A note on multisecants of the Kummer variety of a Jacobian

TL;DR

This paper explores the relationship between multisecants of the Kummer variety of a Jacobian and certain linear systems on algebraic curves, generalizing previous results in algebraic geometry.

Contribution

It introduces a new rational map from symmetric products of curves to linear systems, linking it to Gunning multisecants of the Kummer variety, extending prior work.

Findings

Established a connection between fibers of the rational map and Gunning multisecants.

Generalized previous results on multisecants of Kummer varieties.

Provided a new perspective on the geometry of Jacobians and their Kummer varieties.

Abstract

We show that if $C$ is a smooth projective curve and $\mathfrak{d}$ is a $\mathfrak{g}^{n}_{2n}$ on $C$, then we obtain a rational map $\mathrm{Sym}^{n}(C)\dashrightarrow\mathfrak{d}$ whose fibers can be related in an interesting way to Gunning multisecants of the Kummer variety of $JC$. This generalizes previous work done by the first author with Codogni and Salvati Manni.