Nonsingular hypercubes and nonintersecting hyperboloids

Abstract

Some ten years ago we managed to take a first step in the classification of nonsingular 2$\times$2$\times$2$\times$2 hypercubes over a finite field by resolving the special case where the hypercubes can be written as a product of two 2$\times$2$\times$2 hypercubes, i.e., nonsingular 2$\times$2$\times$2$\times$2 hypercubes of 12-rank two. We have now been able to extend this classification to hypercubes of 12-rank three, based on the connection between nonsingular hypercubes and bundles of nonintersecting quadrics in 3 dimensions. A bit surprisingly, the number of inequivalent nonsingular hypercubes of 12-rank three is only of the same order of magnitude as in the case of 12-rank two. We also made some headway into the remaining case of 12-rank four. In particular, we prove that there are essentially only 2 nonsingular 2$\times$2$\times$2$\times$2 hypercubes that correspond to a hyperbolic fibration of 3-dimensional projective space.