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A note on some formal properties of the infinitesimal Abel-Jacobi mapping. (English) Zbl 0569.14020
Geometry today, Int. Conf., Rome 1984, Prog. Math. 60, 69-73 (1985).
This brief note lists two applications of the derivative of the Abel- Jacobi mapping from a family of curves F on a threefold V to the intermediate Jacobian JV of V. First it is shown that for hypersurfaces in $${\mathbb{C}}{\mathbb{P}}^ 4$$ of degree $$\geq 3$$, the Abel-Jacobi mapping is always non-trivial for families of projectively normal rational curves. Secondly, it is shown that, for threefolds with trivial canonical bundle, the derivative of the Abel-Jacobi mapping at a curve C in V is non- trivial exactly when C does not deform generically with V.

##### MSC:
 14K30 Picard schemes, higher Jacobians 14J30 $$3$$-folds