Quartic and Quintic Hypersurfaces with Dense Rational Points

Let be a quartic hypersurface of dimension over an infinite field k. We show that if either contains a linear subspace of dimension or has double points along a linear subspace of dimension, a smooth k-rational point and is otherwise general, then 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 -folds with a double plane over a number field, and several unirationality results for quintic hypersurfaces over a field.