Cox rings of some symplectic resolutions of quotient singularities

Donten-Bury, Maria; Grab, Maksymilian

doi:10.48550/arxiv.1504.07463

Cited by 5 publications

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“…We provide the Cox rings of X 0 = C 4 /G for chosen representations of G = S 3 and G = D 8 and compute the Cox rings of modifications X → X 0 . We are able to retrieve crepant resolutions that were also found in [10], see Proposition 5.3 and 5. 4.…”

Section: Introductionmentioning

confidence: 79%

“…action (where ε is the 3rd root of unity). For the resolution algorithm to work (compare Example 4.6), we want the generating set to satisfy [10,Condition 3.6], developed in the more general context including the present case in [28]. Roughly speaking, it determines when a generating set of R(X 0 ) is suitable for being extended to the Cox ring of its resolution.…”

Section: Methodsmentioning

confidence: 99%

“…In this article we generalize the methods used in [15]. While finishing a draft of this paper, we also found out about Yamagishi's work [28], which extends [12,10]. However, our approach is different from [28] and our methods are not restricted to the class of crepant resolutions.…”

Section: Introductionmentioning

confidence: 93%

“…Γ (X 0 , O(D)) , see [1] for details on the construction. Cox rings have already been successfully used by various authors to study resolutions of quotient singularities, in particular for symplectic quotients in [12,10], described by a generating set in a simpler ring, and via an algorithmic approach based on toric ambient modifications in [15]. In this article we generalize the methods used in [15].…”

Section: Introductionmentioning

confidence: 99%

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Computing resolutions of quotient singularities

Donten-Bury

Keicher

2017

Journal of Algebra

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Abstract. Let G ⊆ GL(n) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X → C n /G, which is based just on the geometry of the singularity C n /G, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions X → C 3 /G for all G ⊆ GL(3) with the aforementioned property and of order |G| ≤ 12. We also provide examples in dimension 4.

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Section: Introductionmentioning

confidence: 79%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Computing resolutions of quotient singularities

Donten-Bury

Keicher

2017

Journal of Algebra

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“…It is perhaps worth making a philosophical remark about the final point. Any (relative) Mori Dream Space has a finitely generated Cox ring, and in our situation this ring was described by generators and relations by Donten-Bury-Grab [22,Section 5] when n = 2 and Φ is of type A 1 . While we do not make use of the Cox ring in this paper, the fact that our Corollary 1.3 reconstructs all small birational models of X by GIT as quiver varieties for the affine Dynkin graph of Γ suggests that the preprojective algebra Π should be thought of as a kind of 'noncommutative Cox ring' for each of the varieties Hilb [n] (S) with n ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

Birational geometry of symplectic quotient singularities

Bellamy¹,

Craw²

2018

Preprint

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For a finite subgroup Γ ⊂ SL(2, C) and for 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 C 2 /Γ. It is well known that X := Hilb [n] (S) is a projective, crepant resolution of the symplectic singularity C 2n /Γn, where Γn = Γ ≀ Sn is the wreath product. We prove that every projective, crepant resolution of C 2n /Γn can be realised as the fine moduli space of θ-stable Π-modules for a fixed dimension vector, where Π is the framed preprojective algebra of Γ and θ is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of θ-stability conditions to birational transformations of X over C 2n /Γn. As a corollary, we describe completely the ample and movable cones of X over C 2n /Γn, 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 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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ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SL_n(ℂ)

Yamagishi¹

2018

Glasgow Math. J.

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We prove that a quotient singularity C n /G by a finite subgroup G ⊂ SL n (C) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also give a procedure to compute the Cox ring of a minimal model of a given C n /G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities which admit projective symplectic resolutions.

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Cox rings of some symplectic resolutions of quotient singularities

Cited by 5 publications

References 0 publications

Computing resolutions of quotient singularities

Computing resolutions of quotient singularities

Birational geometry of symplectic quotient singularities

ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SL_n(ℂ)

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Cox rings of some symplectic resolutions of quotient singularities

Cited by 5 publications

References 0 publications

Computing resolutions of quotient singularities

Computing resolutions of quotient singularities

Birational geometry of symplectic quotient singularities

ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SLn(ℂ)

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ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SL_n(ℂ)