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 $d\le g-1$ 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 $1$.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.01923
- arXiv:
- arXiv:1710.01923
- Bibcode:
- 2017arXiv171001923H
- Keywords:
-
- Mathematics - Algebraic Geometry;
- 14H51;
- 14M12;
- 14C34
- E-Print:
- 14 pages, to appear in: "Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory", DFG, SPP 1489