The Picard group of the universal moduli space of vector bundles on stable curves

TL;DR

This paper explicitly constructs and analyzes the Picard group of the universal moduli space of vector bundles on stable curves, providing new insights into its structure and related gerbe properties.

Contribution

It explicitly determines the Picard group of the moduli stack of properly balanced vector bundles on semistable curves and describes its implications for universal moduli stacks.

Findings

Explicit description of the Picard groups of universal moduli stacks

Conditions for the existence of Poincaré bundles over the universal curve

Analysis of the gerbe structure over the rigidification

Abstract

We construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of Poincar\'e bundles over the universal curve of an open substack of the rigidification, generalizing a result of Mestrano-Ramanan.