Brill-Noether theory on singular curves and torsion-free sheaves on surfaces

Let C be a smooth curve of genus g. Let W-d(r)(C) be the Brill-Noether locus of line bundles of degree d and with r+1 independent sections. The expected dimension of W-d(r)(C) is rho(r, d)=g-(r+1) (g-d+r). If rho(r, d)>0 then Fulton and Lazarsfeld have proved that W-d(r)(C) is connected. We prove that this is still true if C is a singular irreducible curve lying on a regular surface S with -K-S generated by global sections. We use this result to give a short new proof of the irreducibility of the moduli space of rank 2 semistable torsion-free sheaves (with a generic polarization and low value of c(2)) on a K3 surface (this result was recently proved by a different method by O'Grady).