Density of Noether–Lefschetz loci for surfaces in Fano and Calabi–Yau threefolds

Abstract In this paper we provide applications of general results of Baldi-Klingler-Ullmo and Khelifa-Urbanik on the geometry of the Hodge locus associated to an integral polarized variation of Hodge structures to the case of Noether-Lefschetz loci for families of smooth surfaces. In particular, we consider the family of smooth surfaces in the linear system of a sufficiently ample line bundle on a smooth projective threefold Y , in case Y is a Fano or a Calabi-Yau threefold, and we prove results on the different behaviour of the union of the general, respectively exceptional, components of its Noether-Lefschetz locus.