathrm{Hom}_\Lambda (T_I,x)\cong {\mathbf{R}\text {Hom}}_\Lambda (T_I,x)\) and consider a \(\Gamma \)submodule \(y'\subseteq y\). Since \(\upnu _{\mathbf {b_I}}\upvartheta \cdot {\underline{\mathrm{dim}}\,}y=\upnu _{\mathbf {b_I}}\upvartheta \cdot \upnu _{\mathbf {b_I}}{\underline{\mathrm{dim}}\,}x=\upvartheta \cdot {\underline{\mathrm{dim}}\,}x=0\) by 5.7(1) and 5.6(1), it suffices to show that \(\upnu _{\mathbf {b_I}}\upvartheta \cdot {\underline{\mathrm{dim}}\,}y'\ge 0\).
and so y is \(\upnu _{\mathbf {b_I}}\upvartheta \)semistable, proving the claim.
(3) If \(x\in \mathcal {S}_{\upbeta , \upvartheta }(\Lambda )\) then by 5.10(1) and 5.11(1) \(\mathrm{Ext}^1_\Lambda (T_I,x)=0\). Thus \({\underline{\mathrm{dim}}\,}\mathrm{Hom}_\Lambda (T_I,x)=\upnu _{\mathbf {b_I}}\upbeta \) by 5.7(1). Similarly if \(y\in \mathcal {S}_{\upnu _{\mathbf {b_I}}\upbeta ,\upnu _{\mathbf {b_I}}\upvartheta }(\Gamma )\) then by 5.10(2) and and 5.11(2) \(\mathrm{Tor}_1^\Gamma (T_I,y)=0\) and so \({\underline{\mathrm{dim}}\,}(T_I\otimes _\Gamma y)=\upnu _{\mathbf {b_I}}{\underline{\mathrm{dim}}\,}y=\upnu _{\mathbf {b_I}}\upnu _{\mathbf {b_I}}\upbeta =\upbeta \) by 5.7(2) and 5.6(2). Thus the functors in (5.C) are well defined, and further since \(T_I\) has projective dimension one (on both sides) by 2.20, they are isomorphic to their derived versions. Since the derived versions are an equivalence, we deduce the underived versions are. They are exact by (1) and (2), so it follows that they preserve the Sequivalence classes. It is also clear in the above proof that replacing \(\ge 0\) by \(>0\) throughout shows that under the equivalence, stable modules correspond to stable modules.
(4) Follows immediately from (3). \(\square \)
Corollary 5.13
 (1)
There is an isomorphism \(\mathcal {M}_{\upbeta ,\upvartheta }(\Lambda )\cong \mathcal {M}_{\upnu _{\mathbf {b_I}}\upbeta ,\upnu _{\mathbf {b_I}}\upvartheta }(\Gamma )\).
 (2)
There is an isomorphism \(\mathcal {M}_{\upnu _{\mathbf {c_I}}\upbeta ,\upnu _{\mathbf {c_I}}\upvartheta }(\Lambda )\cong \mathcal {M}_{\upbeta ,\upvartheta }(\Gamma )\).
Proof
(1) It follows immediately from 5.12(3) that there is a bijection on closed points. The fact that 5.12(3) holds after base change, and so there is an isomorphism of schemes, is dealt with in [45, 4.20], noting the small correction in [27, Appendix A].
(2) By A.6 either the assumption (a) or (b) holds for \(\upnu _IM\). Hence we can apply 5.12(3) to the mutation \(\Gamma \mapsto \upnu _I\Gamma \cong \Lambda \). For this mutation, the b’s are given by \(\mathbf {c_I}\), using (A.M) (and the fact that \(W_I\cong V_I\) there). \(\square \)
Recall from the introduction Sect. 1.3 the definition of the dimension vector \(\mathsf{rk}\).
Corollary 5.14
Remark 5.15
We will prove later in 5.22 that \(U_i\cong V_i\) for all \(i\in I\) in the case of cDV singularities, so \(\mathbf {b_I}= \mathbf {c_I}\) in this case. However, even for NCCRs in dimension three with \(I=\{i\}\), \(\mathbf {b_I}\ne \mathbf {c_I}\) in general.
5.3 Chamber structure: reduction to surfaces
In this subsection we revert back to the crepant one dimensional fibre setting of 2.9. Throughout, we restrict to the dimension vector \(\mathsf{rk}\), and show that (for this dimension vector) the chamber structure on the stability parameters can be calculated by passing to a Kleinian singularity.
