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Forum Mathematicum

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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

Issues

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.

Keywords: Vojta’s conjecture; Griffiths’ conjecture; rational surfaces; subspace theorem; Farey sequences

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

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About the article


Received: 2017-04-19

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, DOI: https://doi.org/10.1515/forum-2017-0089.

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