Stratifying quotient stacks and moduli stacks

TL;DR

This paper develops a stratification method for quotient and moduli stacks using recent GIT results for non-reductive group actions, enabling each stratum to have a geometric or coarse moduli space.

Contribution

It introduces a new stratification approach for quotient stacks and moduli stacks based on non-reductive GIT, providing geometric quotients for each stratum.

Findings

Stratification of quotient stacks [X/H] with geometric quotients.

Application to moduli stacks of sheaves with coarse moduli spaces.

Extension of GIT techniques to non-reductive group actions.

Abstract

Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.