$\mathbb Q$-bases of the N\'eron-Severi groups of certain elliptic surfaces

TL;DR

This paper explicitly constructs $Q$-bases for the Néron-Severi groups of certain elliptic surfaces, providing generators and, in some cases, $Z$-bases, advancing understanding of their algebraic structure.

Contribution

The paper explicitly determines $Q$-bases for Néron-Severi groups of specific elliptic surfaces, including rational cases with $Z$-bases, filling a gap left by prior computations.

Findings

Explicit $Q$-bases for Néron-Severi groups of elliptic surfaces

Identification of $Z$-bases in rational cases

Enhanced understanding of the algebraic structure of these groups

Abstract

P. Stiller computed the Picard numbers of several families of elliptic surfaces, the rank of the N\'eron-Severi groups of these surfaces. However he did not give the generators of these groups. In this paper we give $\mathbb Q$-bases of these groups explicitly. If these surfaces are rational, then we also show that they are $\mathbb Z$-bases.