Finite subschemes of abelian varieties and the Schottky problem

Martin G. Gulbrandsen[1]; Martí Lahoz[2]

  • [1] Stord/Haugesund University College, Bjørnsons gate 45 NO-5528 Haugesund (Norway)
  • [2] Universitat de Barcelona Departament d’Àlgebra i Geometria Gran Via, 585, 08007 Barcelona (Spain)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 2039-2064
  • ISSN: 0373-0956

Abstract

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The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties ( A , Θ ) of dimension g , by the existence of g + 2 points Γ ⊂ A in special position with respect to 2 Θ , but general with respect to Θ , and furthermore states that such collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly nonreduced subschemes Γ .

How to cite

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Gulbrandsen, Martin G., and Lahoz, Martí. "Finite subschemes of abelian varieties and the Schottky problem." Annales de l’institut Fourier 61.5 (2011): 2039-2064. <http://eudml.org/doc/219804>.

@article{Gulbrandsen2011,
abstract = {The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties $(A, \Theta )$ of dimension $g$, by the existence of $g + 2$ points $\Gamma \subset A$ in special position with respect to $2\Theta $, but general with respect to $\Theta $, and furthermore states that such collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly nonreduced subschemes $\Gamma $.},
affiliation = {Stord/Haugesund University College, Bjørnsons gate 45 NO-5528 Haugesund (Norway); Universitat de Barcelona Departament d’Àlgebra i Geometria Gran Via, 585, 08007 Barcelona (Spain)},
author = {Gulbrandsen, Martin G., Lahoz, Martí},
journal = {Annales de l’institut Fourier},
keywords = {Principally polarized abelian varieties; Jacobians; Schotty problem; finite schemes; Abel-Jacobi curves; principally polarized abelian varieties},
language = {eng},
number = {5},
pages = {2039-2064},
publisher = {Association des Annales de l’institut Fourier},
title = {Finite subschemes of abelian varieties and the Schottky problem},
url = {http://eudml.org/doc/219804},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Gulbrandsen, Martin G.
AU - Lahoz, Martí
TI - Finite subschemes of abelian varieties and the Schottky problem
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 2039
EP - 2064
AB - The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties $(A, \Theta )$ of dimension $g$, by the existence of $g + 2$ points $\Gamma \subset A$ in special position with respect to $2\Theta $, but general with respect to $\Theta $, and furthermore states that such collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly nonreduced subschemes $\Gamma $.
LA - eng
KW - Principally polarized abelian varieties; Jacobians; Schotty problem; finite schemes; Abel-Jacobi curves; principally polarized abelian varieties
UR - http://eudml.org/doc/219804
ER -

References

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  12. D. Mumford, Curves and their Jacobians, (1975), The University of Michigan Press, Ann Arbor, Mich. Zbl0316.14010MR419430
  13. G. Pareschi, M. Popa, Castelnuovo theory and the geometric Schottky problem, J. Reine Angew. Math. 615 (2008), 25-44 Zbl1142.14030MR2384330
  14. G. Pareschi, M. Popa, Generic vanishing and minimal cohomology classes on abelian varieties, Math. Ann. 340 (2008), 209-222 Zbl1131.14049MR2349774
  15. Z. Ran, On subvarieties of abelian varieties, Invent. Math. 62 (1981), 459-479 Zbl0474.14016MR604839
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