Universal degeneracy classes for vector bundles on [formula omitted] bundles

Abstract

Given a vector bundle on a P1 bundle, the base is stratified by degeneracy loci measuring the splitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. The existence of these formulas is applied to prove the non-emptiness of Brill-Noether splitting loci in [21]. Our results hold over arbitrary fields of any characteristic.