Rational Points on Generic Marked Hypersurfaces

TL;DR

This paper investigates the distribution of rational points on generic marked hypersurfaces over fields of characteristic zero and arbitrary fields, establishing conditions under which all marked points are rational and characterizing rational self-maps.

Contribution

It provides new results on rational points and self-maps of generic hypersurfaces, extending understanding in algebraic geometry over different fields.

Findings

Generic marked hypersurfaces have all marked points rational under specified conditions.

The identity map is the only rational self-map for certain hypersurfaces over arbitrary fields.

Conditions for rational points depend on the dimension and degree of the hypersurface.

Abstract

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 $\mathbb{P}^{n+1}$.