q-bic threefolds and their surface of lines

For any power q of the positive ground field characteristic, a smooth q-bic threefold -- the Fermat threefold of degree $q+1$ for example -- has a smooth surface S of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of S. Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute cohomology of the structure sheaf of S when q is prime.