On the birational geometry of the universal Picard variety

TL;DR

This paper determines the Kodaira dimension of the universal Picard variety for certain degrees and explores its birational properties across different degrees, advancing understanding of its geometric structure.

Contribution

It computes the Kodaira dimension of the universal Picard variety for specific degrees and provides partial results for arbitrary degrees, analyzing its birational nature.

Findings

Kodaira dimension computed for degrees with (d-g+1,2g-2)=1

Partial results on Kodaira dimension for other degrees

Investigation into birationality of universal Picard varieties

Abstract

We compute the Kodaira dimension of the universal Picard variety P_{d,g} parameterizing line bundles of degree d on curves of genus g under the assumption that (d-g+1,2g-2)=1. We also give partial results for arbitrary degrees d and we investigate for which degrees the universal Picard varieties are birational.