The Kodaira dimension of Hilbert modular threefolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910136757387264 |
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| author | Logan, Adam |
| author_facet | Logan, Adam |
| contents | Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of arithmetic genus $0$ and $1$ are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements $x$ of a fractional ideal $I$ in a totally real number field $K$ with the property that $\mathop{\mathrm{tr}} xy < \mathop{\mathrm{min}} I \mathop{\mathrm{tr}} y$ for some $y \gg 0 \in K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Kodaira dimension of Hilbert modular threefolds Logan, Adam Number Theory Algebraic Geometry 14G35, 11F41, 14J30 (Primary), 11Y40, 32S45 (Secondary) Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of arithmetic genus $0$ and $1$ are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements $x$ of a fractional ideal $I$ in a totally real number field $K$ with the property that $\mathop{\mathrm{tr}} xy < \mathop{\mathrm{min}} I \mathop{\mathrm{tr}} y$ for some $y \gg 0 \in K$. |
| title | The Kodaira dimension of Hilbert modular threefolds |
| topic | Number Theory Algebraic Geometry 14G35, 11F41, 14J30 (Primary), 11Y40, 32S45 (Secondary) |
| url | https://arxiv.org/abs/2501.15719 |