On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci

Abstract

For any integer d ≥ 5, the Noether–Lefschetz locus, denoted NL d, parametrizes smooth degree d surfaces in ℙ3 with Picard number at least 2. It is well-known (due to works of Voisin, Green and others) that the largest irreducible component of NL d is of codimension (in the space of all smooth surfaces in ℙ3 of degree d) equal to d-3 and parametrizes surfaces containing a line. In this article we study for an integer 3 ≤ r < d, the sub-locus of NL d, denoted NL r,d, parametrizing surfaces with Picard number at least r. A conjecture of Griffiths and Harris states the largest component of NL r,d is of codimension [Formula: see text] and the irreducible component of NL r,d parametrizing the surfaces containing r - 1 coplanar lines is of this codimension. We prove this statement in the case r ≪ d.