On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties Academic Article uri icon

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 $E o E^{*} otimes L$. 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 $m imes m$ 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.

author list (cited authors)

  • Ilic, B. o., & Landsberg, J. M.

citation count

  • 18

publication date

  • May 1999