The Picard group of the compactified universal Jacobian

TL;DR

This paper explicitly computes the Picard groups of the universal Jacobian stack and its compactification, exploring gerbe structures, Poincaré line bundles, and comparing with existing divisor class group results.

Contribution

It provides the first explicit determination of Picard groups for these stacks and relates gerbe structures to line bundle existence, advancing understanding of universal Jacobians.

Findings

Explicit Picard group descriptions for universal Jacobian stacks

Analysis of gerbe structures and Poincaré line bundles

Comparison with Kouvidakis-Fontanari divisor class computations

Abstract

We determine explicitly the Picard groups of the universal Jacobian stack and of its compactification over the stack of stable curves. Along the way, we prove some results concerning the gerbe structure of the universal Jacobian stack over its rigidification by the natural action of the multiplicative group and relate this with the existence of generalized Poincar\'e line bundles. We also compare our results with Kouvidakis-Fontanari computations of the divisor class group of the universal (compactified) Jacobian scheme.