The AbelJacobi map for a cubic threefold and periods of Fano threefolds of degree 14
Abstract
The AbelJacobi 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 5dimensional projective space for quintics, and a smooth 3dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of MarkushevichTikhomirov, who showed that the first AbelJacobi 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 applicaton, 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1999
 DOI:
 10.48550/arXiv.math/9910058
 arXiv:
 arXiv:math/9910058
 Bibcode:
 1999math.....10058I
 Keywords:

 Mathematics  Algebraic Geometry;
 14J30;
 14J60;
 14J45
 EPrint:
 Latex, 28 pages