Elliptic-elliptic surfaces and the Hesse pencil
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929518531313664 |
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| author | Greer, François Zhang, Yilong |
| author_facet | Greer, François Zhang, Yilong |
| contents | 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 $Γ_0(11)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic-elliptic surfaces and the Hesse pencil Greer, François Zhang, Yilong Algebraic Geometry 14J27, 32G20 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 $Γ_0(11)$. 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. |
| title | Elliptic-elliptic surfaces and the Hesse pencil |
| topic | Algebraic Geometry 14J27, 32G20 |
| url | https://arxiv.org/abs/2409.18927 |