Transplanting Faltings' garden

TL;DR

This paper demonstrates how Faltings' examples of divisors with finitely many integral points can be explained through a theorem by Evertse and Ferretti, by constructing suitable covers that embed the complements into semiabelian varieties.

Contribution

It shows that Faltings' results follow from existing theorems by constructing specific covers, simplifying the understanding of these Diophantine properties.

Findings

Faltings' examples are explained via Evertse and Ferretti's theorem.

Constructing covers embeds complements into semiabelian varieties.

Enables application of Diophantine approximation results.

Abstract

In his contribution to the Baker's Garden book, Faltings gives a family of examples of irreducible divisors $D$ on $\Bbb P^2$ for which $\Bbb P^2\setminus D$ has only finitely many integral points over any given localization of a number ring away from finitely many places. He also notes that neither $\Bbb P^2\setminus D$ nor the \'etale covers used in his proof embed into semiabelian varieties, so his examples do not easily reduce to known results about such subvarieties. In this note, we show how Faltings' results follow directly from a theorem of Evertse and Ferretti; hence these examples can be explained by noting that if one pulls back to a cover of $\Bbb P^2$ \'etale outside of $D$ and then adds components to the pull-back of $D$ then one can embed the complement into a semiabelian variety and obtain useful diophantine approximation results for the original divisor $D$.