Intersection cohomology and Severi varieties of quartic surfaces

Abstract

We give two explicit versions of the decomposition theorem of Beilinson, Bernstein and Deligne applied to the universal family of quartic surfaces of $\mathbb{P}^3$. The starting point of our investigation is the remark that the nodes of a quartic surface impose independent conditions to the linear system $\mid \mathcal{O}_{\mathbb P^3}(4)\mid$. Although this property is known in literature, we provide a different argument more suited to our purposes. By a result of \cite{DGF}, the independence of the nodes implies in turn that each component of Severi's variety is smooth of the expected dimension and that the dual variety is a divisor with normal crossings around Severi's variety. This allows us to study the complex $R\pi{_*}\mathbb{Q}_{\mathcal{X}}$, the derived direct image of the constant sheaf over the universal family of quartic surfaces $ \mathcal{X} \stackrel{\pi}{\longrightarrow} \mathbb P^{34}$, both in the open set parametrizing smooth and nodal quartics and in a tubular neighborhood of the variety of Kummer surfaces. We obtain in both cases an explicit decomposition and a formality result for the complex $R\pi{_*}\mathbb{Q}_{\mathcal{X}}$.