Finite groups, smooth invariants, and isolated quotient singularities

TL;DR

This paper characterizes when invariant rings under finite group actions are polynomial and classifies isolated quotient singularities over algebraically closed fields, linking positive characteristic cases to complex singularities.

Contribution

It provides a classification of invariant rings as polynomial and identifies isolated singularities, connecting positive characteristic cases to known complex singularities.

Findings

Invariant rings are polynomial under specific conditions.

Classification of isolated quotient singularities over algebraically closed fields.

Completion of invariant rings relates to complex singularities via reduction mod p.

Abstract

Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is algebraically closed, all S(V)^G which are isolated singularities. We show that the completion of S(V)^G, at its unique graded maximal ideal, is isomorphic to the completion of S(W)^H, where (H,W) is a reduction mod p of a member of the Zassenhaus-Vincent-Wolf list of complex isolated quotient singularities.