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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 alinear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap\text{Sing}(X_4))-2\}$ or has double points along a linear subspace ofdimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then$X_4$ is unirational over $k$. This improves previous results by A. Predonzanand J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide adensity result for the $k$-rational points of quartic $3$-folds with a doubleplane over a number field, and several unirationality results for quintichypersurfaces over a $C_r$ field.

Quartic and Quintic hypersurfaces with dense rational points