Quartic and Quintic hypersurfaces with dense rational points

Alex Massarenti

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arXiv · 2212.14626

Quartic and Quintic hypersurfaces with dense rational points

Alex Massarenti

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Abstract

Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension $n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap \text{Sing}(X_4))-2\}$ or has double points along a linear subspace of dimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then $X_4$ 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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Alex Massarenti. 2022-12-30. Quartic and Quintic hypersurfaces with dense rational points. https://arxiv.org/abs/2212.14626

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