The Degeneracy Loci for Smooth Moduli of Sheaves

Let S be a smooth projective surface over the complex field. Under certain technical assumptions, we prove that the degeneracy locus of the universal sheaf over the moduli space of stable sheaves is either empty or an irreducible Cohen-Macaulay variety of the expected dimension; we also give a criterion for when the degeneracy locus is nonempty. This result generalizes the work of Bayer, Chen, and Jiang for the Hilbert scheme of points on surfaces. The above statement is a special case of a more general phenomenon: for a two-term complex of locally free sheaves, the geometry of the degeneracy locus is closely related to the geometry of Grassmannians.