Triviality properties of principal bundles on singular curves-II

TL;DR

This paper investigates the conditions under which principal G-bundles on singular curves are trivial outside a divisor, revealing obstructions related to the group's connectivity and providing criteria for local triviality.

Contribution

It introduces a natural obstruction for triviality of principal bundles on singular curves and demonstrates how to vanish it via base change, linking it to local triviality conditions.

Findings

Obstruction is nontrivial for non-simply connected G

Vanishing of obstruction implies etale local triviality

Explicit examples illustrating the obstruction's behavior

Abstract

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.