Nonemptiness of symmetric degeneracy loci

TL;DR

This paper proves that under certain ampleness and dimensional conditions, the degeneracy loci of quadratic forms on vector bundles are guaranteed to be nonempty, with applications to matrix subschemes and projective embeddings.

Contribution

It establishes a new nonemptiness criterion for symmetric degeneracy loci based on vector bundle rank, dimension, and ampleness assumptions.

Findings

Degeneracy loci are nonempty if d > N - r.

Applications to subschemes of matrices and projective embeddings.

Identification of Gysin maps with natural cohomology maps.

Abstract

Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S^2 V^* \otimes L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N-r, then the locus of points where the rank of the quadratic form is at most r-1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin maps. The main result of the appendix identifies a Gysin map with the natural map from ordinary to relative cohomology.