Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves

TL;DR

This paper reformulates the Hilbert-Mumford stability criterion using algebraic stacks, providing geometric conditions for moduli space existence and applying these to construct moduli spaces for torsors over curves.

Contribution

It introduces a stack-based reformulation of the Hilbert-Mumford criterion and establishes conditions for the existence of separated coarse moduli spaces, with applications to $\

Findings

Reformulation of GIT stability on algebraic stacks.

Conditions for separated coarse moduli space existence.

Construction of moduli spaces for torsors under parahoric group schemes.

Abstract

In these notes we reformulate the classical Hilbert-Mumford criterion for GIT stability in terms of algebraic stacks, this was independently done by Halpern-Leinster. We also give a geometric condition that guarantees the existence of separated coarse moduli spaces for the substack of stable objects. This is then applied to construct coarse moduli spaces for torsors under parahoric group schemes over curves.