Computation of GIT quotients of semisimple groups

TL;DR

This paper introduces three efficient algorithms for computing GIT quotients of projective varieties by semisimple groups, with applications to moduli spaces of algebraic varieties and curves.

Contribution

It presents novel algorithms for determining stability loci in GIT quotients, enhancing computational methods in algebraic geometry.

Findings

Algorithms successfully compute stability loci for simple groups

Applications to K-moduli, algebraic curves, and Mukai models demonstrated

Potential improvements and open problems outlined

Abstract

We describe three algorithms to determine the stable, semistable, and torus-polystable loci of the GIT quotient of a projective variety by a reductive group. The algorithms are efficient when the group is semisimple. By using an implementation of our algorithms for simple groups, we provide several applications to the moduli theory of algebraic varieties, including the K-moduli of algebraic varieties, the moduli of algebraic curves and the Mukai models of the moduli space of curves for low genus. We also discuss a number of potential improvements and some natural open problems arising from this work.