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On a conjecture due to Griffiths and Harris

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 $d \gg r$.