Recently A.Verra proved
that the existence of two conic bundle structures (c.b.s.) on the bidegree (2,2) divisor
in the product of two projective planes implies a new counterexample to the
Torelli theorem for Prym varieties. Let J be the jacobian of T. In this
paper we prove that any of the two c.b.s. on T induces a parametrization
of the theta divisor of J by the Abel-Jacobi image of a special family
of elliptic curves of degree 10 (minimal sections of the given c.b.s.) on T.
This result is an analog of the well-known Riemann theorem for curves.
Further we use once again the geometry of curves on T, in order to prove the
Torelli theorem for the bidegree (2,2) threefolds. In the end, we study the
bidegree (2,2) threefold T with one node. It is shown that in this case the
classical Dixon correspondence, between the two discriminant pairs defined by
T, can be represented as a composition of two 4-gonal correspondences of
Donagi.
be the jacobian of 
by the Abel-Jacobi image of a special family
of elliptic curves of degree 