Nodal Uniformization of G-bundles

TL;DR

This paper surveys uniformization results for principal G-bundles on curves, proves uniformization for nodal curves, and constructs compactifications of their moduli spaces, extending known results to singular curves.

Contribution

It provides a proof of uniformization for nodal curves and introduces methods to compactify moduli spaces of G-bundles on such curves.

Findings

Uniformization holds for principal bundles on nodal curves.

A new compactification of the moduli of G-bundles on nodal curves is constructed.

Uses equivariant compactifications to extend moduli stack compactification methods.

Abstract

We give a survey of uniformization results for principal bundles on curves. We provide a proof of uniformization for nodal curves; this result is a special case of work of Belkale and Fakhruddin for uniformization on singular curves. We use the uniformization result to give a construction of a compactification of the moduli of G bundles on a family of nodal curves. Along the way we also explain how to use equivariant compactifactions of a group to compactify the moduli stack of bundles over a fixed nodal curve.