Hypersurfaces passing through the Galois orbit of a point
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| Format: | Preprint |
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2025
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| _version_ | 1866914459595833344 |
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| author | Asgarli, Shamil Love, Jonathan Yip, Chi Hoi |
| author_facet | Asgarli, Shamil Love, Jonathan Yip, Chi Hoi |
| contents | Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_01906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypersurfaces passing through the Galois orbit of a point Asgarli, Shamil Love, Jonathan Yip, Chi Hoi Algebraic Geometry Number Theory Primary: 14G15, 14J70, Secondary: 14N05, 11G25 Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties. |
| title | Hypersurfaces passing through the Galois orbit of a point |
| topic | Algebraic Geometry Number Theory Primary: 14G15, 14J70, Secondary: 14N05, 11G25 |
| url | https://arxiv.org/abs/2501.01906 |