Principal bundles under reductive groups

TL;DR

This paper studies principal bundles with reductive groups over schemes, establishing the existence of universal bundles over complete connected schemes and providing explicit descriptions over genus 0 or 1 curves.

Contribution

It proves the existence of universal principal bundles with reductive groups over complete connected schemes and offers explicit classifications over low-genus curves.

Findings

Universal principal bundles exist over complete connected schemes.

Explicit descriptions of bundles over genus 0 and 1 curves are provided.

Results apply to reductive groups not necessarily of finite type.

Abstract

Let $k$ be a field of characteristic $0$. We consider principal bundles over a $k$-scheme with reductive structure group (not necessarily of finite type). It is showm in particular that for $k$ algebraically closed there exists on any complete connected $k$-scheme a universal such bundle. As a consequence, an explicit description of principal bundles with reductive structure group over curves of genus $0$ or $1$ is obtained.