Abstract We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ 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.