A note on multisecants of the Kummer variety of a Jacobian
Robert Auffarth, Sebastián Rahausen
TL;DR
The paper connects multisecants of the Kummer variety $K(JC)$ to the geometry of Brill–Noether loci on a curve by generalizing previous work on the Gauss map. It replaces the canonical map with a more flexible map $\xi_\mathfrak{d}$ arising from a $\mathfrak{g}^n_{2n}$ on $C$, and studies its fibers on the symmetric product $C^{(n)}$ to relate multisecants to stratifications $\mathfrak{d}_t$ of linear systems. The authors prove dominance and degree formulas for $\xi_\mathfrak{d}$, establish conditions under which Gunning multisecants lying on a fiber impose the restriction that their image lies in $\mathfrak{d}_{n-\ell+1}$, and provide reciprocal multiplicity statements linking preimages to multisecants. Special cases, such as when $n\ge g$, $n=g-1$ with a theta characteristic, or hyperelliptic $C$, reveal structural constraints on the curve and the linear system. These results deepen the understanding of how Kummer geometry encodes the Brill–Noether and cycle geometry of Jacobians.
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.