Generic stabilisers for actions of reductive groups

TL;DR

This paper investigates the properties of stabilisers in reductive group actions on varieties, showing openness of minimal reductive stabilisers and exploring conditions for principal stabilisers.

Contribution

It establishes the openness of points with minimal reductive stabilisers and provides new results on the existence of principal stabilisers in reductive group actions.

Findings

Openness of the set of points with minimal reductive stabilisers

Existence results for principal stabilisers in reductive group actions

Characterization of stabiliser dimensions in algebraic group actions

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $v\in V$ such that ${\rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some results on the existence of principal stabilisers in an appropriate sense.