Elliptic-elliptic surfaces and the Hesse pencil

TL;DR

This paper constructs specific elliptic surfaces from the Hesse pencil, identifies special loci with high Picard number and positive Mordell-Weil rank, and explores their period maps and degenerations.

Contribution

It explicitly constructs elliptic surfaces with particular properties from the Hesse pencil and analyzes their Noether-Lefschetz loci and period maps.

Findings

Identified a component of the higher Noether-Lefschetz locus with positive Mordell-Weil rank.

Constructed a surface with maximal Picard number over .

Extended the period map to boundary degenerations.

Abstract

We construct a family of elliptic surfaces with $p_g=q=1$ that arise from base change of the Hesse pencil. We identify explicitly a component of the higher Noether-Lefschetz locus with positive Mordell-Weil rank, and a particular surface having maximal Picard number and defined over $\mathbb Q$. These examples satisfy the infinitesimal Torelli theorem, providing a second proof of the dominance of period map, which was first obtained by Engel-Greer-Ward. A third proof is provided using the Shioda modular surface associated with $\Gamma_0(11)$. Finally, we find birational models for the degenerations at the boundary of the one-dimensional Noether-Lefschetz locus, and extend the period map at those limit points.