Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916215735189504 |
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| author | Demeio, Julian Lawrence Streeter, Sam Winter, Rosa |
| author_facet | Demeio, Julian Lawrence Streeter, Sam Winter, Rosa |
| contents | We prove that del Pezzo surfaces of degree $2$ over a field $k$ satisfy weak weak approximation if $k$ is a number field and the Hilbert property if $k$ is Hilbertian of characteristic zero, provided that they contain a $k$-rational point lying neither on any $4$ of the $56$ exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado--Testa--Várilly-Alvarado, and Festi--van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_09299 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces Demeio, Julian Lawrence Streeter, Sam Winter, Rosa Algebraic Geometry Number Theory 14G05, 14G12 (Primary), 14G25, 14G27, 11G35 (Secondary) We prove that del Pezzo surfaces of degree $2$ over a field $k$ satisfy weak weak approximation if $k$ is a number field and the Hilbert property if $k$ is Hilbertian of characteristic zero, provided that they contain a $k$-rational point lying neither on any $4$ of the $56$ exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado--Testa--Várilly-Alvarado, and Festi--van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration. |
| title | Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces |
| topic | Algebraic Geometry Number Theory 14G05, 14G12 (Primary), 14G25, 14G27, 11G35 (Secondary) |
| url | https://arxiv.org/abs/2303.09299 |