On the geometry of the compactification of the universal Picard variety

TL;DR

This paper explores the geometric structure of the compactified universal Picard variety, demonstrating how the moduli space of spin curves embeds into it and analyzing its divisor class group.

Contribution

It shows the natural injection of the spin curves moduli space into the compactified Picard variety and extends the understanding of its divisor class group.

Findings

The moduli space of spin curves injects into the compactified Picard variety.

Generators and relations of the divisor class group are explicitly described.

Extension of previous divisor class group results by Kouvidakis.

Abstract

Here we focus on the geometry of $\pdgbar$, the compactification of the universal Picard variety constructed by L. Caporaso. In particular, we show that the moduli space of spin curves constructed by M. Cornalba naturally injects into $\pdgbar$ and we give generators and relations of the rational divisor class group of $\pdgbar$, extending previous work by A. Kouvidakis.