Let X ⊂ ℙ_{ℂ}^{4} be a smooth hypersurface of degree d ≥ 5, and let S ⊂ X be a smooth hyperplane section. Assume that there exists a non trivial cycle Z ∈ Pic(X) of degree 0, whose image in CH_{1}(X) is in the kernel of the Abel-Jacobi map. The family of couples (X, S) containing such Z is a countable union of analytic varieties. We show that it has a unique component of maximal dimension, which is exaclty the locus of couples (X, S) satisfying the following condition: There is line Δ ⊂ S and a plane P ⊂ ℙ_{ℂ}^{4} such that P ∩ X = Δ, and Z = Δ − dh, where h is the class of the hyperplane section in CH_{1}(S).

The image of Z in CH_{1}(X) is 0. Thus, this construction provides evidence for a conjecture by Nori which predicts that the Abel-Jacobi map for 1-cycles on X is injective.