Vojta's conjecture on blowups of \({\mathbb{P}^n}\), greatest common divisors, and the \(abc\) conjecture
type: Item
instance of: scholarly article
title: Vojta's conjecture on blowups of \({\mathbb{P}^n}\), greatest common divisors, and the \(abc\) conjecture
zbMATH Open document ID: 1282.11086
author: Yu Yasufuku
published in: Monatshefte für Mathematik
publication date: 2011-06-03T00:00:00Z
review text: \textit{J. H. Silverman} [Monatsh. Math. 145, No. 4, 333--350 (2005; Zbl 1197.11070)] showed that a gcd inequality due to \textit{P. Corvaja} and \textit{U. Zannier} [Monatsh. Math. 144, No. 3, 203--224 (2005; Zbl 1086.11035)] would in fact follow from a conjecture of the reviewer, applied to the set of integral points on the blowing-up of \(\mathbb P^2\) at one point not lying on any of the coordinate axes, relative to the divisor consisting of the pull-backs of the coordinate axes, plus the exceptional divisor of the blowing-up. The present paper examines the reviewer's conjecture in the more general situation in which all rational points are considered, but in which the blown-up subvariety is a linear subspace contained in exactly one of the coordinate hyperplanes: Let \(k\) be a number field. Let \(Y\subseteq\mathbb P^n\) be a linear subvariety of codimension 2. Let \(H_0,\dots,H_n\) be linearly independent hyperplanes in \(\mathbb P^n\), defined over \(\overline k\), such that \(H_n\) contains \(Y\) but none of the other \(H_i\) do. Let \(X'\) be the blowing-up of \(\mathbb P^n\) along \(Y\) and let \(D\) be the divisor \(D=\widetilde H_1+\dots+\widetilde H_n+E\), where \(\widetilde{\text{ }}\) denotes the strict transform and \(E\) is the exceptional divisor of the blowing-up. Then the reviewer's conjecture holds for rational points on \(X'\) relative to the divisor \(D\). As a special case, the following gcd inequality is noted: For all \(\epsilon>0\) the inequality \[ \gcd(x_1+\dots+x_{n-1}-1,x_n) < C\max(|x_1|,\dots,|x_n|)^\varepsilon|x_1\dotsm x_n| \] holds for all \((x_1,\dots,x_n)\in\mathbb Z^n\) outside of a proper Zariski-closed subset of \(\mathbb A^n\). When \(n=2\), the above theorem also holds for the blowing-up of \(\mathbb P^2\) at finitely many points lying on \(H_3\setminus((H_3\cap H_1)\cup(H_3\cap H_2))\). These results are proved by mapping the blowup into a larger projective space and applying Schmidt's Subspace Theorem (method of Corvaja, …
reviewed by: Paul Vojta
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 11J87
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 5903713
zbMATH Keywords: Vojta's conjecture
zbMATH Keywords: Schmidt's subspace theorem
zbMATH Keywords: greatest common divisor
zbMATH Keywords: abc conjecture
MaRDI profile type: Publication
full work available at URL: s00605-010-0242-8
OpenAlex ID: W2021727703
Wikidata QID: Q122899652
cites work: Q5719099
cites work: A lower bound for the height of a rational function at \(S\)-unit points
cites work: Q3338318
cites work: Q4143433
cites work: Vojta’s Main Conjecture for blowup surfaces
cites work: Linearformen mit algebraischen Koeffizienten
cites work: Linear forms with algebraic coefficients. I
cites work: Generalized greatest common divisors, divisibility sequences, and Vojta's conjecture for blowups
cites work: Diophantine approximations and value distribution theory
cites work: On the abc conjecture and diophantine approximation by rational points
cites work: Integral points and Vojta’s conjecture on rational surfaces
DOI: 10.1007/S00605-010-0242-8
description: scientific article; zbMATH DE number 5903713