Seshadri constants and interpolation on commutative algebraic groups

Stéphane Fischler[1]; Michael Nakamaye[2]

  • [1] Univ Paris-Sud Laboratoire de Mathématiques d’Orsay CNRS, F-91405 Orsay (France)
  • [2] Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131 (U.S.A.)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 3, page 1269-1289
  • ISSN: 0373-0956

Abstract

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In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.

How to cite

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Fischler, Stéphane, and Nakamaye, Michael. "Seshadri constants and interpolation on commutative algebraic groups." Annales de l’institut Fourier 64.3 (2014): 1269-1289. <http://eudml.org/doc/275541>.

@article{Fischler2014,
abstract = {In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.},
affiliation = {Univ Paris-Sud Laboratoire de Mathématiques d’Orsay CNRS, F-91405 Orsay (France); Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131 (U.S.A.)},
author = {Fischler, Stéphane, Nakamaye, Michael},
journal = {Annales de l’institut Fourier},
keywords = {Interpolation estimate; Seshadri constant; ample line bundle; commutative algebraic group; obstruction subgroup; Seshadri exceptional subvariety; interpolation estimate},
language = {eng},
number = {3},
pages = {1269-1289},
publisher = {Association des Annales de l’institut Fourier},
title = {Seshadri constants and interpolation on commutative algebraic groups},
url = {http://eudml.org/doc/275541},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Fischler, Stéphane
AU - Nakamaye, Michael
TI - Seshadri constants and interpolation on commutative algebraic groups
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 1269
EP - 1289
AB - In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.
LA - eng
KW - Interpolation estimate; Seshadri constant; ample line bundle; commutative algebraic group; obstruction subgroup; Seshadri exceptional subvariety; interpolation estimate
UR - http://eudml.org/doc/275541
ER -

References

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  1. Christina Birkenhake, Herbert Lange, Complex abelian varieties, 302 (2004), Springer-Verlag, Berlin Zbl0779.14012MR2062673
  2. Frédéric Campana, Thomas Peternell, Algebraicity of the ample cone of projective varieties, J. Reine Angew. Math. 407 (1990), 160-166 Zbl0728.14004MR1048532
  3. Lawrence Ein, Oliver Küchle, Robert Lazarsfeld, Local positivity of ample line bundles, J. Differential Geom. 42 (1995), 193-219 Zbl0866.14004MR1366545
  4. S. Fischler, Interpolation on algebraic groups, Compos. Math. 141 (2005), 907-925 Zbl1080.14054MR2148195
  5. S. Fischler, M. Nakamaye, Connecting interpolation and multiplicity estimates in commutative algebraic groups Zbl06587939
  6. William Fulton, Intersection theory, 2 (1998), Springer-Verlag, Berlin Zbl0541.14005MR1644323
  7. Robin Hartshorne, Algebraic geometry, (1977), Springer-Verlag, New York-Heidelberg Zbl0531.14001MR463157
  8. Yujiro Kawamata, Katsumi Matsuda, Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985 10 (1987), 283-360, North-Holland, Amsterdam Zbl0672.14006MR946243
  9. F. Knop, H. Lange, Some remarks on compactifications of commutative algebraic groups, Comment. Math. Helv. 60 (1985), 497-507 Zbl0587.14030MR826869
  10. Robert Lazarsfeld, Positivity in algebraic geometry. I and II, 48, 49 (2004), Springer-Verlag, Berlin Zbl1093.14500MR2095471
  11. D. W. Masser, Interpolation on group varieties, Approximations diophantiennes et nombres transcendants (Luminy, 1982) 31 (1983), 151-171, Birkhäuser, Boston, Mass. Zbl0579.14038MR702196
  12. D. W. Masser, G. Wüstholz, Zero estimates on group varieties. I, Invent. Math. 64 (1981), 489-516 Zbl0467.10025MR632987
  13. David Mumford, Abelian varieties, (1970), Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London Zbl0223.14022MR282985
  14. Michael Nakamaye, Multiplicity estimates and the product theorem, Bull. Soc. Math. France 123 (1995), 155-188 Zbl0841.11037MR1340286
  15. Michael Nakamaye, Seshadri constants at very general points, Trans. Amer. Math. Soc. 357 (2005), 3285-3297 Zbl1084.14008MR2135747
  16. Michael Nakamaye, Multiplicity estimates on commutative algebraic groups, J. Reine Angew. Math. 607 (2007), 217-235 Zbl1162.11037MR2338124
  17. Michael Nakamaye, Multiplicity estimates, interpolation, and transcendence theory, Number theory, analysis and geometry: In Memory of Serge Lang, D. Goldfeld et al. (ed) (2012), 475-498, Springer, New York Zbl1268.11097MR2867930
  18. Michael Nakamaye, Nicolas Ratazzi, Lemmes de multiplicités et constante de Seshadri, Math. Z. 259 (2008), 915-933 Zbl1156.11027MR2403749
  19. Patrice Philippon, Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France 114 (1986), 355-383 Zbl0617.14001MR878242
  20. J.-P. Serre, Quelques propriétés des groupes algébriques commutatifs, Astérisque (1978), 191-202 
  21. M. Waldschmidt, Dépendance de logarithmes dans les groupes algébriques, Approximations diophantiennes et nombres transcendants (Luminy, 1982) 31 (1983), 289-328, Birkhäuser, Boston, Mass. Zbl0513.14028MR702187
  22. Michel Waldschmidt, La transformation de Fourier-Borel : une dualité en transcendance 
  23. Michel Waldschmidt, Fonctions auxiliaires et fonctionnelles analytiques. I, II, J. Analyse Math. 56 (1991), 231-254, 255–279 Zbl0742.11036MR1243105
  24. Michel Waldschmidt, Diophantine approximation on linear algebraic groups, 326 (2000), Springer-Verlag, Berlin Zbl0944.11024MR1756786

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