Elliptic-elliptic surfaces and the Hesse pencil

François Greer; Yilong Zhang

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arXiv · 2409.18927

Elliptic-elliptic surfaces and the Hesse pencil

François Greer·Yilong Zhang

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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.

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François Greer, Yilong Zhang. 2024-09-27. Elliptic-elliptic surfaces and the Hesse pencil. https://arxiv.org/abs/2409.18927

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