Cubic Surfaces of Characteristic Two

TL;DR

This paper studies cubic surfaces over fields of characteristic two, revealing the structure of non-Frobenius split surfaces, their classification, and geometric properties related to lines and configurations.

Contribution

It characterizes non-Frobenius split cubic surfaces as a linear subspace, classifies them up to projective equivalence, and describes their defining equations and line configurations.

Findings

Non-Frobenius split cubic surfaces form a codimension four subspace.

There are finitely many non-Frobenius split cubic surfaces up to projective equivalence.

A cubic surface fails to be Frobenius split if and only if no three lines form a triangle.

Abstract

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a "triangle".