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 linear subspace $\Lambda $ $h\geq \max \{2,\dim (\Lambda \cap \operatorname {\mathrm {Sing}}(X_4))+2\}$ or has double points along 3$ , smooth -rational point and is otherwise general, then unirational This improves previous results by A. Predonzan J. Harris, B. Mazur R. Pandharipande for quartics. also provide density result the $3$ -folds with plane number field, several unirationality quintic hypersurfaces $C_r$ field.