Quartic surface, its bitangents and rational points
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866910039355162624 |
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| author | Corvaja, Pietro Zucconi, Francesco |
| author_facet | Corvaja, Pietro Zucconi, Francesco |
| contents | Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_08623 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Quartic surface, its bitangents and rational points Corvaja, Pietro Zucconi, Francesco Number Theory Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense. |
| title | Quartic surface, its bitangents and rational points |
| topic | Number Theory |
| url | https://arxiv.org/abs/2010.08623 |