Non-Density of the Exceptional Components of the Noether–Lefschetz Locus

We study when the Picard group of smooth surfaces of degree d ≥ 5 in P3 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.