On integral points on surfaces
type: Item
instance of: scholarly article
title: On integral points on surfaces
zbMATH Open document ID: 1146.11035
author: Pietro Corvaja
author: Umberto Zannier
published in: Annals of Mathematics. Second Series
publication date: 2005-04-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.annm/1111770732
review text: Let \(k\) be a number field and \(S\) a finite set of places including the archimedean places. The main theorem of the paper imposes geometric conditions on an algebraic surface \(X\) that force all integral solutions \(X(\mathcal{O}_S)\) to be located in a curve \(C \subset X\). Similar work was carried out by the authors in dimension one in [\textit{P.~Corvaja} and \textit{U.~Zannier}, C. R., Math., Acad. Sci. Paris 334, No. 4, 267--271 (2002; Zbl 1012.11051)]. The conditions appear natural when using the Riemann-Roch theorem to find zeros of large order along prescribed divisors. The proofs are not effective, but allow, in principle, quantitative estimations for the degree of the curve \(C\). As an application it is shown that an affine curve with at least five points at infinity has at most finitely many quadratic \(S\)-integral points. Theorem: (Sketch of Main theorem) Let \(\tilde{X}\) be a geometrically irreducible projective surface defined over \(k\), and let \(X\) be an affine open set. Assume that \(\tilde{X} - X=D_1 \cup D_2 \cup \dots \cup D_r\), \(r \geq 2\), where no three of the \(D_i\) shares a common point and also the intersection matrix of the \(D_i\)'s satisfies certain conditions, then there is a curve on \(C \subset X\) containing \(X(\mathcal{O}_S)\). The first thing to note is that the main theorem will follow if it can be proved that for every infinite sequence of integral points on \(X\), there is a curve defined over \(k\) and containing an infinite subsequence. Taking a sequence of \(S\)-integral points \(P_i\) we may assume that for each valuation \(v \in S\), the points \(P_i^v\) converge to \(P^v \in \tilde{X}(k_v)\). The conditions on the intersection matrix for the divisors \(D_i\) allow to build a divisor \(D=\sum_i p_i D_i\) and a vector space \(V=V_N=\{\varphi \in k(X): \text{Div}(\varphi) + ND \geq 0 \}\) with a basis \(\varphi_1,\dots,\varphi_d\). For every \(v \in S\) the authors build linearly independent linear forms \(L_{1v…
reviewed by: Jorge Pineiro
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 2157788
MaRDI profile type: Publication
OpenAlex ID: W2950679619
DOI: 10.4007/ANNALS.2004.160.705
description: scientific article; zbMATH DE number 2157788