Remark 5.16
As the moduli space \(\mathcal {M}_{\mathsf{rk},\upvartheta }(\Lambda )\) parameterises only semistable \(\Lambda \)modules of dimension vector \(\mathsf{rk}\), and such modules x necessarily satisfy \(\upvartheta \cdot \mathsf{rk}=\upvartheta \cdot {\underline{\mathrm{dim}}\,}x=0\) by definition of semistability, henceforth we are only concerned with those stability parameters for which \(\upvartheta \cdot \mathsf{rk}=0\). This subspace of \(\Theta \), which we will temporarily denote by \(\Theta _{\mathsf{rk}}\), has a wall and chamber structure. The nongeneric parameters cut out walls, dividing the generic parameters of \(\Theta _{\mathsf{rk}}\) into chambers.
Each \(\Lambda \) in the general setup of 2.8 has an associated \(\Theta _{\mathsf{rk}}\), and as the chamber structure of \(\Theta _{\mathsf{rk}}\) depends on \(\Lambda \), later care will be required. When it is necessary to emphasise which ring is being considered, we will use the notation \(\Theta _{\mathsf{rk}}(\Lambda )\).
Notation 5.17
Henceforth, until the end of the paper, we will write \(\Theta _{\mathsf{rk}}\) as simply \(\Theta \), and \(\Theta _{\mathsf{rk}}(\Lambda )\) as simply \(\Theta (\Lambda )\), with it being implicit that everywhere walls and chambers are discussed, this involves only working with those stability parameters \(\upvartheta \) such that \(\upvartheta {\cdot }\mathsf{rk}=0\). This is an abuse of notation, but it is required to maintain readability later.
Lemma 5.18
Proof
It is clear that every element of \(C_+\) is generic, and further if \(\upvartheta , \upvartheta '\in C_+\), then x is \(\upvartheta \)stable if and only if it is \(\upvartheta '\)stable. Hence \(C_+\) is contained in some GIT chamber. It suffices to show that for each i, there exists some \(x_i\in \mathcal {M}_{\mathsf{rk},C_+}(\Lambda )\) and an injection \(S_i\hookrightarrow x_i\), since this implies that \(x_i\) is not stable in the limit \(\upvartheta _i\rightarrow 0\) and so \(\upvartheta _i=0\) then defines a wall.
The strategy to describe the chambers of \(\Theta (\Lambda )\) is to track \(C_+\) through mutation, and calculate the combinatorics by passing to surfaces. This requires a special case of the following general result.
Proposition 5.19
 (1)
\((R/xR)\otimes _R\mathrm{End}_R(M)\cong \mathrm{End}_{FR}(FM)\), and FM is indecomposable.
 (2)If further x is \(\mathrm{Ext}^1_R(K_i,K_i)\)regular, the sequenceis exact, and further \(F{a_i}\) is a minimal \(\mathrm{add}\,F{M_{I^c}}\)approximation.$$\begin{aligned} 0\rightarrow F{K}_i\rightarrow F{V}_i\xrightarrow {F{a_i}} F{M}_i\rightarrow 0 \end{aligned}$$(5.I)
Proof
(1) The first statement is wellknown; see for example the argument in [25, 5.24], which uses the fact that x is \(\mathrm{Ext}^1_R(M,M)\)regular. The second follows from the first, since if FM decomposes, since R is complete local we can lift idempotents to obtain a contradiction.
(2) Since \(M_{I^c}\) is a generator and \(\mathrm{Hom}_R(M_{I^c},)\) applied to (2.E) is exact, it follows that \(a_i\) is surjective. Also, since \(M_{I^c}\) is a generator and \(\mathrm{End}_R(M)\in \mathrm{CM}\,R\), necessarily \(M\in \mathrm{CM}\,R\) and since CM modules are closed under kernels of epimorphisms, \(K_I\in \mathrm{CM}\,R\).
Corollary 5.20
 (1)
\(\Lambda /g\Lambda \cong \mathrm{End}_{R/gR}(N/gN)\).
