Quartic and Quintic hypersurfaces with dense rational points

2022/12/30 by Alex Massarenti, Massarenti, Alex
Arts and Humanities · Mathematics · #14M20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #French Historical and Cultural Studies #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 14E08 #Secondary 14G05

paper · pdf · doi:10.48550/arxiv.2212.14626

openalex publication_date 2022/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X₄⊂ℙⁿ⁺¹ be a quartic hypersurface of dimension n≥ 4 over an infinite field k. We show that if either X₄ contains a linear subspace Λ of dimension h≥ max\2,dim(Λ∩ Sing(X₄))-2\ or has double points along a linear subspace of dimension h≥ 3, a smooth k-rational point and is otherwise general, then X₄ is unirational over k. This improves previous results by A. Predonzan and J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide a density result for the k-rational points of quartic 3-folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a C_r field.

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• Rational points on quartic hypersurfaces
• Hypersurfaces quartiques de dimension 3 : non rationalité stable
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• A very general quartic double fourfold or fivefold is not stably rational