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Vojta’s conjecture on rational surfaces and the \(abc\) conjecture. (English) Zbl 1452.11090
In the first of the three main theorems in the paper under review, the author considers three lines \(L_1 , L_2 , L_3\) of \({\mathbb{P}}^2\) defined over \(\overline{\mathbb{Q}}\) in general position. Let \(X_1\) be the blowup of \({\mathbb{P}}^2\) at a point defined over \(\overline{\mathbb{Q}}\) in \(L_1 \setminus (L_2 \cup L_3)\), with \(E_1\) as the exceptional divisor. For \(n \ge 2\), construct \(X_n\) inductively by blowing up \(X_{n-1}\) at (the unique) point of \(E_{n-1} \cap \widetilde{L}_1\), obtaining the exceptional divisor \(E_n\). Then Vojta’s conjecture holds for \(X_n\) with respect to the divisor \[ \widetilde{L}_1 +\widetilde{L}_2 +\widetilde{L}_3 +\widetilde{E}_1 +\cdots+\widetilde{E}_{n-1} +E_n. \] The author remarks that the special case of \(X_1\) had been treated in his earlier work [Monatsh. Math. 163, No. 2, 237–247 (2011; Zbl 1282.11086)]. The proof of this first result is based on Ridout’s Theorem.
For his second main result, the author considers the case of multiple blowups, where he starts from the same \(X_1\), but blows up at a point not in \( \widetilde{L}_1\) at least once. He shows that Vojta’s Conjecture for this situation implies a special case of the \(abc\) conjecture. In the third theorem, he discusses the implication in the other direction. One key ingredient in his proof is a new auxiliary lemma on Farey series organized in the Stern-Brocot tree.
The author points out that his arguments for proving the first and third results carry over to Nevanlinna theory; this enables him to obtain new cases of Griffiths’s conjecture.
MSC:
11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.)
32H30 Value distribution theory in higher dimensions
14G25 Global ground fields in algebraic geometry
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
11J87 Schmidt Subspace Theorem and applications
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14J26 Rational and ruled surfaces
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