Elliptic-elliptic surfaces and the Hesse pencil

2024/09/27 by François Greer, Yilong Zhang, Greer, François +1
Mathematics · #14J27 #32G20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2409.18927

openalex publication_date 2024/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Γ₀(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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