On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties

TL;DR

This paper establishes bounds on the dimension of symmetric matrix spaces with constant even rank and explores their connection to dual varieties in projective geometry, providing new insights and applications.

Contribution

It proves a bound on the dimension of symmetric matrix spaces with constant rank and applies this to solve the constant rank problem for symmetric matrices.

Findings

Maximum dimension of symmetric matrices with constant even rank is m-r+1.

The results relate to the geometry of dual varieties in projective space.

Applications include new bounds and examples in algebraic geometry.

Abstract

Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We use this result to solve the constant rank problem for symmetric matrices, proving that the maximal dimension of a linear subspace of the space of $m\times m$ symmetric matrices such that each nonzero element has even rank $r \ge 2$ is $m-r+1$. We explain how this result relates to the study of dual varieties in projective geometry and give some applications and examples.