Quartic and Quintic hypersurfaces with dense rational points

Abstract

Let $X_4\subset {\mathbb{P}}^{n+1}$ be a quartic hypersurface of dimension $n\ge 4$ over an infinite field $k$. We show that if either $X_4$ contains a linear subspace $\Lambda$ of dimension $h\ge \max \{2,\dim (\Lambda \cap \text{Sing}(X_4))-2\}$ or has double points along a linear subspace of dimension $h\ge 3$, 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.

arXiv comment16 pages