Smooth Surfaces with Maximal Lines

We prove that a smooth projective surface of degree d in P³ contains at most d²(d²-3d+3) lines. We characterize the surfaces containing exactly d²(d²-3d+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^pᵉ+1+y^pᵉ+1+z^pᵉ+1+ w^pᵉ+1 = 0.