Computing GIT-fans with symmetry and the Mori chamber decomposition of $\bar{M}_{0,6}$

TL;DR

This paper introduces an algorithm that computes GIT-fans for torus actions on affine varieties with symmetries, integrating algebraic, geometric, and group-theoretic methods, and applies it to the moduli space of stable genus 0 curves with 6 marked points.

Contribution

The paper presents a novel algorithm and its implementation for computing GIT-fans, enabling the explicit determination of the Mori chamber decomposition of _{0,6}.

Findings

Successfully implemented in Singular library gitfan.lib.

Computed the Mori chamber decomposition of _{0,6}.

Demonstrated the algorithm's effectiveness on a complex moduli space.

Abstract

We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry and group theory. We have implemented our algorithm in the Singular library gitfan.lib. Using our implementation, we compute the Mori chamber decomposition of the cone of movable divisors of $\bar{M}_{0,6}$.