On Enriques-Fano threefolds and a conjecture of Castelnuovo

2021/07/08 by Martello, Vincenzo
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.04089

Abstract

Let W⊂ ℙ¹³ be the image of the rational map defined by the linear system of the sextic surfaces of ℙ³ 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_\bullet⊂ 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_\bullet is birational to an elliptic ruled surface and that the image of W via the rational map defined by L_\bullet is a cubic Del Pezzo surface Δ⊂ ℙ³ with 4 nodes. Interestingly, this fact appears to be related to a conjecture of Castelnuovo.

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