Separating Invariants of Finite Groups

This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac–Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the invariant ring or a separating subalgebra to properties of the group action. All these results are limited to the case of linear actions on vector spaces. The goal of this paper is to lift this restriction by extending these results to the case of (possibly) non-linear actions on affine varieties.