Sections of Jacobian fibrations over lines

TL;DR

This paper investigates the cohomology classes of sections of the universal Jacobian over lines in a linear system on smooth surfaces, with a focus on K3 surfaces where the universal compactified Jacobian forms a hyperk"ahler manifold.

Contribution

It provides new insights into the cohomology classes of sections of the universal Jacobian over lines, especially in the context of K3 surfaces and hyperk"ahler geometry.

Findings

Cohomology classes of sections are characterized in specific geometric contexts.

The universal compactified Jacobian on K3 surfaces is a hyperk"ahler manifold.

Identifies discrepancies with previous results in [BKV25].

Abstract

Let $|H|$ be a linear system on a smooth surface $S$. We study the cohomology classes of sections of the universal Jacobian over lines in $|H|$. When $S$ is a K3 surface, the universal compactified Jacobian is a hyperk\"ahler manifold, and we do not see how to reconcile our results with some of the steps in [BKV25]. v.2: Confusing typo fixed: H^g should be H^{2g}.