Intersection cohomology and Severi's varieties

TL;DR

This paper studies the intersection cohomology of certain local systems associated with smooth projective varieties, focusing on Severi's varieties parametrizing nodal hypersurfaces and their cohomological properties.

Contribution

It provides new insights into the intersection cohomology of local systems over Severi's varieties, linking geometric properties of nodal hypersurfaces to cohomological invariants.

Findings

Cohomology computations over Severi's varieties

Relations between nodal hypersurfaces and intersection cohomology

Conditions under which nodes impose independent conditions

Abstract

Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}\pi{_*}\mathbb{Q}$, where $\pi$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(\mathbb{P} ^N)}^{\vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}\pi{_*}\mathbb{Q})$ over the points of a Severi's variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $\mathbb{P} ^N$.