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...e Homological MMP (proved in 4.15) to give results in GIT, specifically regarding chamber decompositions and later in §6.1 the Craw–Ishii conjecture. Throughout, we use the language of King stability =-=[Ki]-=- since it brings some advantages, but we could use the language of Bridgeland stability instead. After first proving general moduli–tracking results in §5.2, running Figure 2 over all possibilities an...

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...ete local we may factorize f into X g−→ Xcon h−→ SpecR where g(Cj) is a closed point if and only if j ∈ I, and further g∗OX = OXcon , see e.g. [Ko1, p25] or [S, §2]. Further, by the vanishing theorem =-=[KMM, 1-2-5]-=- Rg∗OX = OXcon , which since Rf∗OX = OR in turn implies that Rh∗OXcon = OR. Recall a Cartier divisor D is called g-nef if D · C ≥ 0 for all curves contracted by g, and D is called g-ample if D · C > 0...

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... is an equivalence if and only if the universal sheaf of the noncommutative deformation functor associated to the curves is a perfect complex (equivalently, pdΛΛcon <∞). This would recover Bridgeland =-=[B02]-=- and Chen [C02], since when X has only Gorenstein terminal singularities, the universal sheaf is guaranteed to be perfect [DW1, 7.1]. Whilst mutation needs R to be Gorenstein to ensure that it gives a...

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...g as a corollary of a more general d-dimensional result. Proposition 1.15 (=6.15). Suppose that R is a complete local 3-dimensional normal Gorenstein ring, and suppose that R admits an NCCR (which by =-=[VdB2]-=- implies that the minimal models of SpecR 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 w...

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...the crepant setup of 2.9, and as in the previous subsection assume further that d = 3 and X is Q-factorial. The R admitting such a setup are of course well-known to be precisely the cDV singularities =-=[Re]-=-. Definition 4.8. With R as above, (1) We define the full mutation graph of the MM generators to have as vertices the (isomorphism classes of the) basic MM generators, where each vertex N has an edge ...

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...imC Ext n R(Cd−3, k) ≤ ad−2 + ad−3 for n 0, so cxR Cd−3 ≤ 1. Inducting along the splicing of (6.B) gives cxRX ≤ 1, which implies that cxR k ≤ 1. This is well-known to imply that R is a hypersurface =-=[G]-=-. Corollary 6.15. Suppose that R is a 3-dimensional complete local normal Gorenstein algebra, and suppose that R admits an NCCR. If there are only finitely many basic MM generators up to isomorphism...

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...OMOLOGICAL MINIMAL MODEL PROGRAM MICHAEL WEMYSS Abstract. Suppose that f : X → SpecR is a minimal model of a complete local Gorenstein 3-fold, where the fibres of f are at most one dimensional, so by =-=[VdB]-=- there is a noncommutative ring Λ 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...

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... 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 =-=[A86]-=- to 3-fold compound du Val singularities. One feature is that for 3-folds, unlike for surfaces, there are two correspondences. First, there is a correspondence (1.C) between maximal modifying (=MM) R-...

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... 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 =-=[CI]-=-, originally for quotient singularities and their NCCRs. ASPECTS OF THE HOMOLOGICAL MINIMAL MODEL PROGRAM 7 Conjecture 1.7 (Craw–Ishii). Suppose that S is an arbitrary complete local normal Gorenstein...

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...nce if and only if the universal sheaf of the noncommutative deformation functor associated to the curves is a perfect complex (equivalently, pdΛΛcon <∞). This would recover Bridgeland [B02] and Chen =-=[C02]-=-, since when X has only Gorenstein terminal singularities, the universal sheaf is guaranteed to be perfect [DW1, 7.1]. Whilst mutation needs R to be Gorenstein to ensure that it gives a derived equiva...

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...mmands, and Ω∆ = [−1] on the category CM ∆, it suffices to show that [2] = Id on CM ∆. But by 2.14 ∆ is derived equivalent to Xcon and so CM ∆ ' Dsg(∆) ' Dsg(Xcon) ' ⊕ x∈SingXcon CM ÔXcon,x by Orlov =-=[O09]-=-, since all categories under consideration are idempotent complete. Since each of ÔXcon,x are hypersurfaces, [2] = Id for each of the categories on the right-hand side, so since the above are triangl...

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...ot Gorenstein. There is a version of the Auslander–McKay Correspondence in dimension two when R is not Gorenstein [Wun, Wem], obtained by replacing CM modules by Wuram’s notion of a special CM module =-=[Wun]-=-. There should be a three dimensional analogue of this. Conjecture B.5. There is a notion of ‘special MM generator’ such that the Auslander– McKay Correspondence 4.9 holds for non–Gorenstein rational ...

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... y, z]]S3 for the subgroup S3 := 〈ε3 0 00 ε23 0 0 0 1 , 0 1 01 0 0 0 0 −1 〉 ≤ SL(3,C), there is an MM generator (in fact, CT module) given by the skew group ring Λ := EndR(N), and further by =-=[BSW]-=- it can be presented as Λ ∼= N2 R N1 Aa x y b B l z aB = 0 bA = 0 Aa+ Bb = 2l2 az = xb bz = ya zA = By zB = Ax lz + zl = 0 given by the superpotential Φ := Axb + Bya − zAa − zBb + 2zl2. By 2.15 we rea...

