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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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