Three-dimensional isolated quotient singularities in odd characteristic

TL;DR

This paper extends the classification of isolated quotient singularities to three-dimensional cases over algebraically closed fields of characteristic p>2, including modular cases where p divides the group order.

Contribution

It demonstrates that in three dimensions, the classification aligns with the complex case even in modular settings, expanding understanding of quotient singularities in positive characteristic.

Findings

Classification reduces to Zassenhaus-Vincent-Wolf over complex numbers

Extends classification to three-dimensional modular cases

Provides remarks on other dimensions and characteristics

Abstract

Let a finite group G act linearly on a finite dimensional vector space V over an algebraically closed field k of characteristic p>2. Assume that the quotient V/G is an isolated singularity. In the case when p does not divide the order of G, isolated singularities V/G are completely classified and their classification reduces to Zassenhaus-Vincent-Wolf classification of isolated quotient singularities over the field of complex numbers. In the present paper we show that if dimension of V is 3, then also in the modular case (p divides the order of G) classification of isolated quotient singularities reduces to Zassenhaus-Vincent-Wolf classification. Some remarks on modular quotient singularities in other dimensions and in even characteristic are also given.