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Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen — 2007

Bulletin de la Société Mathématique de France

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.

Finite subschemes of abelian varieties and the Schottky problem

Martin G. GulbrandsenMartí Lahoz — 2011

Annales de l’institut Fourier

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 Γ A in special position with respect to 2 Θ , 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...

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