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Bielliptic abelian surfaces. (English) Zbl 0641.14014

In this paper the following question is considered: Given a polarized abelian surface \((A,H_ A)\) where \(H_ A\) is a polarization of type (1,p), \(p\geq 5\), prime. When is \(H_ A\) very ample? The aim of this paper is to give an answer in terms of period matrices. It turns out that there are precisely two irreducible surfaces \(H_ 1\) and \(H_ 2\) in the moduli space of polarized abelian surfaces where the polarization fails to be very ample. \(H_ 1\) and \(H_ 2\) parametrize the family of split abelian surfaces, resp. bielliptic abelian surfaces. These can be characterized by the existence of a non-trivial involution. The surfaces \(H_ 1\) and \(H_ 2\) are Humbert surfaces whose equations can be given explicitly. These results hold with or without level structure. Finally some geometric properties of bielliptic abelian surfaces are discussed. The starting point of this paper is an article by S. Ramanan [Proc. Lond. Math. Soc., III. Ser. 231-245 (1985; Zbl 0603.14013)].
Reviewer: K.Hulek

MSC:

14K05 Algebraic theory of abelian varieties
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14K10 Algebraic moduli of abelian varieties, classification

Citations:

Zbl 0603.14013
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References:

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