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Integral points and Vojta's conjecture on rational surfaces


Author: Yu Yasufuku
Journal: Trans. Amer. Math. Soc. 364 (2012), 767-784
MSC (2010): Primary 11J97, 14G40; Secondary 14J26
DOI: https://doi.org/10.1090/S0002-9947-2011-05320-1
Published electronically: September 15, 2011
MathSciNet review: 2846352
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Abstract: Using an inequality by Corvaja and Zannier about gcd's of polynomials in $ S$-units, we verify Vojta's conjecture (with respect to integral points) for rational surfaces and triangular divisors. This amounts to a gcd inequality for integral points on $ \mathbb{G}_m^2$. The argument in the proof is generalized to give conditions under which Vojta's conjecture on a variety implies Vojta's conjecture on its blowup.


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Additional Information

Yu Yasufuku
Affiliation: Department of Mathematics, CUNY-Graduate Center, 365 Fifth Avenue, New York, New York 10016
Address at time of publication: Department of Mathematics, Nihon University, 1-8-14 Kanda-Surugadai, Tokyo 101-8308, Japan
Email: yasufuku@post.harvard.edu, yasufuku@math.cst.nihon-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2011-05320-1
Keywords: Vojta’s conjecture, rational surfaces, integral points, blowups, multiplicative groups, greatest common divisors, Farey sequences
Received by editor(s): June 19, 2009
Received by editor(s) in revised form: November 26, 2009, January 26, 2010, and February 13, 2010
Published electronically: September 15, 2011
Additional Notes: This work was supported in part by NSF VIGRE grant number DMS-9977372
Article copyright: © Copyright 2011 American Mathematical Society

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