Somewhat Tautological Bundles and their Degeneracy Loci

Abstract

We define a family of coherent sheaves indexed by the natural numbers on the Hilbert scheme of points on a surface S, and study some of their properties. The first two sheaves are bundles whose degeneracy loci parametrize subsets of singular loci of curves on S, that are members of general linear systems of appropriate dimension. We verify that one degeneracy locus is of expected dimension and compute the total Chern class of the first bundle, as well as relate the Chern classes of the second bundle to the Chern classes of two tautological bundles.