Integral points on threefolds and other varieties
type: Item
instance of: scholarly article
title: Integral points on threefolds and other varieties
zbMATH Open document ID: 1250.11066
author: Pietro Corvaja
author: Aaron Levin
author: Umberto Zannier
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2010-03-30T00:00:00Z
review text: Let \(k\) be an algebraic number field and \(S\) denote a finite set of places of \(k\), including the Archimedean ones. Let \(\tilde{\mathcal X}\) be a projective nonsingular irreducible variety of dimension \(q\) defined over \(k\), \(D_1, \dots, D_r\) be ample divisors such that at most \(q\) of them contain a given point, and put \(\mathcal X = \tilde {\mathcal X} \setminus \bigcup_{i=1}^r D_i\). For \(r\) large enough, one wants to show that no set of \(S\)-integral points of \(\mathcal X\) is Zariski-dense, and that there exists a proper Zariski-closed subset \(\mathcal Y \subset \mathcal X\), independent of \(k\) and \(S\), such that \(\mathcal X \setminus \mathcal Y\) contains only finitely many \(S\)-integral points of \(\mathcal X\). For \(q=2\) this was proved by the \textit{authors} [Ann. Math. (2) 160, No. 2, 705--726 (2004; Zbl 1146.11035)] and [Ann. Math. (2) 170, No. 2, 609--655 (2009; Zbl 1250.11067)], and for \(q \geq 2\) and \(r \geq q^2\) by \textit{P. Autissier} [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 2, 221--239 (2009; Zbl 1173.14016)]. In the paper under review, the latter result is now generalized for \(r > q^2-q\) assuming that \(q > 2\) and \(\sum_{i=1}^r D_i\) is a reduced normal crossing divisor. For the proof, the authors develop some new idea to overcome the problem, when an \(S\)-integral point lies on 3 or more divisors \(D_i\), and then apply Schmidt's subspace theorem.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11J87
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 5687945
zbMATH Keywords: Schmidt Subspace Theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1264084501
OpenAlex ID: W2044940852
cites work: Géométrie, points entiers et courbes entières
cites work: On the number of integral points on algebraic curves
cites work: On integral points on surfaces
cites work: Higher-dimensional algebraic geometry
cites work: Generalizations of Siegel's and Picard's theorems
cites work: A Refinement of Schmidt's Subspace Theorem
cites work: Integral points on subvarieties of semiabelian varieties. I
cites work: On Cartan's theorem and Cartan's conjecture
DOI: 10.2748/TMJ/1264084501
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description: scientific article; zbMATH DE number 5687945