2009/01/14 by Paul Vojta, Vojta, Paul
Computer Science · Mathematics · #11G35 #11J68 (primary) #11J97 #14G05 (secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:11G35 #msc:11J68 #msc:11J97 #msc:14G05
paper · pdf · doi:10.48550/arxiv.0901.2106
11 pages
arxiv created 2009/01/14 · openalex publication_date 2009/01/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his contribution to the Baker's Garden book, Faltings gives a family of examples of irreducible divisors D on \Bbb P2 for which \Bbb P2∖ 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 P2∖ D nor the étale 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 P2 étale 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.