Smooth Surfaces with Maximal Lines

We prove that a smooth projective surface of degree $d$ in ${\mathbb{P}}^3$ contains at most $d^2(d^2-3d+3)$ lines. We characterize the surfaces containing exactly $d^2(d^2-3d+3)$ lines: these occur only in prime characterize $p$ and, up to choice of projective coordinates, are cut out by equations of the form $x^{p^e+1}+y^{p^e+1}+z^{p^e+1}+w^{p^e+1}=0.$