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We study the geometry of smooth projective surfaces defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces extremal surfaces, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces. For instance, extremal surfaces have many lines but no triangles, hence many “star points” analogous to Eckardt points on a cubic surface. We generalize the classical notion of a double six for cubic surfaces to a double 2d on an extremal surface of degree d. We show that, asymptotically in d, smooth extremal surfaces have at least (1/16)d^{14} double 2d's. A key element of the proofs is the large automorphism group of an extremal surface, which we show to act transitively on many associated sets, such as the set of triples of skew lines on the extremal surface. 

We prove that if f f is a reduced homogeneous polynomial of degree d d , then its F F -pure threshold at the unique homogeneous maximal ideal is at least 1 d − 1 \frac {1}{d-1} . We show, furthermore, that its F F -pure threshold equals 1 d − 1 \frac {1}{d-1} if and only if f ∈ m [ q ] f\in \mathfrak m^{[q]} and d = q + 1 d=q+1 , where q q is a power of p p . Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain. 

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”.