Finite subschemes of abelian varieties and the Schottky problem

Gulbrandsen, Martin G.; Lahoz, Martí

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2039-2064.

Le théorème de Castelnuovo-Schottky de Pareschi et Popa caractérise les jacobiennes parmi les variétés abéliennes principalement polarisées $(A,\Theta )$ indécomposables de dimension ${\displaystyle g}$, par l’existence de ${\displaystyle g+2}$ points $\Gamma \subset A$ en position spéciale par rapport à $2\Theta$, mais générale par rapport à $\Theta$. 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 $\Gamma$ finis non nécessairement réduits.

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 $\Gamma$.

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