Nef divisors for moduli spaces of complexes with compact support

Bayer, Arend; Craw, Alastair; Zhang, Ziyu

doi:10.1007/s00029-016-0298-y

Cited by 9 publications

(11 citation statements)

References 46 publications

(48 reference statements)

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“…Following [2], the numerical Grothendieck group for compact support, denoted K num c (X), is defined to be the quotient of K c (X) by the radical of the form χ. [2,Lemma 5.1.1] shows that K num c (X) has finite rank when X is normal and quasi-projective.…”

Section: 1mentioning

confidence: 99%

Gale duality and the linearisation map for noncommutative crepant resolutions

Craw¹

2021

Preprint

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We describe in geometric terms the map that is Gale dual to the linearisation map for quiver moduli spaces associated to noncommutative crepant resolutions in dimension three. This allows us to formulate Reid's recipe in this context in terms of a pair of integer-valued matrices, one of which appears to satisfy an attractive sign-coherence property. We provide some new evidence for a conjecture, known to hold in the toric case, which implies that a minimal generating set of relations between the determinants of the tautological bundles encodes the supports of the images of the vertex simples under the derived equivalence, and vice-versa.C containing the generic stability parameter θ, so we write it aswhere Θ is a corank-one lattice in K(A) and NS(X) is the Néron-Severi group of X. Note that L C (θ) is the polarising ample line bundle on M θ (A, v) given by the GIT quotient construction.On the other hand, the vertex simple modules S i for i ∈ Q 0 determine objects Ψ(S i ) in the bounded derived category of coherent sheaves with compact support on X; here, Ψ is the tilting equivalence determined by the (dual to the) tautological bundle on M θ (A, v). The numerical Grothendieck group K num c (X) for objects with compact support on X admits a dimension filtration, and we demonstrate that the map Gale dual (see Section 4.1) to the linearisation map L C is a map of the formthat is defined in terms of the support (with multiplicities) of the objects Ψ(S i ) for i ∈ Q 0 .This paper describes the relationship between the line bundles det(T i ) and the objects Ψ(S i ) for i ∈ Q 0 via Gale duality between L C and G C . For a specific GIT chamber C + , these two collections of geometric objects have been encoded before, at least in the toric case, via rival interpretations of the combinatorial cookery known as Reid's recipe [31,13]. Here we reconcile these rival interpretations by demonstrating they are equivalent. Our approach via L C + or G C + allows us to formulate Reid's recipe in the non-toric case, and it suggests that a matrix representing G C + is 'sign-coherent'; this surprising phenomenon would follow from a proof of the Trichotomy Conjecture 1.6. 1.2. The main result and an illuminating example. We now present the results in more detail. We assume for now that A is an NCCR 1 satisfying a technical condition (see Assumption 3.2) which implies in particular that the vertex simple modules exist. In addition, assume that the dimension vector v has a component v 0 = 1, and write 0 ∈ Q 0 for the corresponding vertex.

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Section: 1mentioning

confidence: 99%

Gale duality and the linearisation map for noncommutative crepant resolutions

Craw¹

2021

Preprint

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“…n . It would then follow from Bayer-Craw-Zhang [4,Section 7], together with the derived equivalence of [9], that every such X can be realised as a moduli space of Bridgeland-stable objects in the derived category of coherent sheaves on X .…”

Section: Corollary 18mentioning

confidence: 99%

“…Put another way, the framed McKay quiver has too few vertices to be the quiver encoding a tilting bundle on X when n > 1. In particular, the quiver varieties M θ (v, w) that we study here cannot be realised directly as moduli spaces of Bridgelandstable objects in the derived category of coherent sheaves on X in the manner described in [4,Section 7].…”

Section: Relation To Other Workmentioning

confidence: 99%

Birational geometry of symplectic quotient singularities

Bellamy

Craw

2020

Invent. math.

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For a finite subgroup $$\Gamma \subset \mathrm {SL}(2,\mathbb {C})$$ Γ ⊂ SL ( 2 , C ) and for $$n\ge 1$$ n ≥ 1 , we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity $$\mathbb {C}^2/\Gamma $$ C 2 / Γ . It is well known that $$X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S)$$ X : = Hilb [ n ] ( S ) is a projective, crepant resolution of the symplectic singularity $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n , where $$\Gamma _n=\Gamma \wr \mathfrak {S}_n$$ Γ n = Γ ≀ S n is the wreath product. We prove that every projective, crepant resolution of $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n can be realised as the fine moduli space of $$\theta $$ θ -stable $$\Pi $$ Π -modules for a fixed dimension vector, where $$\Pi $$ Π is the framed preprojective algebra of $$\Gamma $$ Γ and $$\theta $$ θ is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of $$\theta $$ θ -stability conditions to birational transformations of X over $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n . As a corollary, we describe completely the ample and movable cones of X over $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n , and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to $$\Gamma $$ Γ by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.

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“…Moduli spaces of sheaves (on a fixed surface) are one kind of moduli space that has been extensively studied (e.g., [BCZ], [BHL+], [CC], [CH3], [DN], [Fog], [Gie], [LQ1], [Mar2], [Mar1], [MO1], [MO2], [MW], [Tak], [Yos3]). In this setting, the geometry of the underlying variety can be used to study the moduli space.…”

Section: Introductionmentioning

confidence: 99%

The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

Ryan¹

2017

Nagoya Math. J.

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Abstract. Let ξ be a stable Chern character on P 1 × P 1 , and let M (ξ) be the moduli space of Gieseker semistable sheaves on P 1 × P 1 with Chern character ξ. In this paper, we provide an approach to computing the effective cone of M (ξ). We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of M (ξ) which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on P 1 × P 1 .

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Nef divisors for moduli spaces of complexes with compact support

Cited by 9 publications

References 46 publications

Gale duality and the linearisation map for noncommutative crepant resolutions

Gale duality and the linearisation map for noncommutative crepant resolutions

Birational geometry of symplectic quotient singularities

The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

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