  For a fixed $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree
$d$ surfaces in $\mathbb{P}^3$ with Picard number greater than $1$. This is a
countable union of proper algebraic varieties. It is known (due to works of
Voisin, Green and others) that the largest irreducible component is of
codimension (in the space of all smooth surface in $\mathbb{P}^3$ of degree
$d$) equal to $d-3$. The main object of study in this article is: For fixed $r$
greater than $2$ and less than $d$, the locus parametrizing degree $d$ surfaces
in $\mathbb{P}^3$ with Picard number at least equal to $r$. It has been
conjectured by Griffiths and Harris that the largest component of this locus is
of codimension equal to $(r-1)(d-3)-\binom{r-3}{2}$. Furthermore, the
irreducible component of this locus parametrizing surfaces with $r$ lines on
the same plane is of this codimension. In this article we prove the statement
for $d \gg r$.
