  Let T be a general bidegree (2,2) divisor in the product of two projective
planes. Recently A.Verra proved that the existence of two conic bundle
structures (c.b.s.) on T implies a new counterexample to the Torelli theorem
for Prym varieties. Let J(T) be the jacobian of T. In this paper we prove that
any of the two c.b.s. on T admits a parametrisation of the theta divisor of
J(T) by the Abel-Jacobi image of a special family of elliptic curves of degree
9 (minimal sections of the given c.b.s.) on T. This result is an analogue of
the well-known Riemann theorem for curves. In particular, this implies some
results about K3 surfaces and plane sextics with vanishing theta-null. Further
we use once again the geometry of curves on T, in order to prove the Torelli
theorem for the bidegree (2,2) threefolds.
