The Degeneracy Loci for Smooth Moduli of Sheaves

Let S be a smooth projective surface over $\mathbb{C}$. We prove that, under certain technical assumptions, 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 provide a criterion for when the degeneracy locus is non-empty. This result generalizes the work of Bayer, Chen, and Jiang for the Hilbert scheme of points on surfaces. The above result is a special case of a general phenomenon: for a perfect complex of Tor-amplitude [0,1], the geometry of the degeneracy locus is closely related to the geometry of the derived Grassmannian. We analyze their birational geometry and relate it to the incidence varieties of derived Grassmannians. As a corollary, we prove a statement previously claimed by the author in arXiv:2408.06860.