The CompGIT package: a computational tool for Geometric Invariant Theory quotients

TL;DR

CompGIT is a SageMath package that automates the computation of Geometric Invariant Theory quotients for projective varieties under simple groups, aiding algebraic geometry research.

Contribution

It introduces a practical implementation of algorithms for GIT quotients, enabling the description of stable, semi-stable, and unstable loci in a computational setting.

Findings

Successfully describes GIT quotients for simple groups

Identifies stable, semi-stable, and unstable loci computationally

Facilitates applications in algebraic geometry

Abstract

We describe CompGIT, a SageMath package to describe Geometric Invariant Theory (GIT) quotients of projective space by simple groups. The implementation is based on algorithms described by Gallardo--Martinez-Garcia--Moon--Swinarski. In principle the package is sufficient to describe any GIT quotient of a projective variety by a simple group -- in practice it requires that the user can construct an equivariant embedding of the polarised variety into projective space. The package describes the non-stable and unstable loci up to conjugation by the group, as well as describing the strictly polystable loci. We discuss potential applications of the outputs of CompGIT to algebraic geometry problems, a well as suggesting directions for future developments.