A census of cubic fourfolds over ${\mathbb{F}}_2$

Abstract

We compute a complete set of isomorphism classes of cubic fourfolds over ${\mathbb{F}}_2$. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all zeta functions of the smooth cubic fourfolds over ${\mathbb{F}}_2$; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over ${\mathbb{F}}_2$, which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of $K3$ surfaces. We present a refinement to the standard method of orbit enumeration that leverages filtrations and gives a significant speedup. In the case of cubic fourfolds, the relevant filtration is determined by Waring representation and the method brings the problem into the computationally tractable range.

21 pages, 6 figures