## Integral points and Vojta’s conjecture on rational surfaces

HTML articles powered by AMS MathViewer

- by Yu Yasufuku PDF
- Trans. Amer. Math. Soc.
**364**(2012), 767-784 Request permission

## 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.## References

- Enrico Bombieri and Walter Gubler,
*Heights in Diophantine geometry*, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR**2216774**, DOI 10.1017/CBO9780511542879 - P. Corvaja and U. Zannier,
*On integral points on surfaces*, Ann. of Math. (2)**160**(2004), no. 2, 705–726. MR**2123936**, DOI 10.4007/annals.2004.160.705 - Pietro Corvaja and Umberto Zannier,
*A lower bound for the height of a rational function at $S$-unit points*, Monatsh. Math.**144**(2005), no. 3, 203–224. MR**2130274**, DOI 10.1007/s00605-004-0273-0 - G. Faltings,
*Endlichkeitssätze für abelsche Varietäten über Zahlkörpern*, Invent. Math.**73**(1983), no. 3, 349–366 (German). MR**718935**, DOI 10.1007/BF01388432 - Gerd Faltings,
*Diophantine approximation on abelian varieties*, Ann. of Math. (2)**133**(1991), no. 3, 549–576. MR**1109353**, DOI 10.2307/2944319 - William Fulton,
*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037**, DOI 10.1515/9781400882526 - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157** - Marc Hindry and Joseph H. Silverman,
*Diophantine geometry*, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR**1745599**, DOI 10.1007/978-1-4612-1210-2 - Serge Lang,
*Fundamentals of Diophantine geometry*, Springer-Verlag, New York, 1983. MR**715605**, DOI 10.1007/978-1-4757-1810-2 - David McKinnon,
*Vojta’s main conjecture for blowup surfaces*, Proc. Amer. Math. Soc.**131**(2003), no. 1, 1–12. MR**1929015**, DOI 10.1090/S0002-9939-02-06784-9 - Charles F. Osgood,
*A number theoretic-differential equations approach to generalizing Nevanlinna theory*, Indian J. Math.**23**(1981), no. 1-3, 1–15. MR**722894** - Hans Peter Schlickewei,
*Linearformen mit algebraischen koeffizienten*, Manuscripta Math.**18**(1976), no. 2, 147–185. MR**401665**, DOI 10.1007/BF01184304 - Wolfgang M. Schmidt,
*Linear forms with algebraic coefficients. I*, J. Number Theory**3**(1971), 253–277. MR**308061**, DOI 10.1016/0022-314X(71)90001-1 - Joseph H. Silverman,
*Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups*, Monatsh. Math.**145**(2005), no. 4, 333–350. MR**2162351**, DOI 10.1007/s00605-005-0299-y - Paul Vojta,
*Diophantine approximations and value distribution theory*, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR**883451**, DOI 10.1007/BFb0072989 - Yu Yasufuku,
*Vojta’s conjecture on blowups of $\Bbb P^n$, greatest common divisors, and the $abc$ conjecture*, Monatsh. Math.**163**(2011), no. 2, 237–247. MR**2794199**, DOI 10.1007/s00605-010-0242-8 - —,
*Vojta’s conjecutre and blowups*, Ph.D. Dissertation at Brown University.

## 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
- MR Author ID: 681581
- Email: yasufuku@post.harvard.edu, yasufuku@math.cst.nihon-u.ac.jp
- 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
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2846352