 (2)Let \(N_I\) be a summand as in 2.13, and consider the exchange sequencesThen \(a_i\) and \(b_i\) are surjective, and$$\begin{aligned} 0\rightarrow K_i\xrightarrow {c_i} V_i\xrightarrow {a_i} N_i\\ 0\rightarrow J_i\xrightarrow {d_i} U_i^*\xrightarrow {b_i} N_i^* \end{aligned}$$are exact, with \(Fa_i\) and \(Fb_i\) being minimal right approximations.$$\begin{aligned} 0\rightarrow FK_i\xrightarrow {Fc_i} FV_i\xrightarrow {Fa_i} FN_i\rightarrow 0\\ 0\rightarrow FJ_i\xrightarrow {Fd_i} FU^*_i\xrightarrow {Fb_i} FN_i^*\rightarrow 0 \end{aligned}$$
 (3)We have that \(0\rightarrow J_i\xrightarrow {d_i} U_i^*\xrightarrow {b_i} N_i^*\rightarrow 0\) is exact, inducing an exact sequencewhere \(F(b_i^*)\) is a minimal left \(\mathrm{add}\,FM_{I^c}\)approximation.$$\begin{aligned} 0\rightarrow FN_i\xrightarrow {F(b_i^*)} FU_i\xrightarrow {F(d_i^*)} FJ_i^*\rightarrow 0 \end{aligned}$$
Proof
(1)(2) Since \(\mathrm{End}_R(N)\in \mathrm{CM}\,R\), \(\mathrm{depth}\,\mathrm{Ext}^1_R(N,N)>0\) and so if an element g acts on \(E_N:=\mathrm{Ext}^1_R(N,N)\) as a zero divisor, then it is contained in one of the finitely many associated primes of \(E_N\), which are all nonmaximal. We can apply the same logic to both \(K_I\) and \(J_I^*\), and thus the finite number of associated primes of \(E_N\oplus E_{K_I}\oplus E_{J_I^*}\) are nonmaximal. Hence we can find g sufficiently generic to be \(E_N\oplus E_{K_I}\oplus E_{J_I^*}\)regular, so the first two parts follow from 5.19.
(3) This is just the dual of (2), and follows by A.1, part (2) and isomorphisms such as \( \mathrm{Hom}_{FR}(FN_i^*,FR)\cong F\,\mathrm{Hom}_{R}(N_i^*,R)\cong FN_i\). \(\square \)
The proof of the next lemma, 5.22, requires a little knowledge regarding knitting on AR quivers, which we briefly review in an example. We refer the reader to [22, §4] for full details.
Example 5.21
The following is an extension of the above observation.
Lemma 5.22
Proof
As in 5.20, let g be a sufficiently generic hyperplane section and let \(F:=(R/gR)\otimes _R()\). As before, decompose \(U_i\cong \bigoplus _{j\notin I}N_j^{\oplus b_{i,j}}\) and \(V_i\cong \bigoplus _{j\notin I}N_j^{\oplus c_{i,j}}\), from which it is clear that \(FU_i\cong \bigoplus _{j\notin I}(FN_j)^{\oplus b_{i,j}}\) and \(FV_i\cong \bigoplus _{j\notin I}(FN_j)^{\oplus c_{i,j}}\). By 5.19(1) the \(FN_j\) are indecomposable, so by Krull–Schmidt to prove both parts it suffices to show that \(b_{i,j}=c_{i,j}\) for all \(i\in I\), \(j\notin I\).
Both \(FU_i\) and \(FV_i\) can be calculated by knitting on the AR quiver of the ADE singularity FR. As in 5.21, the calculation for \(FU_i\) begins by placing a 1 in the place of \(FN_i\), and proceeds by counting to the left, using the usual knitting rules, and records the numbers in the circles whilst treating them as zero for the next step. At the end of the calculation, we read off the \(b_{i,j}\) by adding the numbers in the circled vertices.
On the other hand, the calculation for \(FV_i\) begins by placing a 1 in the place of \(FN_i\), then proceeds by counting to the right. In exactly the same way, we read off the \(c_{i,j}\) by adding the numbers in the circled vertices. Since the AR quiver of ADE surface singularities coincides with the McKay quiver [1], which is symmetric, we can obtain one calculation from the other by reflecting in the line through the original boxed vertex. Thus both calculations return the same numbers, so \(c_{i,j}=b_{i,j}\) for all \(i\in I\) and \(j\notin I\). \(\square \)
Corollary 5.23
 (A)
\(f:X\rightarrow \mathrm{Spec}\,R\) is a minimal model, or
 (B)
\(f:X\rightarrow \mathrm{Spec}\,R\) is a flopping contraction.
Proof
Pick a curve \(C_i\) (i.e. consider \(I=\{i\}\)), and mutate at the indecomposable summand \(N_i\) of N. By 5.18, \(\upvartheta _i=0\) is a wall. Since we are mutating only at indecomposable summands, in situation (A) 2.23 shows that the assumptions of 5.12 are satisfied. In situation (B), 2.25 together with 3.5 shows that the assumptions of 5.12 are satisfied. Thus, in either case, since \(U_i\cong V_i\) by 5.22, provided that \(\upvartheta _i\ne 0\) it is possible to track moduli using 5.14.
