Barth, Wolf 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 Cited in 4 ReviewsCited in 18 Documents MSC: 14K05 Algebraic theory of abelian varieties 14K10 Algebraic moduli of abelian varieties, classification 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:(1,2)-polarization; integrable Hamiltonian systems; quadratic equations for abelian surfaces; (2,4)-polarization; Heisenberg group; coarse moduli space; Kummer variety; Prym variety Citations:Zbl 0628.00007; Zbl 0545.58027; Zbl 0521.58042 PDFBibTeX XML