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 $R\pi{}\mathbb{Q}_{\mathcal{X}}$.