Hilbert squares of degeneracy loci

TL;DR

This paper studies the Hilbert square of a degeneracy locus defined by a general matrix of linear forms, showing it is isomorphic to a zero locus of a section of a vector bundle on a product of Grassmannians, under positivity conditions.

Contribution

It provides a new geometric description of the Hilbert square of degeneracy loci using vector bundles on Grassmannians and constructs an explicit isomorphism involving associated Fano varieties.

Findings

Hilbert square of degeneracy locus is isomorphic to a zero locus of a section of a vector bundle.

Explicit construction of the isomorphism involving Fano varieties.

Conditions under which the isomorphism holds are established.

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.