Secant varieties and successive minima

Author:
Christophe Soulé

Journal:
J. Algebraic Geom. **13** (2004), 323-341

DOI:
https://doi.org/10.1090/S1056-3911-03-00351-5

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.

[A]A A. Arakelov: Intersection theory of divisors on an arithmetic surface, *Math. USSR, Izv.* **8**, 1974, 1167-1180.
[ACGH]ACGH E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris: Geometry of Algebraic Curves, Vol. I, 1985, Springer-Verlag.
[B]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]BHP R.C. Barker, G. Harman, J. Pintz: The difference between consecutive primes, II, *Proc. London Math. Soc.* **83**, 2001, 532-562.
[Bo-Va]BoVa E. Bombieri, J. Vaaler: On Siegel’s lemma, *Invent. Math.* **73**, 1983, 11-32.
[BGS]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]E R. Elkik: Fonctions de Green, Volumes de Faltings, Application aux surfaces arithmétiques, *Astérisque* **127**, 1985, 89-112.
[I]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]N J. Nagura: On the interval containing at least one prime number, *Proc. Japan Acad.* **28**, 1952, 177-181.
[R]R P. Ribenboim: The book of prime number records, New York, Springer-Verlag, 1988.
[S]S C. Soulé: A vanishing theorem on arithmetic surfaces, *Invent. Math.* **116**, 1994, 577-599.
[V]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]Z S. Zhang: Positive line bundles on arithmetic surfaces, *Annals of Maths.* **136**, 1992, 569-587.

[A]A A. Arakelov: Intersection theory of divisors on an arithmetic surface, *Math. USSR, Izv.* **8**, 1974, 1167-1180.
[ACGH]ACGH E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris: Geometry of Algebraic Curves, Vol. I, 1985, Springer-Verlag.
[B]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]BHP R.C. Barker, G. Harman, J. Pintz: The difference between consecutive primes, II, *Proc. London Math. Soc.* **83**, 2001, 532-562.
[Bo-Va]BoVa E. Bombieri, J. Vaaler: On Siegel’s lemma, *Invent. Math.* **73**, 1983, 11-32.
[BGS]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]E R. Elkik: Fonctions de Green, Volumes de Faltings, Application aux surfaces arithmétiques, *Astérisque* **127**, 1985, 89-112.
[I]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]N J. Nagura: On the interval containing at least one prime number, *Proc. Japan Acad.* **28**, 1952, 177-181.
[R]R P. Ribenboim: The book of prime number records, New York, Springer-Verlag, 1988.
[S]S C. Soulé: A vanishing theorem on arithmetic surfaces, *Invent. Math.* **116**, 1994, 577-599.
[V]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]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

Received by editor(s):
November 20, 2001

Published electronically:
December 4, 2003