The theta divisor of the bidegree \((2,2)\) threefold in \(\mathbb{P}^ 2 \times \mathbb{P}^ 2\) (Q1816530)

Recently A. Verra proved that the existence of two conic bundle structures (c.b.s.) on the bidegree (2,2) divisor \(T\) in the product of two projective planes 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 \(T\) induces a parametrization of the theta divisor of \(J(T)\) by the Abel-Jacobi image of a special family of elliptic curves of degree 10 (minimal section 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.