On Enriques-Fano threefolds and a conjecture of Castelnuovo

Abstract

Let Wsubset 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 Sin 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.