Triviality properties of principal bundles on singular curves

TL;DR

This paper proves that principal bundles for semisimple groups on certain singular affine curves are trivial under specific conditions, with implications for conformal blocks bundles on moduli stacks.

Contribution

It establishes triviality results for principal bundles on singular affine curves and connects these to conformal blocks bundles on moduli stacks.

Findings

Principal bundles are trivial on affine curves with semi-normal singularities under certain conditions.

The results extend to cases where the fundamental group order is invertible in the field.

Applications include realizing conformal blocks bundles as pushforwards of line bundles.

Abstract

We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $\pi_1$ of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve.