Verlinde bundles of families of hypersurfaces and their jumping lines

TL;DR

This paper investigates the splitting behavior and jumping lines of Verlinde bundles associated with families of hypersurfaces, providing explicit calculations of their cohomology classes in specific cases.

Contribution

It analyzes the splitting properties of Verlinde bundles over universal hypersurface families and computes the cohomology class of jumping lines for certain dimensions.

Findings

Determined the splitting behavior of Verlinde bundles for hypersurfaces.

Calculated the cohomology class of jumping lines for $V_{d+1}$ in cases $n=2,3$.

Provided explicit geometric descriptions of jumping loci.

Abstract

Verlinde bundles are vector bundles $V_k$ arising as the direct image $\pi_*(\mathcal L^{\otimes k})$ of polarizations of a proper family of schemes $\pi\colon \mathfrak X \to S$. We study the splitting behavior of Verlinde bundles in the case where $\pi$ is the universal family $\mathfrak X \to |\mathcal O(d)|$ of hypersurfaces of degree $d$ 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 $n=2,3$.