A census of cubic fourfolds over 𝔽₂

Auel, Asher; Kulkarni, Avinash; Petok, Jack; Weinbaum, Jonah

doi:10.1090/mcom/4010

A census of cubic fourfolds over $\mathbb {F}_2$
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by Asher Auel, Avinash Kulkarni, Jack Petok and Jonah Weinbaum;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4010

Published electronically: September 24, 2024

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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.

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