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On Mumford's construction of degenerating abelian varieties

Valery Alexeev·Iku Nakamura
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Abstract

We prove that a 1-dimnl family of abelian varieties with an ample sheaf defining principal polarization can be canonically compactified (after a finite base change) to a projective family with an ample sheaf. We show that the central fiber (P,L), which we call an SQAV, has a canonical Cartier theta divisor. We give a combinatorial definition for SQAVs and describe their geometrical properties, in particular compute cohomologies of L^n, n\ge0.

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Valery Alexeev, Iku Nakamura. 1999-05-18. On Mumford's construction of degenerating abelian varieties. https://arxiv.org/abs/alg-geom/9608014

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