Hilbert squares of degeneracy loci

Abstract

Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in ${\mathbb{P}}^s$. We prove that, under certain positivity conditions, its Hilbert square ${\mathrm{Hilb}}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.

20 pages, comments welcome!