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Verlinde bundles are vector bundles \(V_k\) arising as the direct image \(\pi _*(\mathscr {L}^{\otimes k})\) of polarizations of a proper family of schemes \(\pi :\mathfrak X \rightarrow S\). We study the splitting behavior of Verlinde bundles in the case where \(\pi \) is the universal family \(\mathfrak X \rightarrow \left|{\mathscr {O}(d)}\right|\) of hypersurfaces of degree d in \(\left|{\mathscr {O}(d)}\right|\) and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles \(V_{d+1}\) in the cases \(n=2,3\).

Verlinde bundles of families of hypersurfaces and their jumping lines