Verlinde bundles of families of hypersurfaces and their jumping lines (Q2316755)

The universal family \(\pi:\mathcal{X}\rightarrow|\mathcal{O}_{\mathbb{P}}^n(d)|\) of hypersurfaces of degree \(d\) in the complex projective plane \(\mathbb{P}^n\) comes equipped with a polarization \(\mathcal{L}\). Furthermore, the sheaf \(\pi_*\mathcal{L}^{\otimes^k}\), with \(k\geq 1\), is locally free and is called the \(k\)-th Verlinde bundle of the family \(\pi\), denoted by \(V_k\). In the present paper, the author studies the splitting behavior of such Verlinde bundles. Let \(Z\) be the set of jumping lines of \(V_{d+1}\) in the Grassmannian of lines in \(|\mathcal{O}(d)|\), \(\mathrm{Gr}(1,|\mathcal{O}(d)|)\). In the main theorem of this paper, the author calculates for \(n\leq 3\) the dimension of \(Z\) as well as the class of \(Z\) in the Chow ring \(CH(\mathrm{Gr}(1),|\mathcal{O}(d)|)\). [\textit{J. N. Iyer}, ``Bundles of verlinde spaces and group actions'', Preprint \url{arXiv:1309.7562}]