English

Transplanting Faltings' garden

Number Theory 2009-01-15 v1 Algebraic Geometry

Abstract

In his contribution to the Baker's Garden book, Faltings gives a family of examples of irreducible divisors DD on P2\Bbb P^2 for which P2D\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 P2D\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 P2\Bbb P^2 \'etale outside of DD and then adds components to the pull-back of DD then one can embed the complement into a semiabelian variety and obtain useful diophantine approximation results for the original divisor DD.

Cite

@article{arxiv.0901.2106,
  title  = {Transplanting Faltings' garden},
  author = {Paul Vojta},
  journal= {arXiv preprint arXiv:0901.2106},
  year   = {2009}
}

Comments

11 pages

R2 v1 2026-06-21T12:00:55.941Z