On Enriques-Fano threefolds and a conjecture of Castelnuovo

TL;DR

This paper investigates a special linear system of hyperplane sections of a threefold related to Enriques surfaces, revealing a birational relation to elliptic ruled surfaces and a connection to Castelnuovo's conjecture.

Contribution

It demonstrates that hyperplane sections with a triple point are birational to elliptic ruled surfaces and links the geometry to a cubic Del Pezzo surface with four nodes, addressing Castelnuovo's conjecture.

Findings

General elements of the sublinear system are birational to elliptic ruled surfaces.

The image of the threefold under the sublinear system is a cubic Del Pezzo surface with four nodes.

The results establish a geometric connection related to Castelnuovo's conjecture.

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.