Focal schemes to families of secant spaces to canonical curves

This article is a generalisation of results of Ciliberto and Sernesi. For a general canonically embedded curve $C$ of genus $g\ge 5$, let $d\le g-1$ be an integer such that the Brill–Noether number $\rho (g,d,1)=g-2(g-d+1)\ge 1$. We study the family of $d$-secant ${\mathbb{P}}^{d-2}$'s to $C$ induced by the smooth locus of the Brill–Noether locus $W_d^1(C)$. Using the theory of foci and a structure theorem for the rank one locus of special $1$-generic matrices by Eisenbud and Harris, we prove a Torelli-type theorem for general curves by reconstructing the curve from its Brill–Noether loci $W_d^1(C)$ of dimension at least $1$.