Gorensteinness and iteration of Cox rings for Fano type varieties

Braun, Lukas

doi:10.1007/s00209-021-02946-w

Cited by 4 publications

(15 citation statements)

References 35 publications

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“…Moreover, they induce an action of a solvable reductive group. This generalizes observations made in [ABHW18,Bra19a]. We start with the following lemma slightly generalizing [AG10, Theorem 5.1].…”

Section: Boundedness Of Iteration Of Cox Ringsmentioning

confidence: 56%

“…Iteration of Cox rings for relative Mori dream spaces. In this subsection, we define the iteration of Cox rings and generalize some results from [Bra19a] to the case of relative Mori dream spaces. The setting is the following.…”

Section: Boundedness Of Iteration Of Cox Ringsmentioning

confidence: 99%

“…Thus, it is natural to iterate the Cox construction for Fano type varieties, or more generally, for klt singularities. The first author made this observation in [Bra19a], where he proves the existence and termination of the iteration of Cox rings for Fano type varieties and klt quasi-cones.…”

Section: Introductionmentioning

confidence: 96%

“…It is known that Fano type varieties are a special class of Mori dream spaces [BCHM10]. Furthermore, the Cox ring of a Fano type variety is an affine Gorenstein canonical quasi-cone [GOST15,Bro13,Bra19a]. In particular, it is an affine model of a klt singularity.…”

Section: Introductionmentioning

confidence: 99%

“…In this way, we obtain a sequence of finite solvable Galois covers of the starting singularity (or Fano variety). This method was initiated in [Bra19a]. Using the Jordan property for the regional fundamental group of klt singularities [BFMS20], we prove that the number of iterations is bounded from above by a constant which only depends on the dimension.…”

Section: Introductionmentioning

confidence: 99%

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Iteration of Cox rings of klt singularities

Braun,

Moraga

2021

Preprint

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Given a klt singularity (X, ∆; x), we define the iteration of Cox rings of (X, ∆; x). The first result of this article is that the iteration of Cox rings Cox (k) (X, ∆; x) of a klt singularity stabilizes for k large enough. The second result is a boundedness one, we prove that for a n-dimensional klt singularity (X, ∆; x) the iteration of Cox rings stabilizes for k ≥ c(n), where c(n) only depends on n. Then, we use Cox rings to establish the existence of a simply connected factorial canonical cover (or scfc cover) of a klt singularity. We prove that the scfc cover dominates any sequence of quasi-étale finite covers and reductive abelian quasi-torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano type morphisms.

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Section: Boundedness Of Iteration Of Cox Ringsmentioning

confidence: 56%

Section: Boundedness Of Iteration Of Cox Ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Iteration of Cox rings of klt singularities

Braun,

Moraga

2021

Preprint

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Reductive quotients of klt singularities

Braun,

Greb,

Langlois

et al. 2024

Invent. math.

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We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety $X$ X endowed with the action of a reductive group $G$ G and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$ X → X / / G , we can find a boundary $B$ B on $X/\!/G$ X / / G so that the pair $(X/\!/G,B)$ ( X / / G , B ) is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$ G -manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$ n -dimensional K-polystable smooth Fano varieties of volume $v$ v has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

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Orbifold Kähler–Einstein metrics on projective toric varieties

Braun

2023

Bulletin of London Math Soc

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Gorensteinness and iteration of Cox rings for Fano type varieties

Cited by 4 publications

References 35 publications

Iteration of Cox rings of klt singularities

Iteration of Cox rings of klt singularities

Reductive quotients of klt singularities

Orbifold Kähler–Einstein metrics on projective toric varieties

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