Universal degeneracy classes for vector bundles on $\mathbb{P}^1$ bundles

TL;DR

This paper derives universal formulas for degeneracy classes of vector bundles on $\

Contribution

It provides the first universal formulas for degeneracy classes of vector bundles on $\

Findings

Classes are computed in expected codimension cases

Formulas are valid over arbitrary fields

Results characterize degenerations of vector bundles

Abstract

Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by degeneracy loci measuring the spitting 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. Our results hold over arbitrary fields of any characteristic.