In either (A) or (B), if \(\dim _{\mathbb {C}}\Lambda _i<\infty \) then \(C_i\) flops, in which case \(\upnu _iN\cong H^0(\mathcal {V}_{X_i^+})\) by 4.20. Thus \(\mathcal {M}_{\mathsf{rk},\upphi }(\upnu _i\Lambda )\cong X_i^+\) for all \(\upphi \in C_+(\upnu _i\Lambda )\) by 4.19(2), so the result then follows by moduli tracking 5.14. The only remaining case is when \(\dim _{\mathbb {C}}\Lambda _i=\infty \) in situation (A), but then \(\upnu _iN\cong N\) by 2.23 and so the result is obvious. \(\square \)
The main result of this subsection needs the following result, which may be of independent interest. The case when Y is the minimal resolution is well known [9, 36].
Theorem 5.24
 (1)
The walls of \(\Theta (\Gamma )\) are obtained by intersecting the subspace L of \(\Theta (\Lambda )\) spanned by \(\upvartheta _1,\ldots ,\upvartheta _r\) with the walls of \(\Theta (\Lambda )\) that do not contain L.
 (2)
\(\Theta (\Gamma )\) has a finite number of chambers, and the walls are given by a finite collection of hyperplanes containing the origin. The coordinate hyperplanes \(\upvartheta _i=0\) are included in this collection.
 (3)
Considering iterated mutations at indecomposable summands, tracking the chamber \(C_+\) on \(\upnu _{i_1}\ldots \upnu _{i_t}(\Gamma )\) back to \(\Theta (\Gamma )\) gives all the chambers of \(\Theta (\Gamma )\).
Proof
As notation, \(\Theta (\Lambda )\) has coordinates \(\upvartheta _1,\ldots ,\upvartheta _{n}\), and we let S be the subspace spanned by \(\upvartheta _1,\ldots ,\upvartheta _{n1}\). By abuse of notation, we let \(\upvartheta _1,\ldots ,\upvartheta _{n1}\) also denote the coordinates of \(\Theta (\Lambda _{n1})\), so that we identify \(\Theta (\Lambda _{n1})\) with S. We let \(W_S\) be the set of walls of \(\Theta (\Lambda )\) not containing S, then the intersection \(S\cap W_S\) partitions S into a finite number of regions. We claim that these are precisely the chambers of \(\Theta (\Lambda _{n1})\).
First, since by 5.18 \(\{\upvartheta \in \Theta (\Lambda )\mid \upvartheta _i>0 \hbox { for all }1\le i\le n\}\) is a chamber of \(\Theta (\Lambda )\), certainly no walls of \( \Theta (\Lambda )\) intersect \(\{\upvartheta \in S\mid \upvartheta _i>0 \hbox { for all }1\le i\le n1\}\). Thus we may identify this region of \(S\backslash (S\cap W_S)\) with \(C_+\) in \(\Theta (\Lambda _{n1})\).
The proof then proceeds by induction. By applying the argument above, tracking \(C_+\) from \(\upnu _{\{2,n\}}\upnu _{\{1,n\}}\Lambda \) to \(\upnu _{\{1,n\}}\Lambda \), implies that tracking \(C_+\) from \(\upnu _2\upnu _1\Lambda _{n1}\) to \(\upnu _1\Lambda _{n1}\) gives a chamber in \(\Theta (\upnu _1\Lambda _{n1})\), adjacent to \(C_+\), cut out by intersecting walls from \(\Theta (\Lambda )\). In particular, the plane \(x_1=0\) does not cut through this chamber, so by 5.14 we can track the full chamber all the way back to \(\Theta (\Lambda _{n1})\) to obtain a chamber adjacent to (5.K). Again, the same argument shows that its walls are given by intersecting S with the elements of \(W_S\). By symmetry, all the walls of all the chambers bordering all the chambers that border \(C_+\) in \(\Theta (\Lambda _{n1})\) are given by intersecting S with the elements of \(W_S\).
Since \(\Theta (\Lambda )\) has finitely many walls [9, 36], so does \(S\cap W_S\), so continuing the above process all the walls of \(\Theta (\Lambda _{n1})\) are given by intersecting S with the elements of \(W_S\), and each region is the tracking of \(C_+\) under iterated mutation. This proves (1), (2) and (3) for \(X_{n1}\).
Next, consider \(f_2:X_{n1}\rightarrow X_{n2}\). Since by above \(\Theta (\Lambda _{n1})\), and all other \(\Lambda '_{n1}\) obtained from X by contracting only a single curve, have walls given by a finite collection of hyperplanes passing through the origin, the above argument can be repeated to \(\Lambda _{n2}\cong (1e_{n1})\Lambda _{n1}(1e_{n1})\) to show that \(\Theta (\Lambda _{n2})\) can be obtained from \(\Theta (\Lambda _{n1})\) by intersecting (and thus from \(\Theta (\Lambda )\) by intersecting), and each region is the tracking of \(C_+\) under iterated mutation. By induction, parts (1), (2) and (3) follow. \(\square \)
The following is the main result of this subsection.
