Brill-Noether theory on singular curves and torsion-free sheaves on surfaces
type: Item
instance of: scholarly article
title: Brill-Noether theory on singular curves and torsion-free sheaves on surfaces
zbMATH Open document ID: 1028.14010
author: Tomas L. Gómez
published in: Communications in Analysis and Geometry
publication date: 2003-08-17T00:00:00Z
review text: If \(C\) is a smooth curve of genus \(g\), the Brill-Noether locus \(W^r_d(C)\) parametrizes the line bundles \(L\) on \(C\) of degree \(d\) with \(h^0(L)\geq r+1\). It has a natural structure of determinantal variety and its ``expected'' dimension is the Brill-Noether number \(\rho(g,r,d)=g-(r+1)(g-d+r)\). It has been proved by Fulton-Lazarsfeld that \(W^r_d(C)\) is connected if \(\rho(g,r,d)>0\). In the paper under review, the author extends this result to a class of singular curves, precisely to any irreducible curve \(C\) lying on a smooth regular surface \(S\) such that the anticanonical divisor \(-K_S\) is generated by its global sections. Now \(W^r_d(C)\) is replaced by the generalized Brill-Noether locus \(\overline W^r_d(C)\), parametrizing rank one torsion-free sheaves \(A\) on \(C\), of degree \(d\) with at least \(r+1\) independent sections. The rough idea of the proof is to deform \(C\) to a smooth curve and then use the connectedness theorem of Fulton-Lazarsfeld. As an application of this result, the author gives a new proof of a theorem of \textit{K. G. O'Grady} [J. Algebr. Geom. 6, 599-644 (1997; Zbl 0916.14018)] concerning the irreducibility of certain moduli spaces of rank 2 Gieseker-semistable torsion-free sheaves with low \(c_2\) on a \(K3\) surface \(S\). Precisely, the moduli space considered are of the form \({\mathfrak M}_H(L,c_2)\), where \(L\) is a primitive nef and big line bundle, \(c_2\leq {1\over 2}L^2+3\) and \(H\) is an \((L,c_2)\)-generic polarization.
reviewed by: Emilia Mezzetti
Mathematics Subject Classification ID: 14H51
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 1833138
zbMATH Keywords: Brill-Noether number
zbMATH Keywords: torsion-free sheaves
zbMATH Keywords: K3 surfaces
zbMATH Keywords: singular curves
zbMATH Keywords: generalized Brill-Noether locus
zbMATH Keywords: moduli spaces
MaRDI profile type: Publication
DOI: 10.4310/CAG.2001.V9.N4.A3
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description: scientific article; zbMATH DE number 1833138