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

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 $S^2(E^{\ast })\otimes L$ is an ample vector bundle and that there is a constant even rank $r\ge 2$ symmetric bundle map $E\to E^{\ast }\otimes L$. 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.

arXiv commentAMS-TeX, 15 pages

Cited by in corpus (1)

• Algebraic geometry and projective differential geometry, Seoul National University concentrated lecture series, 1997