Finite separating sets and quasi-affine quotients

TL;DR

This paper explores the structure of invariant rings in algebraic geometry, introducing new methods to identify quasi-affine varieties with specific invariant rings and criteria for separating algebras.

Contribution

It presents a novel approach for constructing quasi-affine varieties corresponding to invariant rings and provides a new criterion for recognizing separating algebras.

Findings

New method for constructing quasi-affine varieties with given invariant rings

Criterion for recognizing separating algebras

Application to known examples and new constructions

Abstract

Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-affine, and there always exist finite separating sets. In this paper, we give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and we give a criterion to recognize separating algebras. The method and criterion are used on some known examples and in a new construction.