Rational Points on Generic Marked Hypersurfaces

Over fields of characteristic zero, we show that for $n=1,d\geq4$ or $n=2,d\geq5$ or $n\geq3, d\geq 2n$, 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 $n=1,d\geq4$ or $n\geq2, d\geq 2n+3$, the identiy map is the only rational self-map of the generic degree-$d$ hypersurface in $\mathbb{P}^{n+1}$.