Geometry of Smooth Extremal Surfaces

TL;DR

This paper explores the geometry of extremal surfaces defined by Frobenius forms in prime characteristic, revealing their rich structure, line configurations, and connections to Hermitian geometries, with implications for algebraic geometry.

Contribution

It introduces extremal surfaces defined by Frobenius forms, analyzes their geometric properties, and generalizes classical cubic surface configurations to higher degrees.

Findings

Extremal surfaces contain a quadratic number of lines in degree d.

Number of star points on extremal surfaces is quintic in degree.

Asymptotically, extremal surfaces have at least (1/16)d^{14} double 2d's.

Abstract

We study the geometry of the smooth projective surfaces that are 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 $\textit{extremal surfaces}$, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces of characteristic two, which are examples of extremal surfaces. For example, we show that an extremal surface $X$ contains $d^2(d^2-3d+3)$ lines where $d$ is the degree, which is notable since the number of lines on a complex surface is bounded above by a quadratic function in $d$. Whenever two of those lines meet, they determine a $d$-tangent plane to $X$ which consists of a union of $d$ lines meeting in one point; we count the precise number of such "star points" on $X$, showing that it is quintic in the degree, which recovers the fact that there are exactly 45 Eckardt points on an extremal cubic surface. Finally, 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 $\frac{1}{16}d^{14}$ double $2d$'s. A key element of the proofs is using the large automorphism group of extremal surfaces which we show acts transitively on many sets, such as the set of (triples of skew) lines on the extremal surface. Extremal surfaces are closely related to finite Hermitian geometries, which we recover as the $\mathbb F_{q^2}$-rational points of special extremal surfaces defined by Hermitian forms over $\mathbb F_{q^2}$.