On a conjecture of Harris.

For d ≥ 4 , the Noether–Lefschetz locus NL d parametrizes smooth, degree d surfaces in ℙ 3 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 d , but is true for d ≤ 5. She also showed that for d = 6 , 7 , there are finitely many reduced, irreducible components of NL d of non-maximal codimension. In this paper, we prove that for any d ≥ 6 , there are infinitely many non-reduced irreducible components of NL d of non-maximal codimension. [ABSTRACT FROM AUTHOR]