Finite subschemes of abelian varieties and the Schottky problem
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2039-2064.

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 ΓA in special position with respect to 2Θ, 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.

Le théorème de Castelnuovo-Schottky de Pareschi et Popa caractérise les jacobiennes parmi les variétés abéliennes principalement polarisées (A,Θ) indécomposables de dimension g, par l’existence de g+2 points ΓA en position spéciale par rapport à 2Θ, mais générale par rapport à Θ. Il affirme par ailleurs que ces collections de points doivent être contenues dans une courbe d’Abel-Jacobi. En s’appuyant sur les idées contenues dans l’article de Pareschi et Popa, nous donnons ici une preuve autonome qui utilise le point de vue schématique et permet d’étendre le résultat aux sous-schémas Γ finis non nécessairement réduits.

DOI: 10.5802/aif.2665
Classification: 14H42, 14H40, 14K05, 14K99
Keywords: Principally polarized abelian varieties, Jacobians, Schotty problem, finite schemes, Abel-Jacobi curves.
Mot clés : variétés abéliennes principalement polarisées, Jacobiennes, problème de Schottky, schémas finis, courbes d’Abel-Jacobi
Gulbrandsen, Martin G. 1; Lahoz, Martí 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)
@article{AIF_2011__61_5_2039_0,
     author = {Gulbrandsen, Martin G. and Lahoz, Mart{\'\i}},
     title = {Finite subschemes of abelian varieties and the {Schottky} problem},
     journal = {Annales de l'Institut Fourier},
     pages = {2039--2064},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     doi = {10.5802/aif.2665},
     zbl = {1239.14026},
     mrnumber = {2961847},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2665/}
}
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
SP  - 2039
EP  - 2064
VL  - 61
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2665/
DO  - 10.5802/aif.2665
LA  - en
ID  - AIF_2011__61_5_2039_0
ER  - 
%0 Journal Article
%A Gulbrandsen, Martin G.
%A Lahoz, Martí
%T Finite subschemes of abelian varieties and the Schottky problem
%J Annales de l'Institut Fourier
%D 2011
%P 2039-2064
%V 61
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2665/
%R 10.5802/aif.2665
%G en
%F AIF_2011__61_5_2039_0
Gulbrandsen, Martin G.; Lahoz, Martí. Finite subschemes of abelian varieties and the Schottky problem. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2039-2064. doi : 10.5802/aif.2665. http://www.numdam.org/articles/10.5802/aif.2665/

[1] Birkenhake, C.; Lange, H. Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 302, Springer-Verlag, Berlin, 2004 | MR | Zbl

[2] Debarre, O. Fulton-Hansen and Barth-Lefschetz theorems for subvarieties of abelian varieties, J. Reine Angew. Math., Volume 467 (1995), pp. 187-197 | DOI | MR | Zbl

[3] Eisenbud, D.; Green, M.; Harris, J. Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.), Volume 33 (1996) no. 3, pp. 295-324 | DOI | MR | Zbl

[4] Eisenbud, D.; Harris, J. Finite projective schemes in linearly general position, J. Algebraic Geom., Volume 1 (1992) no. 1, pp. 15-30 | MR | Zbl

[5] Eisenbud, D.; Harris, J. An intersection bound for rank 1 loci, with applications to Castelnuovo and Clifford theory, J. Algebraic Geom., Volume 1 (1992) no. 1, pp. 31-60 | MR | Zbl

[6] Griffiths, P.; Harris, J. Residues and zero-cycles on algebraic varieties, Ann. of Math. (2), Volume 108 (1978) no. 3, pp. 461-505 | DOI | MR | Zbl

[7] Grushevsky, S. Erratum to “Cubic equations for the hyperelliptic locus”, Asian J. Math., Volume 9 (2005) no. 2, pp. 273-274 | MR | Zbl

[8] Gunning, R. C. Some curves in abelian varieties, Invent. Math., Volume 66 (1982) no. 3, pp. 377-389 | DOI | MR | Zbl

[9] Krichever, I. Characterizing Jacobians via trisecants of the Kummer Variety, 2006 (arXiv:math/0605625v4 [math.AG]) | arXiv | Zbl

[10] Le Barz, P. Formules pour les trisécantes des surfaces algébriques, Enseign. Math. (2), Volume 33 (1987) no. 1-2, pp. 1-66 | MR | Zbl

[11] Mukai, S. Duality between D(X) and D(X ^) with its application to Picard sheaves, Nagoya Math. J., Volume 81 (1981), pp. 153-175 http://projecteuclid.org/getRecord?id=euclid.nmj/1118786312 | MR | Zbl

[12] Mumford, D. Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., 1975 | MR | Zbl

[13] Pareschi, G.; Popa, M. Castelnuovo theory and the geometric Schottky problem, J. Reine Angew. Math., Volume 615 (2008), pp. 25-44 | DOI | MR | Zbl

[14] Pareschi, G.; Popa, M. Generic vanishing and minimal cohomology classes on abelian varieties, Math. Ann., Volume 340 (2008) no. 1, pp. 209-222 | DOI | MR | Zbl

[15] Ran, Z. On subvarieties of abelian varieties, Invent. Math., Volume 62 (1981) no. 3, pp. 459-479 | DOI | MR | Zbl

[16] Welters, G. E. A criterion for Jacobi varieties, Ann. of Math. (2), Volume 120 (1984) no. 3, pp. 497-504 | DOI | MR | Zbl

Cited by Sources: