Nonemptiness of symmetric degeneracy loci

Abstract. 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 S2V∗⊗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, 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, which may be of independent interest, identifies a Gysin map with the natural map from ordinary to relative cohomology. 1. Introduction The main result of this paper is a nonemptiness result for symmetric degeneracy loci. Before stating the result, we illustrate it with an application. Let Sr(N) denote the projectivized variety of symmetric N × N complex matrices of rank at most r. Because the codimension of Sr−1(N) in Sr(N) is N − r + 1, there exist N − r-dimensional closed subvarieties of Sr(N) which d