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...roup, even up to an appropriate change of parameters, and this has implications to the braiding of flops and the Bridgeland stability manifold. It also means, for example, that any naive extension of =-=[T1]-=- or [BIKR, DH] is not possible, since root systems and Weyl groups do not necessarily appear. However, this phenomenon will come as no surprise to Pinkham [P, p366]. Second, we are able to give minima...

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...ed g-nef if D · C ≥ 0 for all curves contracted by g, and D is called g-ample if D · C > 0 for all curves contracted by g. There are many (equivalent) definitions of flops in the literature, see e.g. =-=[Ko2]-=-. We will use the following. Definition 2.6. Suppose that f : X → SpecR is a crepant projective birational morphism, where R is complete local, with at most one-dimensional fibres. Choose ⋃ i∈I Ci in ...

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...diagram, via the Auslander–McKay correspondence. This then allows us to obtain the numbers bj using Auslander–Reiten (=AR) theory, using the knitting–type constructions on the known AR quivers, as in =-=[IW1]-=-. We refer the reader to §5.4 for details, in particular the example 5.24. Once we have obtained the bj for all exchange sequences, which in particular depends only on the curves which appear in the p...

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...the particular example u2 = xyz was computed independently, using entirely different methods, by Craw and King in 2000. Indeed, [C01, 5.31, footnote 5 p117] computes the first four chambers. See also =-=[MT]-=-. 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 alr...

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...the module Mi cuts to Mi/gMi, which is precisely one of the CM modules corresponding to a vertex in an ADE Dynkin diagram, via the original Auslander–McKay correspondence. Following the notation from =-=[K]-=-, we encode X2 pictorially by simply describing which curves are blown down from the minimal resolution. The diagrams 38 MICHAEL WEMYSS represent, respectively, the minimal resolution of theD5 surface...

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...)=i head(a)6=i Mhead(a). However these morphisms are not approximations in general. In other words, the mutation defined in 2.18 above is not in general a vertex tilt in the sense of Bridgeland–Stern =-=[BS]-=-, and in full generality there is no simple combinatorial description of the decomposition of UI or VI . In the case of cDV singularities, we do give a combinatorial description later in §5.3 and §5.4...

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...ed isolated cAn singularities with smooth minimal models and observed the connection to the Weyl group Sn, [NS] which considered specific quotient singularities, again with smooth minimal models, and =-=[IW3]-=- which considered general cAn singularities. All these previous works relied heavily on direct calculation, manipulating explicit forms. Based on the above results, we offer the following conjecture. ...

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...of artn consisting of those objects that are commutative rings is denoted cartn. Given Γ ∈ artn, the morphism i produces n idempotents e1, . . . , en ∈ Γ, and we denote Γij := eiΓej . Definition 3.1. =-=[L02]-=- Fix a finite collection S := {Si | i ∈ I} of left Λ-modules. Then (1) For Γ ∈ art|I|, we say that M ∈ Mod Λ⊗C Γop (i.e. a Λ-Γ bimodule) is Γ-matricfree if M ∼= (Si ⊗C Γij) as Γop-modules, where the r...

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...the fact that the Bridgeland–Chen flop functor is only known to be an equivalence in the setting of Gorenstein terminal singularities. This paper began by trying to lift the reconstruction algebra of =-=[Wem]-=- to 3-folds, and through the analysis of many non-Gorenstein examples. There is evidence to suggest the following. Conjecture B.1. Suppose that X → Xcon is a flopping contraction of 3-folds, where X h...

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...t is known [IR, 3.10] that R is d-sCY if and only if R is Gorenstein and equi-codimensional with dimR = d. One noncommutative source of d-sCY algebras are maximal modification algebras, introduced in =-=[IW2]-=- as the notion of a noncommutative minimal model. Definition 2.2. Suppose that R is a normal d-sCY algebra. We call N ∈ ref R a modifying module if EndR(N) ∈ CMR, and we say that N ∈ ref R is a maxima...

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...for example, that any naive extension of [T1] or [BIKR, DH] is not possible, since root systems and Weyl groups do not necessarily appear. However, this phenomenon will come as no surprise to Pinkham =-=[P, 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 X2. The Homological MMP enriches the GI...

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...y this subsection restricts to minimal models and mutates at only indecomposable summands, even although the results can be stated generally at the expense of technical hypotheses. It was observed in =-=[DW1]-=- that in the case dimC Λi <∞, applying mutation Φi twice leads to a spherical-like twist around the finite dimensional self-injective algebra Λi, and that this functor is isomorphic to the inverse of ...

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.... We also remark that the above greatly generalises and simplifies [BIKR, DH], which considered isolated cAn singularities with smooth minimal models and observed the connection to the Weyl group Sn, =-=[NS]-=- which considered specific quotient singularities, again with smooth minimal models, and [IW3] which considered general cAn singularities. All these previous works relied heavily on direct calculation...

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... if and only if the minimal models of SpecR have one-dimensional fibres (equivalently, R is cDV). The direction (⇐) is true by 1.13. Although we cannot yet prove (⇒), by strengthening some results in =-=[B13]-=- to cover non-isolated singularities, we do show the following as a corollary of a more general d-dimensional result. Proposition 1.15 (=6.15). Suppose that R is a complete local 3-dimensional normal ...

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...r 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 =-=[DW3]-=-. 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 Λop as (the completion of) a quiver with relations, ...