The moduli stack of $G$-bundles

TL;DR

This paper provides a comprehensive geometric analysis of the moduli stack of G-bundles, establishing its algebraic structure, properties, and smoothness under certain conditions, with implications for algebraic geometry and bundle theory.

Contribution

It offers a detailed proof that the moduli stack of G-bundles is an algebraic stack with desirable properties, including smoothness over the base scheme under specific conditions.

Findings

Bun_G is an algebraic stack locally of finite presentation over S.

Bun_G has schematic, affine diagonal.

Bun_G is smooth over S when G is smooth and X→S is a relative curve.

Abstract

In this paper, we give an expository account of the geometric properties of the moduli stack of $G$-bundles. For $G$ an algebraic group over a base field and $X \to S$ a flat, finitely presented, projective morphism of schemes, we give a complete proof that the moduli stack $Bun_G$ is an algebraic stack locally of finite presentation over $S$ with schematic, affine diagonal. In the process, we prove some properties of $BG$ and Hom stacks. We then define a level structure on $Bun_G$ to provide alternative presentations of quasi-compact open substacks. Finally, we prove that $Bun_G$ is smooth over $S$ if $G$ is smooth and $X \to S$ is a relative curve.