On the extendability of elliptic surfaces of rank two and higher

TL;DR

This paper investigates the embedding properties of elliptic surfaces of rank two and higher within threefolds, providing new theorems on their possible embeddings and conditions under which they form cones.

Contribution

It establishes general embedding theorems for elliptic surfaces with Picard number two and analyzes the extendability of Weierstrass fibrations of various ranks.

Findings

Rank two Weierstrass fibrations not being K3 surfaces cannot be hyperplane sections of LCI threefolds.

Certain embeddings of Weierstrass fibrations imply the threefold must be a cone.

General theorems on embeddings of elliptic surfaces with Picard number two.

Abstract

We study threefolds X in a projective space having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.