Quartic and Quintic Hypersurfaces with Dense Rational Points

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 $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ 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.