On the discriminant locus of a rank $n-1$ vector bundle on ${\mathbb{P}}^{n-1}$

Abstract

In this paper, we show that if a vector bundle of rank $n-1$ on projective $(n-1)$-space is very ample and if its top Chern class is greater than one, then the discriminant locus of the vector bundle, the locus of singular sections of the vector bundle, is an irreducible hypersurface and that the degree of the hypersurface can be expressed as a function of invariants of the vector bundle. As applications, we study the eigendiscriminant locus (the locus of tensors whose eigenschemes are singular) and the discriminant locus of the generic graded matrix (the locus of singular determinantal schemes).