Brill-Noether theory on singular curves and torsion-free sheaves on surfaces
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.
scientific article; zbMATH DE number 1833138