Rational points on generic marked hypersurfaces

Over fields of characteristic zero, we show that for \(n=1,d\ge 4\) $n=1,d\ge 4$ or \(n=2,d\ge 5\) $n=2,d\ge 5$ or \(n\ge 3, d\ge 2n\) $n\ge 3,d\ge 2n$ , the generic m-marked degree-d hypersurface in \(\mathbb {P}^{n+1}\) ${\mathbb{P}}^{n+1}$ admits the m marked points as all the rational points. Over arbitrary fields, we show that for \(n=1,d\ge 4\) $n=1,d\ge 4$ or \(n\ge 2, d\ge 2n+3\) $n\ge 2,d\ge 2n+3$ , the identity map is the only rational self-map of the generic degree-d hypersurface in \(\mathbb {P}^{n+1}\) ${\mathbb{P}}^{n+1}$ .