Vojta’s conjecture on rational surfaces and the \(abc\) conjecture.

*(English)*Zbl 1452.11090In 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.

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.

Reviewer: Michel Waldschmidt (Paris)

##### 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 |

##### Keywords:

Vojta’s conjecture; Griffiths’ conjecture; rational surfaces; subspace theorem; \(abc\) conjecture; Farey sequences
Full Text:
DOI

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