A stacky approach to identifying the semistable locus of bundles

TL;DR

This paper characterizes the semistable locus within the moduli stack of principal bundles over a curve, highlighting its uniqueness for rank 2 bundles and the existence of other loci for higher ranks.

Contribution

It introduces a stacky approach to identify the semistable locus and distinguishes its properties across different ranks of vector bundles.

Findings

Semistable locus is the unique maximal open substack with a schematic moduli space.

For rank 2, the semistable locus is the only maximal open substack with a separated moduli space.

Higher ranks admit other open substacks with separated moduli spaces.

Abstract

We show that the semistable locus is the unique maximal open substack of the moduli stack of principal bundles over a curve that admits a schematic moduli space. For rank $2$ vector bundles it coincides with the unique maximal open substack that admits a separated moduli space, but for higher rank there exist other open substacks that admit separated moduli spaces.