Sur la jacobienne intermédiaire du double solide d'indice deux. (On the intermediate Jacobian of the double solid of index two)
Let S be a quartic surface in \({\mathbb{P}}_ 3\) not containing any line and B the double cover of \({\mathbb{P}}_ 3\) ramified in S. Its intermediate Jacobian \(J=J^ 2(B)\) is a principally polarized abelian variety of dimension 10. Let \(\Theta\) denote a theta divisor of J and \(\Phi: F\to J\) the Abel-Jacobi map, where F is the surface of lines in B. A translate \(F_ u\) of F by a point \(u\in J\) is by definition the surface \(\Phi(F)+u\) in J. The singularities of the theta divisor can be described by restricting the linear system \(| \Theta |_{\Theta}|\) to the translates \(F_ u\) of u contained in \(\Theta\). In this way a component Z of the quadratic singularities \(Sing^ 2(\Theta)\) of \(\Theta\) is constructed, which leads to a geometric interpretation of the projectivized tangent cones of the corresponding singular points: They are the quadrics of rank 5 in \({\mathbb{P}}_ 5\) intersecting the image of the surface S under the Veronese embedding \({\mathbb{P}}_ 3{\mathbb{P}}_ 9\). Since for a general threefold B one can recover the component Z among all components of Sing(\(\Theta\)) in a constructive way (the proof of this fact is not included in the paper), one obtains in this way a constructive Torelli theorem for generic double covers B.
scientific article; zbMATH DE number 4144155
scientific article