Torsors over Luna strata

TL;DR

This paper investigates the structure of Luna strata in quotient spaces of rational G-modules, focusing on torsors, divisor class groups, and Cox rings, revealing reasons for their singularities.

Contribution

It introduces methods to compute divisor class groups and Cox rings of Luna strata using torsors, explaining their boundary singularities.

Findings

Computed divisor class groups of Luna strata

Determined Cox rings under mild assumptions

Identified causes of boundary singularities

Abstract

Let G be a reductive group and X be a Luna stratum on the quotient space V//G of a rational G-module V. We consider torsors over X with both non-commutative and commutative structure groups. It allows us to compute the divisor class group and the Cox ring of a Luna stratum under mild assumptions. This techniques gives a simple cause why many Luna strata are singular along their boundary.