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Density of the Noether- Lefschetz locus for the hyperplane sections of Calabi-Yau threefolds. (Densité du lieu de Noether-Lefschetz pour les sections hyperplanes des variétés de Calabi-Yau de dimension 3.) (French) Zbl 0772.14015
The subject of this paper is related with the problem of studying the image of the Abel-Jacobi map for a projective manifold \(X\). In a previous paper the author proved that it has infinite rank when \(X\) is a generic hypersurface of degree 5 in \(\mathbb{P}^ 4\), and a fundamental step in the proof was the density of the irreducible components of maximal codimension of the Noether-Lefschetz locus \(S(U)\), where \(U\subseteq\mathbb{P}(H^ 0({\mathcal O}_ X(1)))\) parametrizes the smooth hyperplane sections of \(X\). The same result is established here for a Calabi-Yau 3-fold \(X\) and for \(U\subseteq\mathbb{P}(H^ 0({\mathcal O}_ X(n)))\) \((n\gg 0)\). In this case the density statement is reduced to the study of a system of quadrics in \(\mathbb{P}(H^ 0({\mathcal O}_ \Sigma(n)))\) for a generic \(\Sigma\in U\).

14J30 \(3\)-folds
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C22 Picard groups
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
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