Cubic surfaces of characteristic two

Kadyrsizova, Zhibek; Kenkel, Jennifer; Page, Janet; Singh, Jyoti; Smith, Karen; Vraciu, Adela; Witt, Emily

doi:10.1090/tran/8341

Cubic surfaces of characteristic two
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by Zhibek Kadyrsizova, Jennifer Kenkel, Janet Page, Jyoti Singh, Karen E. Smith, Adela Vraciu and Emily E. Witt PDF

Trans. Amer. Math. Soc. 374 (2021), 6251-6267

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”.

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