The theta divisior of the bidegree (2,2) threefold in ${\bf P}^{2} \times {\bf P}^{2}$

Details The theta divisior of the bidegree (2,2) threefold in ${\bf P}^{2} \times {\bf P}^{2}$ The theta divisior of the bidegree (2,2) threefold in ${\bf P}^{2} \times {\bf P}^{2}$

1994-02-03

arxiv.org/abs/alg-geom/9402001v1

Mathematics

Algebraic Geometry

32 pages, LaTeX

Scientific paper

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.

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