On a conjecture of Harris

Abstract

For [Formula: see text], the Noether–Lefschetz locus [Formula: see text] parametrizes smooth, degree [Formula: see text] surfaces in [Formula: see text] with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the Noether–Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large [Formula: see text], but is true for [Formula: see text]. She also showed that for [Formula: see text], there are finitely many reduced, irreducible components of [Formula: see text] of non-maximal codimension. In this paper, we prove that for any [Formula: see text], there are infinitely many non-reduced irreducible components of [Formula: see text] of non-maximal codimension.