A note on multisecants of the Kummer variety of a Jacobian

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.