The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties of dimension , by the existence of points in special position with respect to , but general with respect to , 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 .
Le théorème de Castelnuovo-Schottky de Pareschi et Popa caractérise les jacobiennes parmi les variétés abéliennes principalement polarisées indécomposables de dimension , par l’existence de points en position spéciale par rapport à , mais générale par rapport à . 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 finis non nécessairement réduits.
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
@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}, mrnumber = {2961847}, zbl = {1239.14026}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.2665/
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