Corollary 5.25
 (A)
\(f:X\rightarrow \mathrm{Spec}\,R\) is a minimal model, or
 (B)
\(f:X\rightarrow \mathrm{Spec}\,R\) is a flopping contraction.
 (1)
The chamber structure of \(\Theta (\Lambda )\) is the same as the chamber structure of \(\Theta (\Lambda /g\Lambda )\).
 (2)
\(\Theta (\Lambda )\) has a finite number of chambers, and the walls are given by a finite collection of hyperplanes containing the origin. The coordinate hyperplanes \(\upvartheta _i=0\) are included in this collection.
 (3)
Considering iterated mutations at indecomposable summands, tracking the chamber \(C_+\) on \(\upnu _{i_1}\ldots \upnu _{i_t}\Lambda \) back to \(\Theta (\Lambda )\) gives all the chambers of \(\Theta (\Lambda )\).
5.4 Surfaces chamber structure via AR theory
Having in 5.24 and 5.25 reduced the problem to tracking the chamber \(C_+\) under iterated mutation for partial crepant resolutions of Kleinian singularities, in this subsection we illustrate the combinatorics in two examples, summarising others in Sect. 7.1, and give some applications.
The intersection in 5.24(1) is in practice very cumbersome to calculate, since the full root systems are very large and contain much redundant information. In addition to giving an easy, direct way of calculating the chamber structure, the benefit of working with mutation is that we also obtain, in 5.28, a lower bound for the number of minimal models on the 3fold.
Example 5.26
However, often the dual graph does change under mutation.
Example 5.27
The following is an extension of the above observation. In each chamber of \(\Theta (\Lambda /g\Lambda )\), we draw the curve configuration appearing on the surface mutation calculation, as calculated in the above example. We refer to this as the enhanced chamber structure of \(\Theta (\Lambda /g\Lambda )\). See 7.3 for an example.
Lemma 5.28
Suppose that R is a cDV singularity, with a minimal model \(X\rightarrow \mathrm{Spec}\,R\). Set \(\Lambda :=\mathrm{End}_R(N)\), where \(N:=H^0(\mathcal {V}_X)\). Calculating \(\Theta (\Lambda )\) by passing to a general hyperplane section g, the number of minimal models of \(\mathrm{Spec}\,R\) is bounded below by the number of different curve configurations obtained in the enhanced chamber structure of \(\Theta (\Lambda /g\Lambda )\).
Proof
Certainly if two minimal models X and Y cut under generic hyperplane section to two different curve configurations, then X and Y must be different minimal models. Thus it suffices to show that every curve configuration in the enhanced chamber structure of \(\Theta (\Lambda /g\Lambda )\) does actually appear as the cut of some minimal model. By 5.24(3) and 5.25(3) it is possible to reach any such configuration starting at \(C_+\) by mutating at indecomposable summands. Since by 4.6 at each wall crossing either the moduli stays the same, or some curve flops, each chamber in \(\Theta (\Lambda )\) gives a minimal model of \(\mathrm{Spec}\,R\). Hence if a curve configuration is in a chamber D of \(\Theta (\Lambda /g\Lambda )\), consider the minimal model given by the chamber D of \(\Theta (\Lambda )\). This minimal model cuts to the desired curve configuration, by 5.20. \(\square \)
6 First applications
6.1 The Craw–Ishii conjecture
Combining moduli tracking from Sect. 5 with the Homological MMP in Fig. 2 leads immediately to a proof of the Craw–Ishii conjecture for cDV singularities. To prove a slightly more precise version, the following terminology will be convenient.
Definition 6.1
The skeleton of the GIT chamber decomposition of \(\Theta \) is defined to be the graph obtained by placing a vertex in every chamber, and two vertices are connected by an edge if and only if the associated chambers share a codimension one wall.
The following is the main result of this subsection.
Theorem 6.2
 (1)
In the skeleton of \(\Theta (\Lambda )\), there exists a connected path, containing the chamber \(C_+\), where every minimal model can be found. Furthermore, each wall crossing in this path corresponds to a flop.
 (2)
The Craw–Ishii Conjecture 1.7 is true for cDV singularities, namely for any fixed MMA \(\Gamma :=\mathrm{End}_R(M)\) with \(R\in \mathrm{add}\,M\), every projective minimal model can be obtained as the moduli space of \(\Gamma \) for some stability parameter \(\upvartheta \).
