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  Journal of Algebraic Geometry
Journal of Algebraic Geometry
  
Online ISSN 1534-7486; Print ISSN 1056-3911
 

Secant varieties and successive minima


Author: Christophe Soulé
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 323-341
Published electronically: December 4, 2003
MathSciNet review: 2047701
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Abstract | References | Additional Information

Abstract: Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C. Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A. Granville on the divisibility properties of binomial coefficients in a given line of Pascal's triangle.


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

  • [A] A. Arakelov: Intersection theory of divisors on an arithmetic surface, Math. USSR, Izv. 8, 1974, 1167-1180.
  • [ACGH] E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris: Geometry of Algebraic Curves, Vol. I, 1985, Springer-Verlag.
  • [B] A. Bertram: Moduli of rank $2$ vector bundles, theta divisors, and the geometry of curves in projective space, J. Diff. Geom. 35, 1992, 429-469.
  • [BHP] R.C. Barker, G. Harman, J. Pintz: The difference between consecutive primes, II, Proc. London Math. Soc. 83, 2001, 532-562.
  • [Bo-Va] E. Bombieri, J. Vaaler: On Siegel's lemma, Invent. Math. 73, 1983, 11-32.
  • [BGS] J.-B. Bost, H. Gillet, C. Soulé: Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7, 1994, 903-1027.
  • [E] R. Elkik: Fonctions de Green, Volumes de Faltings, Application aux surfaces arithmétiques, Astérisque 127, 1985, 89-112.
  • [I] A. Ivic: The Riemann zeta-function. The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, New York, 1985, John Wiley & Sons.
  • [N] J. Nagura: On the interval containing at least one prime number, Proc. Japan Acad. 28, 1952, 177-181.
  • [R] P. Ribenboim: The book of prime number records, New York, Springer-Verlag, 1988.
  • [S] C. Soulé: A vanishing theorem on arithmetic surfaces, Invent. Math. 116, 1994, 577-599.
  • [V] C. Voisin: Appendix to ``Secant varieties and successive minima" by C. Soulé: On linear subspaces contained in the secant varieties of a projective curve, J. Alg. Geom., this volume.
  • [Z] S. Zhang: Positive line bundles on arithmetic surfaces, Annals of Maths. 136, 1992, 569-587.


Additional Information

Christophe Soulé
Affiliation: IHES, Le Bois Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, France
Email: soule@ihes.fr

DOI: http://dx.doi.org/10.1090/S1056-3911-03-00351-5
PII: S 1056-3911(03)00351-5
Received by editor(s): November 20, 2001
Published electronically: December 4, 2003


Journal of Algebraic Geometry
The Journal of Algebraic Geometry
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