Points rationnels dans leur fibre: compl\'ements \`a un th\'eor\`eme de Poonen

TL;DR

This paper extends Poonen's theorem by showing the existence of points with specific residue field properties in integral schemes, including codimension 1 points and schemes of higher dimension, under various conditions.

Contribution

It generalizes Poonen's result to include points over codimension 1 and higher-dimensional schemes, with additional separability conditions when schemes are smooth.

Findings

Existence of points with residue fields purely inseparable over images

Points can be chosen with images of codimension 1 in the base scheme

Results extend to noetherian integral schemes of dimension ≥ 2

Abstract

Soit $f:X\to S$ un morphisme surjectif et de pr\'esentation finie de sch\'emas int\`egres. Lorsque $S$ est de type fini sur un corps ou sur $\mathbb{Z}$, et de dimension $>0$, Poonen a montr\'e qu'il existe un point $x\in X$ dont le corps r\'esiduel $\kappa(x)$ est une extension radicielle de $\kappa(f(x))$. Dans ce travail, on prolonge ce r\'esultat de plusieurs fa\c{c}ons. D'abord, on peut choisir $x$ tel que $f(x)$ soit un point de codimension $1$ de $S$; si $S$ est lisse sur un corps $k$, on peut exiger que $\kappa(f(x))$ soit s\'eparable sur $k$. Dans une autre direction, on montre des r\'esultats analogues pour d'autres classes de sch\'emas $S$, par exemple les sch\'emas noeth\'eriens int\`egres de dimension $\geq2$. Let $f:X\to S$ be a morphism of integral schemes, which is surjective and of finite presentation. When $S$ is of finite type over a field or over $\mathbb{Z}$, of positive dimension, Poonen has shown that there is a point $x\in X$ whose residue field $\kappa(x)$ is purely inseparable over $\kappa(f(x))$. In this paper, we extend this result in several ways. First we prove that we can take $x$ such that $f(x)$ is a codimension $1$ point of $S$; if $S$ is smooth over a fiel $k$, we can require $\kappa(f(x))$ to be separable over $k$. In another direction, we prove that similar results hold for other schemes $S$, such as noetherian integral schemes of dimension $\geq2$.