Quartic and Quintic Hypersurfaces with Dense Rational Points
Forum of Mathematics, Sigma, Tome 11 (2023)
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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 $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ 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 and 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.
@article{10_1017_fms_2023_55,
author = {Alex Massarenti},
title = {Quartic and {Quintic} {Hypersurfaces} with {Dense} {Rational} {Points}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.55},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.55/}
}
Alex Massarenti. Quartic and Quintic Hypersurfaces with Dense Rational Points. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.55
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