Intersection cohomology and Severi's varieties

Let $X^{2n}\subseteq {\mathbb{P}}^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}\pi {}_{\ast }\mathbb{Q}$, 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 $IC(R^{2n-1}\pi {}_{\ast }\mathbb{Q})$ 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 ${\mathbb{P}}^N$.