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Abstract We study when the Picard group of smooth surfaces of degree d 5 in P^3 acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable (Y, L), where Y is a smooth projective three-fold and L a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.
Baldi et al. (Tue,) studied this question.