On Enriques-Fano threefolds and a conjecture of Castelnuovo

Abstract

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