Variation of Non-reductive Geometric Invariant Theory

TL;DR

This paper explores how non-reductive geometric invariant theory (GIT) quotients vary with linearisation and grading choices, extending known reductive GIT wall-and-chamber structures to non-reductive cases.

Contribution

It provides a survey of recent non-reductive GIT results and analyzes the dependence of non-reductive quotients on linearisation and grading parameters, revealing a similar structure to reductive GIT.

Findings

Non-reductive GIT quotients depend on linearisation and grading.

A wall-and-chamber structure analogous to reductive GIT is established.

The dependence on grading parameters is characterized.

Abstract

The wall-and-chamber structure of the dependence of the reductive GIT quotient on the choice of linearisation is well known. In this article, we first give a brief survey of recent results in non-reductive GIT, which apply when the unipotent radical is 'graded'. We then examine the dependence of these non-reductive quotients on the linearisation and an additional parameter, the choice of one-parameter subgroup grading the unipotent radical, and arrive at a picture similar to the reductive one.