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The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold. (English) Zbl 0987.14028
The 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 $$X$$.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14H10 Families, moduli of curves (algebraic) 14J30 $$3$$-folds 14J10 Families, moduli, classification: algebraic theory 14H40 Jacobians, Prym varieties 14J45 Fano varieties
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