Let $X$ be a nonsingular surface in $P^3$ of degree $d$. Then the Picard number does not exceed $h^{1,1}\approx 2/3 d^3$, whereas the maximal number attained that I know is that of a Fermat surface, which is $\approx 3d^2$. What is the true maximum? Is there any literature on this subject?

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