Vojta's conjecture on blowups of \({\mathbb{P}^n}\), greatest common divisors, and the \(abc\) conjecture
\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, Zannier, Evertse, and Ferretti). In addition, it is also shown how the reviewer's conjecture, even on very simple varieties, already is closely related to the abc conjecture: Let \(D\) be a divisor consisting of three lines on \(\mathbb P^2\) not all passing through a point. Let \(X^{(1)}\) be the blowing-up of \(\mathbb P^2\) at a smooth point of \(D\), and let \(E^{(1)}\) be the exceptional divisor. Let \(X=X^{(2)}\) be the blowing-up of \(X^{(1)}\) along a point of \(E^{(1)}\) not lying on the strict transform of \(D\), and let \(E^{(2)}\) be the exceptional divisor. Then the reviewer's conjecture on \(X\) with the divisor \(\widetilde D+\widetilde{E^{(1)}}+E^{(2)}\) implies the special case of the abc conjecture in which \(a\) is an \(S\)-unit integer, \(b=1-a\), and \(c=1\).
scientific article; zbMATH DE number 5903713
scientific article