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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

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Volume 30, Issue 3

# Vojta’s conjecture on rational surfaces and the abc conjecture

Yu Yasufuku
• Corresponding author
• Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
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Published Online: 2017-08-30 | DOI: https://doi.org/10.1515/forum-2017-0089

## Abstract

We prove Vojta’s conjecture for some rational surfaces. Moreover, for similar but different rational surfaces, we show that their Vojta’s conjecture is related to the abc conjecture. More specifically, we prove that Vojta’s conjecture on these surfaces implies a special case of the abc conjecture, while the abc conjecture implies Vojta’s conjecture on these surfaces. The argument carries over to the holomorphic case, so we unconditionally obtain Griffiths’ conjecture for the same situation. To prove these results, we prove and use some properties of Farey sequences.

MSC 2010: 11J97; 32H30; 14G25; 11B57; 11J87; 14G40; 14J26

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Revised: 2017-08-01

Published Online: 2017-08-30

Published in Print: 2018-05-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 15K17522

Funding Source: Nihon University

Award identifier / Grant number: 15K17522

Supported in part by JSPS Grant-in-Aid 15K17522 and by Nihon University College of Science and Technology Grant-in-Aid for Fundamental Science Research.

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 631–649, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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