The Abel-Jacobi map for cubic threefold and periods of Fano threefolds of degree $14$

Abstract

The Abel{Jacobi maps of the families of elliptic quintics and rational quartics lying on a smooth cubic threefold are studied. It is proved that their generic b er is the 5-dimensional projective space for quintics, and a smooth 3-dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of Markushevich{Tikhomirov, who showed that the rst Abel{Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c1 = 0;c2 = 2 obtained by Serre's construction from elliptic quintics, and that the factorizing map from the moduli space to the intermediate Jacobian is etale. The above result implies that the degree of the etale map is 1, hence the moduli component of vector bundles is birational to the intermediate Jacobian. As an application, it is shown that the generic b er of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold. 1991 Mathematics Subject Classication: 14J30,14J60,14J45