2003/05/11 by William Graham, Graham, William
Mathematics · #14N05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG #msc:14N05
paper · pdf · doi:10.48550/arxiv.math/0305159
31 pages
arxiv created 2003/05/11 · openalex publication_date 2003/05/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S2 V^* ⊗ L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N-r, then the locus of points where the rank of the quadratic form is at most r-1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin maps. The main result of the appendix identifies a Gysin map with the natural map from ordinary to relative cohomology.