Verlinde bundles of families of hypersurfaces and their jumping lines

Verlinde bundles are vector bundles $V_k$ arising as the direct image ${\pi }_{\ast }({\mathcal{L}}^{\otimes k})$ of polarizations of a proper family of schemes $\pi \text{\hspace{0.17em}⁣}:\mathfrak{X}\to S$. We study the splitting behavior of Verlinde bundles in the case where $\pi$ is the universal family $\mathfrak X \to |\mathcal O(d)|$ofhypersurfacesofdegree$d$in$\mathbb P^n$ and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles $V_{d+1}$ in the cases $n=2,3$.