Proof
(1) We run Fig. 2 whilst picking only single curves satisfying \(\dim _{\mathbb {C}}\Lambda _i<\infty \). As in 4.10, this produces all minimal models. By 4.19 we can view all these minimal models as the \(C_+ \) moduli on their corresponding algebra \(\upnu _{i_1}\ldots \upnu _{i_t}\Lambda \). By 5.25(3) it is possible to track all these back to give chambers in \(\Theta (\Lambda )\), and the proof of 5.24 shows that this combinatorial tracking gives a connected path. The fact that each wall gives a flop is identical to 5.23, since at each stage \(\dim _{\mathbb {C}}\Lambda _i<\infty \).
(2) Consider an MMA \(\Gamma :=\mathrm{End}_R(M)\) as in the statement. By the Auslander–McKay correspondence 4.10 \(M\cong H^0(\mathcal {V}_Y)\) for some minimal model \(Y\rightarrow \mathrm{Spec}\,R\). The result then follows by applying (1) to \(Y\rightarrow \mathrm{Spec}\,R\). \(\square \)
We remark that flops of multiple curves can also be easily described. The following is the multicurve version of 5.23.
Lemma 6.3
With the \(d=3\) crepant setup of 2.9, set \(N=H^0(\mathcal {V}_X)\), \(\Lambda =\mathrm{End}_R(N)\), and pick a subset of curves I above the origin. If \(\dim _{\mathbb {C}}\Lambda _I<\infty \), then \(\mathcal {M}_{\mathsf{rk},\,\upnu _{\mathbf {b_I}}\upvartheta }(\Lambda )\) is the flop of \(\bigcup _{i\in I}C_i\), for all \(\upvartheta \in C_+(\Lambda )\)
6.2 Auslander–McKay revisited
In the \(d=3\) crepant setup of 2.9, in this subsection we use the extra information of the GIT chamber decomposition of \(\Theta (\Lambda )\) from Sect. 5 to extend aspects of the Auslander–McKay Correspondence from Sect. 4.2.
Theorem 6.4
 (5)
The simple mutation graph of MM generators can be viewed as a subgraph of the skeleton of the GIT chamber decomposition of \(\Theta (\Lambda )\).
 (6)
The number of MM generators is bounded above by the number of chambers in the GIT chamber decomposition of any of the MMAs, and is bounded below by the number of different curve configurations obtained in the enhanced chamber structure of \(\Theta (\Lambda /g\Lambda )\).
6.3 Root systems
We observed in 5.26 that the chamber structure of partial resolutions of Kleinian singularities, and thus by 5.25 also the corresponding cDV singularities, cannot in general be identified with the root system of a semisimple Lie algebra. In special cases, however, they can.
Lemma 6.5
With the crepant setup \(f:X\rightarrow \mathrm{Spec}\,R\) of 2.9, suppose that \(d=2\) and R is a type A Kleinian singularity. Set \(\Lambda :=\mathrm{End}_X(\mathcal {V}_X)\cong \mathrm{End}_R(N)\). If there are t curves above the unique closed point, then the chamber structure for \(\Theta (\Lambda )\) can be identified with the root system of \(\mathfrak {s}\mathfrak {l}_t\).
Proof
Label the CM Rmodules corresponding to the curves in the minimal resolution by \(N_1,\ldots , N_n\), from left to right. Since X is dominated by the minimal resolution, it is obtained by contracting curves, to leave CM modules \(N_{j_1},\ldots ,N_{j_t}\) say, so that \(N=H^0(\mathcal {V}_X)=N_{j_1}\oplus \cdots \oplus N_{j_t}\).
and so the combinatorics that determine the tracking negates \(\upvartheta _{j_i}\) and adds \(\upvartheta _{j_i}\) to each of its neighbours.
On the other hand, if we consider the minimal resolution of the Type A singularity \(\frac{1}{t+1}(1,1)\), which also has t curves above the origin, the combinatorics that governs tracking \(C_+\) in this case is also the rule that negates \(\upvartheta _j\) and adds \(\upvartheta _j\) to its neighbours. Hence since the chamber structure for the minimal resolution of \(\frac{1}{t+1}(1,1)\) can be identified with the root system of \(\mathfrak {sl}_t\) [9, 36], so too can the chamber structure of \(\Theta (\Lambda )\). \(\square \)
By combining 5.25 and 6.5, the following is immediate.
Corollary 6.6
 (A)
\(f:X\rightarrow \mathrm{Spec}\,R\) is a minimal model, or
 (B)
\(f:X\rightarrow \mathrm{Spec}\,R\) is a flopping contraction.
