Secant varieties and successive minima
Author:
Christophe Soulé
Translated by:
Journal:
J. Algebraic Geom. 13 (2004), 323341
Published electronically:
December 4, 2003
MathSciNet review:
2047701
Fulltext PDF
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.
 [A]
A. Arakelov: Intersection theory of divisors on an arithmetic surface, Math. USSR, Izv. 8, 1974, 11671180.
 [ACGH]
E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris: Geometry of Algebraic Curves, Vol. I, 1985, SpringerVerlag.
 [B]
A. Bertram: Moduli of rank vector bundles, theta divisors, and the geometry of curves in projective space, J. Diff. Geom. 35, 1992, 429469.
 [BHP]
R.C. Barker, G. Harman, J. Pintz: The difference between consecutive primes, II, Proc. London Math. Soc. 83, 2001, 532562.
 [BoVa]
E. Bombieri, J. Vaaler: On Siegel's lemma, Invent. Math. 73, 1983, 1132.
 [BGS]
J.B. Bost, H. Gillet, C. Soulé: Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7, 1994, 9031027.
 [E]
R. Elkik: Fonctions de Green, Volumes de Faltings, Application aux surfaces arithmétiques, Astérisque 127, 1985, 89112.
 [I]
A. Ivic: The Riemann zetafunction. The theory of the Riemann zetafunction with applications, A WileyInterscience Publication, New York, 1985, John Wiley & Sons.
 [N]
J. Nagura: On the interval containing at least one prime number, Proc. Japan Acad. 28, 1952, 177181.
 [R]
P. Ribenboim: The book of prime number records, New York, SpringerVerlag, 1988.
 [S]
C. Soulé: A vanishing theorem on arithmetic surfaces, Invent. Math. 116, 1994, 577599.
 [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, 569587.
 [A]
 A. Arakelov: Intersection theory of divisors on an arithmetic surface, Math. USSR, Izv. 8, 1974, 11671180.
 [ACGH]
 E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris: Geometry of Algebraic Curves, Vol. I, 1985, SpringerVerlag.
 [B]
 A. Bertram: Moduli of rank vector bundles, theta divisors, and the geometry of curves in projective space, J. Diff. Geom. 35, 1992, 429469.
 [BHP]
 R.C. Barker, G. Harman, J. Pintz: The difference between consecutive primes, II, Proc. London Math. Soc. 83, 2001, 532562.
 [BoVa]
 E. Bombieri, J. Vaaler: On Siegel's lemma, Invent. Math. 73, 1983, 1132.
 [BGS]
 J.B. Bost, H. Gillet, C. Soulé: Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7, 1994, 9031027.
 [E]
 R. Elkik: Fonctions de Green, Volumes de Faltings, Application aux surfaces arithmétiques, Astérisque 127, 1985, 89112.
 [I]
 A. Ivic: The Riemann zetafunction. The theory of the Riemann zetafunction with applications, A WileyInterscience Publication, New York, 1985, John Wiley & Sons.
 [N]
 J. Nagura: On the interval containing at least one prime number, Proc. Japan Acad. 28, 1952, 177181.
 [R]
 P. Ribenboim: The book of prime number records, New York, SpringerVerlag, 1988.
 [S]
 C. Soulé: A vanishing theorem on arithmetic surfaces, Invent. Math. 116, 1994, 577599.
 [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, 569587.
Additional Information
Christophe Soulé
Affiliation:
IHES, Le Bois Marie, 35 route de Chartres, F91440 BuressurYvette, France
Email:
soule@ihes.fr
DOI:
http://dx.doi.org/10.1090/S1056391103003515
PII:
S 10563911(03)003515
Received by editor(s):
November 20, 2001
Published electronically:
December 4, 2003
