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Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified bydegeneracy loci measuring the spitting type of the vector bundle restricted toeach fiber. The classes of these degeneracy loci in the Chow ring or cohomologyring of the base are natural invariants characterizing the degenerations of thevector bundle. When these degeneracy loci occur in the expected codimension, wefind their classes. This yields universal formulas for degeneracy classes interms of naturally arising vector bundles on the base. Our results hold overarbitrary fields of any characteristic.

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