A census of cubic fourfolds over 𝔽₂

Abstract

We compute a complete set of isomorphism classes of cubic fourfolds over F 2 \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 F 2 \mathbb {F}_2 ; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over F 2 \mathbb {F}_2 , which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of K 3 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.