$q$-bic threefolds and their surface of lines

Abstract

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.