A conjecture on rational approximations to rational points
Author:
David McKinnon
Journal:
J. Algebraic Geom. 16 (2007), 257-303
DOI:
https://doi.org/10.1090/S1056-3911-06-00458-9
Published electronically:
November 8, 2006
MathSciNet review:
2274515
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Abstract |
References |
Additional Information
Abstract: In this paper, we examine how well a rational point $P$ on an algebraic variety $X$ can be approximated by other rational points. We conjecture that if $P$ lies on a rational curve, then the best approximations to $P$ on $X$ can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta’s Main Conjecture.
References
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References
- Beauville, A., Complex Algebraic Surfaces, Cambridge University Press, 1996. MR 1406314 (97e:14045)
- Drozd, E., “Curves on a nonsingular Del Pezzo Surface in $P^4_k$”, preprint, arXiv.org/math.AG/0410518, 2004.
- Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
- Hosoh, T., “Automorphism groups of quartic del Pezzo surfaces”, J. Algebra 185 (1996), 374–389. MR 1417377 (97i:14026)
- Kovács, S., “The cone of curves of a K3 surface”, Math. Annalen 300 (1994), no. 4, 681–691. MR 1314742 (96a:14044)
- Manin, Yu., Cubic Forms (trans. M. Hazewinkel), Elsevier Science Publishers, 1986. MR 0833513 (87d:11037)
- McKinnon, David, “Counting Rational Points on Ruled Varieties”, Canad. Math. Bull. 47 (2004), no. 2, 264–270. MR 2059421 (2005c:11088)
- Tschinkel, Yu., “Fujita’s Program and Rational Points”, in Higher Dimensional Varieties and Rational Points, Springer-Verlag, 2003. MR 2011749 (2004g:14024)
- Vojta, P., Diophantine Approximations and Value Distribution Theory, Springer Lecture Notes in Mathematics, 1239, Springer-Verlag, 1987. MR 0883451 (91k:11049)
Additional Information
David McKinnon
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 Canada
MR Author ID:
667698
Email:
dmckinnon@math.uwaterloo.ca
Received by editor(s):
May 16, 2005
Received by editor(s) in revised form:
April 24, 2006
Published electronically:
November 8, 2006
Additional Notes:
This research was supported in part by NSERC grant 250196-02