There are also other cases in which root systems appear. Consider the following assumption, made throughout in [46].
Setup 6.7
In the \(d=3\) crepant setup of 2.9, suppose that R is isolated and there is a hyperplane section which cuts X to give the minimal resolution.
The setup is restrictive, for example in the case of a minimal model of \(\mathrm{Spec}\,R\) with only one curve above the origin, it forces R to be Type A. Nevertheless, under the setup 6.7, associated to R is some ADE Dynkin diagram. The following recovers [47, §5.1].
Lemma 6.8
 (1)
The chamber structure of \(\Theta (\Lambda )\) can be identified with the root system of the corresponding Dynkin diagram.
 (2)
There are precisely W chambers, where W is the corresponding Weyl group.
Proof
(1) Since R is isolated \(\mathrm{Ext}^1_R(N,N)=0\). Thus the \(\mathrm{Ext}^1_R(N,N)\)regular condition in 5.19 is redundant, so 5.20 holds for the particular g in 6.7. Appealing to this directly in the proof of 5.25 shows that the chamber structure of \(\Theta (\Lambda )\) and \(\Theta (\Lambda /g\Lambda )\) coincide. Since by 6.7 the pullback of the hyperplane section is the full minimal resolution, it follows that \(\Lambda /g\Lambda \) is the preprojective algebra of the corresponding Dynkin diagram, and its chamber structure is wellknown [9, 36]. Part (2) is immediate. \(\square \)
6.4 Auslander–McKay for isolated singularities
With the crepant setup of 2.9, the case when R is in addition an isolated singularity is particularly important for two reasons. First, it aligns well with cluster theory, since in this setting \({\underline{\mathrm{CM}}}\,R\) is a Homfinite 2CY category, with maximal rigid objects the MM generators, and cluster tilting objects (if they exist) the CT modules. Second, the minimal models are easier to count, thus we have finer control over the mutation graph.
Corollary 6.9
 (5)
The simple mutation graph of the maximal rigid (respectively, cluster tilting) objects in \({\underline{\mathrm{CM}}}\,R\) is precisely the skeleton of the GIT chamber structure.
 (6)
The number of basic maximal rigid (respectively, cluster tilting) objects in \({\underline{\mathrm{CM}}}\,R\) is precisely the number of chambers in the GIT chamber decomposition.
 (7)
There are precisely W maximal rigid objects in \(\mathcal {C}={\underline{\mathrm{CM}}}\,R\), where W is the corresponding Weyl group.
Proof
Since R is isolated, M is a maximal rigid object in the category \(\mathrm{CM}\,R\) if and only if M is an MM generator [23, 5.12], so the first bijection is a special case of the bijection in 4.10. The second bijection is similar, using [23, 5.11]. Further, since R is isolated, it follows that always \(\dim _{\mathbb {C}}\Lambda _i<\infty \), so all curves flop, and all summands nontrivially mutate. Thus (5) follows from 4.10(3), using the argument of 6.2(1). Part (6) follows immediately from (5), and part (7) follows from (6) together with 6.8. \(\square \)
We refer the reader to Sect. 7.1 for examples of chamber structures and mutation graphs. The following is a nonexplicit proof of [7, 4.15], extended from crepant resolutions to also cover minimal models.
Corollary 6.10
Consider an isolated cDV singularity \(R:=\mathbb {C}[[u,v,x,y]]/(uvf_1\ldots f_n)\) where each \(f_i\in \mathfrak {m}:=(x,y)\subseteq \mathbb {C}[[x,y]]\). Then there are precisely n! maximal rigid objects in \({\underline{\mathrm{CM}}}\,R\), and all are connected by mutation.
Proof
As in [7, 6.1(e)], R is a \(cA_m\) singularity, where \(m=\mathrm{ord}(f_1\ldots f_n)1\), and it is well known (see e.g. the calculation in [24, §5.1]) that the minimal models of \(\mathrm{Spec}\,R\) have n curves above the origin. But by 6.6 the GIT chamber decomposition of any of the MMAs \(\mathrm{End}_R(M)\) with \(R\in \mathrm{add}\,M\) has precisely n! chambers, so the result follows from 6.9. \(\square \)
6.5 Partial converse
Let R be a complete local Gorenstein 3fold. By the Auslander–McKay correspondence, if R is cDV then there are only finitely many basic MM modules up to isomorphism. Recall from 1.14 that we conjecture the converse to be true. Since such R are known to be hypersurfaces, the corollary of the following result, although it does not prove the conjecture, does give it some credibility.
The following extends [2, §3] to cover notnecessarilyisolated singularities.
