Degeneracy loci of twisted differential forms and linear line complexes

TL;DR

This paper studies the geometric properties of degeneracy loci of twisted differential forms on projective space, showing unirationality, dominance by a Grassmannian, and birationality in a special case.

Contribution

It establishes the unirationality and dominance of the Hilbert scheme of degeneracy loci, providing a constructive method to realize pencils of linear line complexes with given centers.

Findings

Hilbert scheme of degeneracy loci is unirational.

The scheme is dominated by the Grassmannian of lines in skew-symmetric forms.

The map is birational when n=4.

Abstract

We prove that the Hilbert scheme of degeneracy loci of pairs of global sections of Omega(2), the twisted cotangent bundle on P^(n-1), is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n=4.