Integral points on varieties with infinite \'etale fundamental group

TL;DR

This paper investigates the finiteness of integral points on varieties with infinite étale fundamental groups, establishing conditions under which integral points are finite for certain divisors on these varieties.

Contribution

It proves that for varieties with infinite étale fundamental groups, there exist ample divisors with finite integral points sets, expanding understanding of integral points in this context.

Findings

Existence of ample divisors with finite integral points on varieties with infinite étale fundamental groups

Finiteness results for integral points on general members of linear systems on covers of varieties

Application to new examples of varieties with finite integral points

Abstract

We study integral points on varieties with infinite \'etale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi\colon Y \to X$ of degree at least $2\dim(X)^2$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi(D)$-integral points on $X$ is finite. We apply this result to varieties with infinite \'etale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.