On a conjecture of harris
For d ≥ 4, the Noether-Lefschetz locus NLd parametrizes smooth, degree d sur- faces in P3 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 codimen- sion. Voisin showed that the conjecture is false for sufficiently large d, but is true for d ≤ 5. She also showed that for d = 6, 7, there are finitely many reduced, irreducible components of NLd of non-maximal codimension. In this article, we prove that for any d ≥ 6, there are infinitely many non-reduced irreducible components of NLd of non-maximal codimension.