  For $d \ge 4$, the Noether-Lefschetz locus $\mathrm{NL}_d$ parametrizes
smooth, degree $d$ surfaces in $\mathbb{P}^3$ 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
$d \le 5$. She also showed that for $d=6, 7$, there are finitely many
\emph{reduced}, irreducible components of $\mathrm{NL}_d$ of non-maximal
codimension. In this article, we prove that for any $d \ge 6$, there are
infinitely many \emph{non-reduced} irreducible components of $\mathrm{NL}_d$ of
non-maximal codimension.
