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 the cohomology of the structure sheaf of S when q is prime.

Oort–Zink proved that a p-divisible group over a normal base in characteristic p with constant Newton polygon is isogenous to a p-divisible group admitting a slope filtration. In this article, we generalize this result to log p-divisible groups.

In this article, by utilizing the properties of elliptic functions, we characterize the meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb {C}$, where $L(z)$ is a nonconstant entire function, and m and n are two positive integers. As applications, we also investigate the meromorphic solutions of Fermat-type difference and q-difference equations.