Abelian surfaces with (1,2)-polarization

Description

In an earlier paper [Invent. Math. 67 (1982), no. 2, 297331; MR0665159] M. Adler and P. van Moerbeke described three algebraically integrable cases of geodesic flows on SO(4). In these cases there are 4 integrals of the motion defining in C6 affine abelian surfaces as the complete intersection of 4 hypersurfaces. In an appendix to the above- cited paper Mumford observed that the smooth complete model A of the surface of the first case is of type (2, 4) in P7 . L. Haine showed [Math. Ann. 263 (1983), no. 4, 435472; MR0707241] that A is described in P^7 by 6 quadratic equations. He also showed that A is isomorphic to the Prym variety of an elliptic double cover of a curve D of genus 3. The paper in question applies the technique of Heisenberg groups to give an algebraic proof of Haines result. The isomorphism A Prym(D|E) is reduced to standard geometric constructions: In the P5 parametrizing quadrics passing through A the Kummer variety K of the dual abelian surface A parametrizes all quadrics of rank 4. The elliptic curve E lies on K and D parametrizes the pencils of P5 contained in these quadrics. This gives the geometric map D A which induces Haines isomorphism. Apart from this the paper gives a fairly complete description of abelian surfaces of type (2, 4) and its moduli space.