Sections of Jacobian fibrations over lines

Let $\mathrm{\mid }H\mathrm{\mid }$ be a linear system on a smooth surface $S$. We study the cohomology classes of sections of the universal Jacobian over lines in $\mathrm{\mid }H\mathrm{\mid }$. When $S$ is a K3 surface, the universal compactified Jacobian is a hyperkähler 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}.