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Martin G. Gulbrandsen; Martí Lahoz
Finite subschemes of abelian varieties and the Schottky problem
(Sous-schémas finis de variétés abéliennes et le problème de Schottky)
Annales de l'institut Fourier, 61 no. 5 (2011), p. 2039-2064, doi: 10.5802/aif.2665
Article PDF | Reviews MR 2961847 | Zbl 1239.14026
Class. Math.: 14H42, 14H40, 14K05, 14K99
Keywords: Principally polarized abelian varieties, Jacobians, Schotty problem, finite schemes, Abel-Jacobi curves.

Résumé - 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 $.

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