The Degeneracy Loci for Smooth Moduli of Sheaves

TL;DR

This paper investigates the geometric properties of degeneracy loci in the moduli space of stable sheaves on smooth projective surfaces, establishing conditions for their nonemptiness and irreducibility, and relating them to Grassmannian geometry.

Contribution

It generalizes previous results on Hilbert schemes to broader moduli spaces, providing new criteria and geometric insights into degeneracy loci of universal sheaves.

Findings

Degeneracy locus is either empty or an irreducible Cohen-Macaulay variety.

Provided a criterion for nonemptiness of the degeneracy locus.

Linked the geometry of degeneracy loci to Grassmannian structures.

Abstract

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.