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Abelian surfaces with (1,2)-polarization. (English) Zbl 0639.14023

Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 41-84 (1987).
[For the entire collection see Zbl 0628.00007.]
The study of certain integrable Hamiltonian systems coming from geodesic flows on SO(4) made by M. Adler and P. van Moerbeke [Proc. Natl. Acad. Sci. USA 81, 4613-4616 (1984; Zbl 0545.58027)] and by L. Haine [Math. Ann. 263, 435-472 (1983; Zbl 0521.58042)] leads to affine complete intersection surfaces in \({\mathbb{C}}^ 6,\) which are affine parts of abelian surfaces, hence singular at infinity when extended to \({\mathbb{P}}^ 6.\) The corresponding smooth projective model is an abelian surface \(A\subset {\mathbb{P}}^ 7 \)of degree 16, whose hyperplane sections provide a (2,4) polarization. The paper under review is devoted to the study of quadratic equations for abelian surfaces embedded in \({\mathbb{P}}^ 7 \)by a linear system corresponding to a (2,4)-polarization. The author determines explicitly the quadratic equations for \(A\subset {\mathbb{P}}^ 7 \)in terms of the symmetries of a suitable Heisenberg group. This also allows him to identify the coarse moduli space for such abelian surfaces with an open subset of \({\mathbb{P}}\) \(1\times {\mathbb{P}}\) \(1\times {\mathbb{P}}^ 1.\) Moreover, through the study of the Kummer variety of the dual surface of A and some related geometric constructions, the author provides a new explanation for the description of A as a Prym variety, given by Haine.
Reviewer: A.Lanteri

MSC:

14K05 Algebraic theory of abelian varieties
14K10 Algebraic moduli of abelian varieties, classification
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems