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

1996/11/20 by Bo Ilic, Ilic, Bo, J. M. Landsberg +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #alg-geom #dg-ga #math.AG #math.DG

paper · pdf · doi:10.48550/arxiv.alg-geom/9611025

AMS-TeX, 15 pages

arxiv created 1996/11/20 · openalex publication_date 1996/11/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 S²(E^*) ⊗ L is an ample vector bundle and that there is a constant even rank r ≥ 2 symmetric bundle map E → E^* ⊗ L. We prove that m ≤ 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× m symmetric matrices such that each nonzero element has even rank r ≥ 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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