Geometry of smooth extremal surfaces

Abstract

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 116d14 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.