Quartic and Quintic Hypersurfaces with Dense Rational Points

Let X-4 subset of Pn+1 be a quartic hypersurface of dimension n >= 4 over an infinite field k. We show that if either X-4 contains a linear subspace Lambda of dimension h >= max{2, dim(Lambda boolean AND Sing(X-4)) + 2} or has double points along a linear subspace of dimension h >= 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.