Hypersurfaces passing through the Galois orbit of a point

[2501.01906] Hypersurfaces passing through the Galois orbit of a point Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Algebraic Geometry arXiv:2501.01906 (math) [Submitted on 3 Jan 2025 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Hypersurfaces passing through the Galois orbit of a point Authors: Shamil Asgarli , Jonathan Love , Chi Hoi Yip View a PDF of the paper titled Hypersurfaces passing through the Galois orbit of a point, by Shamil Asgarli and 2 other authors View PDF HTML (experimental) Abstract: 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. Comments: 29 pages Subjects: Algebraic Geometry (math.AG) ; Number Theory (math.NT) MSC classes: Primary: 14G15, 14J70, Secondary: 14N05, 11G25 Cite as: arXiv:2501.01906 [math.AG] (or arXiv:2501.01906v2 [math.AG] for this version) https://doi.org/10.48550/arXiv.2501.01906 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Shamil Asgarli [ view email ] [v1] Fri, 3 Jan 2025 17:22:18 UTC (29 KB) [v2] Thu, 9 Apr 2026 06:12:52 UTC (31 KB) Full-text links: Access Paper: View a PDF of the paper titled Hypersurfaces passing through the Galois orbit of a point, by Shamil Asgarli and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2025-01 Change to browse by: math math.NT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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