On Enriques-Fano threefolds and a conjecture of Castelnuovo

Abstract

Let $W\subset \mathbb{P}^{13}$ be the image of the rational map defined by the linear system of the sextic surfaces of $\mathbb{P}^3$ having double points along the edges of a tetrahedron. Let $\mathcal{L}$ be the linear system of the hyperplane sections of $W$. It is known that a general $S\in \mathcal{L}$ is an Enriques surface. The aim of this paper is to study the sublinear system $\mathcal{L}_{\bullet}\subset \mathcal{L}$ of the hyperplane sections of $W$ having a triple point at a general point $w \in W$. We will show that a general element of $\mathcal{L}_{\bullet}$ is birational to an elliptic ruled surface and that the image of $W$ via the rational map defined by $\mathcal{L}_{\bullet}$ is a cubic Del Pezzo surface $\Delta\subset \mathbb{P}^3$ with $4$ nodes. Interestingly, this fact appears to be related to a conjecture of Castelnuovo.