Rational curves and the Hilbert Property on Jacobian Kummer varieties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916888371527680 |
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| author | Gvirtz-Chen, Damián Huang, Zhizhong |
| author_facet | Gvirtz-Chen, Damián Huang, Zhizhong |
| contents | A conjecture by Corvaja and Zannier predicts that smooth, projective, simply connected varieties over a number field with Zariski dense set of rational points have the Hilbert Property; this was proved by Demeio for Kummer surfaces which are associated to products of two elliptic curves.
In this article, over a finitely generated field of characteristic zero, we establish the Hilbert Property for all Kummer surfaces associated to the Jacobian of a genus $2$ curve. In general we prove that all Jacobian Kummer varieties associated to a hyperelliptic curve of genus $\geq 2$ of odd degree also have the Hilbert Property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04364 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rational curves and the Hilbert Property on Jacobian Kummer varieties Gvirtz-Chen, Damián Huang, Zhizhong Number Theory Algebraic Geometry 14G05, 11G35, 14J28 A conjecture by Corvaja and Zannier predicts that smooth, projective, simply connected varieties over a number field with Zariski dense set of rational points have the Hilbert Property; this was proved by Demeio for Kummer surfaces which are associated to products of two elliptic curves. In this article, over a finitely generated field of characteristic zero, we establish the Hilbert Property for all Kummer surfaces associated to the Jacobian of a genus $2$ curve. In general we prove that all Jacobian Kummer varieties associated to a hyperelliptic curve of genus $\geq 2$ of odd degree also have the Hilbert Property. |
| title | Rational curves and the Hilbert Property on Jacobian Kummer varieties |
| topic | Number Theory Algebraic Geometry 14G05, 11G35, 14J28 |
| url | https://arxiv.org/abs/2205.04364 |