Quiver GIT for varieties with tilting bundles

Karmazyn, Joseph

doi:10.1007/s00229-016-0914-3

Cited by 10 publications

(17 citation statements)

References 38 publications

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“…The fact that G is injective is just 4.4, and so G is bijective. The fact that its inverse is given by F is precisely [Kar,5.2.5], with the small caveat that [Kar,5.2.5] works with the opposite algebra, but only since his conventions for composing morphisms are opposite to ours.…”

Section: Under This Correspondencementioning

confidence: 99%

“…Proof. Part (1) follows immediately from [Kar,5.2.5] applied to X. By 4.2, part (2) follows from [Kar,5.2.5] applied to X + .…”

Section: Flops and Mutationmentioning

confidence: 99%

“…There are various other outputs to the Homological MMP that for clarity have not been included in Figure 2. One such output, when the curve does flop, is obtained by combining 1.2(2) with [Kar,5.2.5]. This shows that it is possible to output the flop as a fixed, specified, GIT moduli space of the mutated algebra.…”

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confidence: 99%

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Flops and clusters in the homological minimal model programme

Wemyss

2017

Invent. math.

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Suppose that f : X → Spec R is a minimal model of a complete local Gorenstein 3-fold, 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 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 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 is finite dimensional. These results then allow us to jump between all the minimal models of 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 R-module 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.

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Section: Under This Correspondencementioning

confidence: 99%

“…Proof. Part (1) follows immediately from [Kar,5.2.5] applied to X. By 4.2, part (2) follows from [Kar,5.2.5] applied to X + .…”

Section: Flops and Mutationmentioning

confidence: 99%

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Flops and clusters in the homological minimal model programme

Wemyss

2017

Invent. math.

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“…The statement of part (i) is due originally to Karmazyn [23,Corollary 5.4.5], while an analogue of part (ii) in the complete local setting can be deduced by combining Iyama-Kalck-Wemyss-Yang [26,Theorem 4.6] with [23,Corollary 5.2.5]. Note however that our approach is completely different in each case, and is closer in spirit to the geometric construction of the Special McKay correspondence for cyclic subgroups of GL(2, k) given by Craw [16].…”

Section: Theorem 12 Let G ⊂ Gl(2 K) Be a Finite Subgroup Without Psmentioning

confidence: 99%

Multigraded linear series and recollement

2017

Self Cite

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Given a scheme Y equipped with a collection of globally generated vector bundles E 1 , . . . , E n , we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E :This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G ⊂ GL(2, k), every sub-minimal partial resolution of A 2 k /G is isomorphic to a fine moduli space M(E C ) where E C is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.B Alastair Craw a.craw@bath.ac.uk

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“…Remark 7.4.3. When M ∼ = Y and E = O ∆ , equation (7.9) gives T = E ∨ ; see Karmazyn [Kar14] and references therein for many examples where this is known to hold.…”

Section: On Schemes Admitting a Tilting Bundlementioning

confidence: 99%

Nef divisors for moduli spaces of complexes with compact support

Bayer

Craw

Zhang

2016

Sel. Math. New Ser.

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In Bayer and Macrì (J Am Math Soc 27(3): 2014), the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgelandstable objects on a smooth projective variety X . In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a θ -stability condition for the endomorphism algebra.

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Quiver GIT for varieties with tilting bundles

Cited by 10 publications

References 38 publications

Flops and clusters in the homological minimal model programme

Flops and clusters in the homological minimal model programme

Multigraded linear series and recollement

Nef divisors for moduli spaces of complexes with compact support

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