Linearly Reductive Quotient Singularities

TL;DR

This paper investigates quotient singularities arising from linearly reductive group schemes, establishing their properties, classifications, and deformations, and connecting them to classical results and conjectures in algebraic geometry.

Contribution

It introduces the concept of lrq singularities, classifies subgroup schemes related to them, and extends classical results to positive and mixed characteristic settings.

Findings

Lrq singularities can be recovered from their group scheme and quotient presentation.

Canonical lifts to characteristic zero establish a bijection with characteristic zero counterparts.

Lrq singularities in dimension ≥4 are infinitesimally rigid.

Abstract

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 ${\mathbf{GL}}_d$ and ${\mathbf{SL}}_d$ that correspond to lrq singularities. For $d=2$, this generalises results of Klein, Brieskorn, and Hashimoto. 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 ${\mathbf{GL}}_2$ (resp. ${\mathbf{SL}}_2$). This generalises results of Klein and Du Val. (5) Lrq singularities in dimension $\geq 4$ 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 to positive and mixed characteristic. Finally, we study Riemenschneider's conjecture in this context, that is, whether lrq singularities deform to lrq singularities.