Universal formulas for degeneracy classes of vector bundles on P^1 bundles in terms of vector bundles on the base, valid in any characteristic when loci are in expected codimension.
Sacchiero, On the varieties parameterizing rational space curves with fixed normal bundle , Manuscripta Math., 37 (1982), 2170–228
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Universal degeneracy classes for vector bundles on $\mathbb{P}^1$ bundles
Universal formulas for degeneracy classes of vector bundles on P^1 bundles in terms of vector bundles on the base, valid in any characteristic when loci are in expected codimension.