Quartic and Quintic hypersurfaces with dense rational points

TL;DR

This paper proves conditions under which certain quartic and quintic hypersurfaces over fields have dense rational points, improving previous results and establishing new unirationality criteria for these complex algebraic varieties.

Contribution

It establishes new unirationality conditions for quartic hypersurfaces with linear subspaces or double points, and extends results to quintic hypersurfaces over specific fields.

Findings

Quartic hypersurfaces with linear subspaces are unirational over the base field.

Density of rational points on quartic 3-folds with a double plane over number fields.

Unirationality results for quintic hypersurfaces over $C_r$ fields.

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.