q-bic Hypersurfaces

Let’s count: 1, q, q+1; here, q is a power of a prime p. In this talk, I will sketch an analogy between the geometry of a class of hypersurfaces over a field of positive characteristic p, which I call q-bic hypersurfaces, and the geometry of low degree hypersurfaces, such as quadrics and cubics, over the complex numbers. For instance, a smooth q-bic threefold has a smooth Fano surface of lines, and the intermediate Jacobian of the threefold is isogenous to the Albanese of the Fano surface.