The Chevalley–Shephard–Todd theorem for finite linearly reductive group schemes

Satriano, Matthew

doi:10.2140/ant.2012.6.1

Cited by 11 publications

(13 citation statements)

References 12 publications

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“…It is easy to see that pseudo-reflections generate a normal subgroup scheme N G. By the theorem of Chevalley-Satriano-Shephard-Todd [Sa09], the quotient A d /G is smooth if and only if G is generated by pseudo-reflections, that is, if and only if N = G. In particular, after replacing G by G/N and A d by A d /N ∼ = A d , we may assume that G acts without pseudo-reflections and that the G-action is free outside a closed subset of codimension ≥ 2.…”

Section: Properties Of Lrq Singularitiesmentioning

confidence: 99%

“…Then, the toric variety Spec k[M ] has no torus factors, so the Cox construction (see, for example, [GS15, Section 3.1]) realizes Spec k[M ] as a quotient of A d by a diagonal action of the abelian group scheme G = Cl(k[M ]) D = Cl(X) D (see [GS15, Corollary 5.9] for the latter equality) with fixed locus of codimension at least 2. In fact, by the Chevalley-Shephard-Todd Theorem for linearly reductive group schemes [Sa09], the G-action on A d is very small, since (X, x) is an isolated singularity.…”

Section: Properties Of Lrq Singularitiesmentioning

confidence: 99%

“…image) of the induced homomorphism G → Aut E . By the Chevalley-Shephard-Todd theorem [Sa09], Y ′ := Y /K is smooth and the image E ′ of E in Y ′ is still isomorphic to P 1 . The action of H on Y ′ preserves E ′ and acts faithfully on it.…”

Section: F-regular Surfaces Are Lrq-surfacesmentioning

confidence: 99%

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Linearly Reductive Quotient Singularities

Liedtke,

Martin,

Matsumoto

2021

Preprint

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We study isolated quotient singularities by finite and linearly reductive group schemes (lrq singularities for short) and show that they satisfy many, but not all, of the known properties of finite quotient singularities in characteristic zero:(1) From the lrq singularity we can recover the group scheme and the quotient presentation.(2) We establish canonical lifts to characteristic zero, which leads to a bijection between lrq singularities and certain characteristic zero counterparts.(3) We classify subgroup schemes of GL d and SL d that correspond to lrq singularities. For d = 2, this generalises results of Klein, Brieskorn, and Hashimoto [Kl84,Br67,Ha15]. Also, our classification is closely related to the spherical space form problem. (4) F-regular (resp. F-regular and Gorenstein) surface singularities are precisely the lrq singularities by finite and linearly reductive subgroup schemes of GL2 (resp. SL2). This generalises results of Klein and Du Val [Kl84, DV34]. (5) Lrq singularities in dimension ≥ 4 are infinitesimally rigid. We classify lrq singularities in dimension 3 that are not infinitesimally rigid and compute their deformation spaces. This generalises Schlessinger's rigidity theorem [Sc71] to positive and mixed characteristic. Finally, we study Riemenschneider's conjecture [Ri74] in this context, that is, whether lrq singularities deform to lrq singularities.

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Section: Properties Of Lrq Singularitiesmentioning

confidence: 99%

Section: Properties Of Lrq Singularitiesmentioning

confidence: 99%

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Linearly Reductive Quotient Singularities

Liedtke,

Martin,

Matsumoto

2021

Preprint

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“…By this we mean that each point has an étale neighbourhood with for some smooth variety and finite diagonalisable group . In this situation, there exists a canonical stack which is smooth and has as coarse space [Vis89, Sat12]. By applying the functorial destackification algorithm on , we obtain a functorial desingularisation algorithm.…”

Section: Introduction and Main Theoremsmentioning

confidence: 99%

Functorial destackification of tame stacks with abelian stabilisers

Bergh¹

2017

Compositio Math.

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We give an algorithm for removing stackiness from smooth, tame Artin stacks with abelian stabilisers by repeatedly applying stacky blow-ups. The construction works over a general base and is functorial with respect to base change and compositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how the result can be used for destackifying general Deligne-Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over an arbitrary field, the method can be used to obtain a functorial algorithm for desingularising varieties with simplicial toric quotient singularities, without assuming the presence of a toroidal structure

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“…In addition to having a moduli interpretation, the stacky resolution f has the property that it is an isomorphism away from codimension 2 on the source. This property was particularly useful, for example, in [Sa2]: in [Sa1,Thm 1.10], we constructed analogous stacky resolutions for schemes with linearly reductive singularities ([Sa1,Def 5.1]), and then used such resolutions in [Sa2,Thm 4.8] to show the degeneracy of a variant of the Hodge-de Rham spectral sequence in characteristic p.…”

Section: Introductionmentioning

confidence: 99%

Canonical Artin stacks over log smooth schemes

Satriano

2012

Math. Z.

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Abstract. We develop a theory of toric Artin stacks extending the theories of toric Deligne-Mumford stacks developed by Borisov-Chen-Smith, Fantechi-Mann-Nironi, and Iwanari. We also generalize the Chevalley-Shephard-Todd theorem to the case of diagonalizable group schemes. These are both applications of our main theorem which shows that a toroidal embedding X is canonically the good moduli space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky structure is supported on the singular locus of X.

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The Chevalley–Shephard–Todd theorem for finite linearly reductive group schemes

Cited by 11 publications

References 12 publications

Linearly Reductive Quotient Singularities

Linearly Reductive Quotient Singularities

Functorial destackification of tame stacks with abelian stabilisers

Canonical Artin stacks over log smooth schemes

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