Focal schemes to families of secant spaces to canonical curves

TL;DR

This paper generalizes previous results to study families of secant spaces to canonical curves, reconstructing the curve from Brill--Noether loci using advanced algebraic geometry techniques.

Contribution

It introduces a Torelli-type theorem for general curves by reconstructing them from Brill--Noether loci via secant space families and focus theory.

Findings

Reconstruction of canonical curves from Brill--Noether loci.

Extension of previous results to broader genus and degree ranges.

Application of focus theory and matrix rank loci to curve reconstruction.

Abstract

This article is a generalisation of results of Ciliberto and Sernesi. For a general canonically embedded curve $C$ of genus $g\geq 5$, let $d\le g-1$ be an integer such that the Brill--Noether number $\rho(g,d,1)=g-2(g-d+1)\geq 1$. We study the family of $d$-secant $\mathbb{P}^{d-2}$'s to $C$ induced by the smooth locus of the Brill--Noether locus $W^1_d(C)$. 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 $W^1_d(C)$ of dimension at least $1$.