The Picard group of the universal moduli stack of principal bundles on pointed smooth curves II

TL;DR

This paper studies the Picard group of the universal moduli stack of principal G-bundles over pointed smooth curves, providing new presentations, restrictions, and computations of divisor class groups, extending previous work.

Contribution

It introduces new functorial descriptions and computes the Picard group and divisor class group for various moduli stacks of G-bundles, advancing understanding of their geometric structure.

Findings

New functorial presentations of Picard groups

Determination of Picard group of the rigidified stack

Computation of divisor class group of semistable G-bundles

Abstract

In this paper, which is a sequel of arXiv:2002.07494, we investigate, for any reductive group $G$ over an algebraically closed field $k$, the Picard group of the universal moduli stack $\mathrm{Bun}_{G,g,n}$ of $G$-bundles over $n$-pointed smooth projective curves of genus $g$. In particular: we give new functorial presentations of the Picard group of $\mathrm{Bun}_{G,g,n}$; we study the restriction homomorphism onto the Picard group of the moduli stack of principal $G$-bundles over a fixed smooth curve; we determine the Picard group of the rigidification of $\mathrm{Bun}_{G,g,n}$ by the center of $G$ as well as the image of the obstruction homomorphism of the associated gerbe. As a consequence, we compute the divisor class group of the moduli space of semistable $G$-bundles over $n$-pointed smooth projective curves of genus $g$.