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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^{n}A$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^{n}A$ carries a divisor with vanishing Beauville-Bogomolov square.

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...