Rational Points on Generic Marked Hypersurfaces

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