The Abel-Jacobi map for cubic threefold and periods of Fano threefolds of degree \(14\).

*(English)*Zbl 0938.14021The 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 fiber 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 D. Markushevich and A. S. Tikhomirov [“The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold” (e-print arXiv: math.AG/9809140, to appear in J. Algebr. Geom.)], who showed that the first Abel-Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers \(c_1=0\), \(c_2=2\) obtained by Serre’s construction from elliptic quintics, and that the factorizing map from the moduli space to the intermediate Jacobian is étale. The above result implies that the degree of the étale 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 fiber of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold.

The paper is a continuation of the recent work of D. Markushevich and A. S. Tikhomirov [“The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold” (e-print arXiv: math.AG/9809140, to appear in J. Algebr. Geom.)], who showed that the first Abel-Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers \(c_1=0\), \(c_2=2\) obtained by Serre’s construction from elliptic quintics, and that the factorizing map from the moduli space to the intermediate Jacobian is étale. The above result implies that the degree of the étale 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 fiber of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold.

##### MSC:

14J30 | \(3\)-folds |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14J45 | Fano varieties |

14H50 | Plane and space curves |