Proposition 6.11
Suppose that R is a ddimensional complete local Gorenstein algebra. If R admits only finitely many basic CT modules up to isomorphism, then R is a hypersurface.
Proof
Now, by general mutation theory, \(\Omega ^iM\) are CT modules for all \(i\in \mathbb {N}\) [23, 6.11], and since by assumption there are only finitely many basic CT modules, \(\Omega ^iM\cong \Omega ^jM\) for some \(i\ne j\), which by taking cosyzygies implies that \(\Omega ^tM\cong M\) for some \(t\ge 1\). Consequently, \({\mathrm {cx}}_RM\le 1\).
Corollary 6.12
Suppose that R is a 3dimensional complete local normal Gorenstein algebra, and suppose that R admits an NCCR. If there are only finitely many basic MM generators up to isomorphism, then R is a hypersurface.
7 Examples
In this section we summarise the GIT chamber decompositions of some crepant partial resolutions of ADE surface singularities, and give the corresponding applications to cDV singularities. We also illustrate how to run Fig. 2 in some explicit cases.
7.1 GIT chamber structures
Throughout this subsection, \(Y\rightarrow \mathrm{Spec}\,S\) denotes a crepant partial resolution, where S is a complete local ADE surface singularity, and \(X\rightarrow \mathrm{Spec}\,R\) denotes a crepant partial resolution, where R is cDV.
Example 7.1
Example 7.2
Example 7.3
By 5.28, it follows that any cDV singularity with a minimal model that cuts under generic hyperplane section to (7.A) has at least 5, and at most 10, minimal models.
Example 7.4
For the 3curve configuration Open image in new window , the chamber structure is illustrated in Fig. 6, whereas for the 3curve configuration Open image in new window , the chamber structure is illustrated in Fig. 7.
Tracking the dual graph through mutation, as in 5.27 and 5.28, any cDV singularity with a minimal model that cuts to the above \(D_4\) configuration has at least 4 and at most 32 minimal models. Any cDV singularity with a minimal model that cuts to the above \(E_6\) configuration has at least 5 and at most 60 minimal models.
Remark 7.5
The singularity \(R:=\mathbb {C}[[u,x,y,z]]/(u^2xyz)\) in 3.4 is in fact \(cD_4\) with a three curve configuration, so the chamber structure is precisely Fig. 6. The chamber structure for the particular example \(u^2=xyz\) was computed independently, using entirely different methods, by Craw and King in 2000. Indeed, [12, 5.31, footnote 5 p117] computes the first four chambers (Fig. 8). See also [38].
7.2 Running the algorithm
This subsection illustrates how to run the Homological MMP in two examples. For the aid of the reader, we begin with the toric example in 3.4, since the geometry will already be familiar.
Example 7.6
We next claim that this is precisely the simple mutation graph of the MM generators, equivalently we have already found all minimal models of \(\mathrm{Spec}\,R\).
Step 1b: Contractions We plug in the mutated algebra \(\mathrm{End}_R(\upnu _{2}N)\) into Step 1, and repeat. Due to the relations in the algebra \(\upnu _{2}\Lambda =\mathrm{End}_R(\upnu _{2}N)\), it follows that \(\dim _\mathbb {C}(\upnu _{2}\Lambda )_{\{1\}}=\infty =\dim _{\mathbb {C}}(\upnu _{2}\Lambda )_{\{3\}}\), thus in \(\upnu _{2}\Lambda \) the only curve we can nontrivially mutate is the middle one, which gives us back our original N. Thus the Homological MMP stops, and we have reached all minimal models.
Example 7.7
Remark 7.8
The mutation trees of quivers are usually quite easy to write down, and this then determines all the geometry. We refer the reader to [39, §4.1] for the calculation of the mutation trees for some other quotient singularities, in particular [39, 4.4]. We remark that it follows from Fig. 2 that [39, §4.1] is now enough to establish we have all minimal models. In particular, we can immediately read off the dual graph and whether curves flop from the quivers there, avoiding all the hard explicit calculations in [39, §5–6].
Notes
Acknowledgements
It is a pleasure to thank Alastair Craw and Alastair King for many discussions and comments over the eight years since this work began, and to thank Will Donovan and Osamu Iyama for many discussions and ideas, some of which are included here, and for collaborating on the foundational aspects [15, 16, 17, 23, 24, 25]. It is also a pleasure to thank the referee for many helpful comments and suggestions. Thanks are also due to Gavin Brown, Alvaro Nolla de Celis, Hailong Dao, Eleonore Faber, Martin Kalck, Joe Karmazyn, Miles Reid and Yuhi Sekiya.
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