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

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

1996-11-20

arxiv.org/abs/alg-geom/9611025v1

Mathematics

Algebraic Geometry

AMS-TeX, 15 pages

Scientific paper

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.

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