Abstract
Let X ⊂ ℙ4 ℂ 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 CH1(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 exists a line Δ ⊂ S and a plane P ⊂ ℙ4 ℂ such that P ∩ X = Δ, and Z = Δ − dh, where h is the class of the hyperplane section in CH1(S). The image of Z in CH1(X) is thus 0. This construction provides evidence for a conjecture by Nori which predicts that the Abel–Jacobi map for 1–cycles on X is injective.
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Otwinowska, A. Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz (Components of Maximal Dimension of an Analogue of the Noether-Lefschetz Locus) . Compositio Mathematica 131, 31–50 (2002). https://doi.org/10.1023/A:1014751331345
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DOI: https://doi.org/10.1023/A:1014751331345