Graded linearisations

TL;DR

This paper extends Mumford's geometric invariant theory to actions of non-reductive linear algebraic groups using graded linearisations, enabling the construction of moduli spaces for unstable objects.

Contribution

It introduces a method to apply GIT to non-reductive groups via graded linearisations, broadening the scope of moduli space constructions beyond stable objects.

Findings

Allows construction of moduli spaces of unstable objects

Generalizes GIT to non-reductive group actions

Enables analysis of objects with singularities or instability

Abstract

When the action of a reductive group on a projective variety has a suitable linearisation, Mumford's geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford's GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford's GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder--Narasimhan type).