Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz (Components of Maximal Dimension of an Analogue of the Noether-Lefschetz Locus)

Abstract

Let X [subset or is implied by] $\mathbb P$4$\mathbb _C$ be a smooth hypersurface of degree d [gt-or-equal, slanted] 5, and let S [subset or is implied by] X be a smooth hyperplane section. Assume that there exists a non trivial cycle Z [set membership] 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 [subset or is implied by] S and a plane P [subset or is implied by] $\mathbb P$4$_{\mathbb C}$ such that P [cap B: intersection] 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.