Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
Full-text PDF

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 [Enhancements On Off] (What's this?)

  • [Be] Beauville, A., Complex Algebraic Surfaces, Cambridge University Press, 1996. MR 1406314 (97e:14045)
  • [Dr] Drozd, E., ``Curves on a nonsingular Del Pezzo Surface in $ P^4_k$'', preprint, arXiv.org/math.AG/0410518, 2004.
  • [Ha] Hartshorne, R., Algebraic Geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [Ho] Hosoh, T., ``Automorphism groups of quartic del Pezzo surfaces'', J. Algebra 185 (1996), 374-389. MR 1417377 (97i:14026)
  • [Ko] Kovács, S., ``The cone of curves of a K3 surface'', Math. Annalen 300 (1994), no. 4, 681-691. MR 1314742 (96a:14044)
  • [Ma] Manin, Yu., Cubic Forms (trans. M. Hazewinkel), Elsevier Science Publishers, 1986. MR 0833513 (87d:11037)
  • [M1] McKinnon, David, ``Counting Rational Points on Ruled Varieties'', Canad. Math. Bull. 47 (2004), no. 2, 264-270. MR 2059421 (2005c:11088)
  • [Tsch] Tschinkel, Yu., ``Fujita's Program and Rational Points'', in Higher Dimensional Varieties and Rational Points, Springer-Verlag, 2003. MR 2011749 (2004g:14024)
  • [Vo] 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
Email: dmckinnon@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S1056-3911-06-00458-9
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

American Mathematical Society