Focal schemes to families of secant spaces to canonical curves
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 $d\le g1$ be an integer such that the BrillNoether number $\rho(g,d,1)=g2(gd+1)\geq 1$. We study the family of $d$secant $\mathbb{P}^{d2}$'s to $C$ induced by the smooth locus of the BrillNoether 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 Torellitype theorem for general curves by reconstructing the curve from its BrillNoether loci $W^1_d(C)$ of dimension at least $1$.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.01923
 arXiv:
 arXiv:1710.01923
 Bibcode:
 2017arXiv171001923H
 Keywords:

 Mathematics  Algebraic Geometry;
 14H51;
 14M12;
 14C34
 EPrint:
 14 pages, to appear in: "Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory", DFG, SPP 1489