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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.01669 |
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| _version_ | 1866911247496118272 |
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| author | Chen, Nathan Church, Ben Pasten, Hector Vogt, Isabel |
| author_facet | Chen, Nathan Church, Ben Pasten, Hector Vogt, Isabel |
| contents | Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover $π\colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $π$ are dense. We show that Vojta's Conjecture implies that, once the canonical bundle of $X$ is sufficiently positive, there are no other sources of Zariski dense quadratic points. This is complemented by several examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic points on double planes Chen, Nathan Church, Ben Pasten, Hector Vogt, Isabel Number Theory Algebraic Geometry 11G35 Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover $π\colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $π$ are dense. We show that Vojta's Conjecture implies that, once the canonical bundle of $X$ is sufficiently positive, there are no other sources of Zariski dense quadratic points. This is complemented by several examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points. |
| title | Quadratic points on double planes |
| topic | Number Theory Algebraic Geometry 11G35 |
| url | https://arxiv.org/abs/2511.01669 |