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.
Smith, Vector bundles on trees of smooth rational curves (2019), preprint
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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.