The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold.

*(English)*Zbl 0987.14028The celebrated paper by C. H. Clemens and P A. Griffiths [Ann. Math. (2) 95, 281-356 (1972; Zbl 0231.14004)], begun, by means of the Abel-Jacobi map, the study of connections between abelian varieties associated to families of curves on a Fano variety \(X\), and Abelian varieties related to \(X\). In the present paper it is addressed the question of understanding the fibres of the Abel-Jacobi map on the parametrising variety itself. Let \(H\) be the variety parametrising elliptic quintic curves on a general cubic threefold. The authors prove that \(H\) is irreducible and determine, generically, the Abel-Jacobi map, \(AJ_H:H\rightarrow J_1(X)\), for this family. The result, interesting and quite unexpected, is the following.

There is an open \(U\subset H\) where the map is smooth, and factors on the moduli space \(M^0\) of stable vector bundles on \(X\) of rank 2 and Chern numbers \(c_1=0\), \(c_2=2\). Then \(f:U\rightarrow M^0\) is isomorphic to the projectivisation of a rank 6 vector bundle, locally in the étale topology over \(M^0\).

It has to be stressed that the generic vector bundle in \(M^0\) is obtained by Serre’s construction from elliptic quintics on \(X\).

There is an open \(U\subset H\) where the map is smooth, and factors on the moduli space \(M^0\) of stable vector bundles on \(X\) of rank 2 and Chern numbers \(c_1=0\), \(c_2=2\). Then \(f:U\rightarrow M^0\) is isomorphic to the projectivisation of a rank 6 vector bundle, locally in the étale topology over \(M^0\).

It has to be stressed that the generic vector bundle in \(M^0\) is obtained by Serre’s construction from elliptic quintics on \(X\).

Reviewer: Massimiliano Mella (Ferrara)