Geometry of Smooth Extremal Surfaces

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 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 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 F q 2 -rational points of special extremal surfaces defined by Hermitian forms over F q 2 .

Lower bounds on the F-pure threshold and extremal singularities

“…For example, regardless of the characteristic, a smooth cubic surface can contain at most eighteen Eckardt points-unless its defining equation achieves the minimal F -pure threshold of bound (1), in which case this extremal cubic surface contains exactly forty-five Eckardt points-one in each and every of the forty-five tri-tangent planes [KKP + 21]. As another example, in characteristic zero (or when p > d), the maximal number of lines on a smooth projective surface of degree d is bounded above by a quadratic function in d [Seg43, RS15,BR20]; however, the number of lines on a smooth projective extremal surface-one whose defining polynomial achieves the lower bound (1)-is quartic in its degree [BPRS21,3.2.3]. These examples confirm that singularities can be much worse in characteristic p than in characteristic zero, reflecting the fact that our sharp lower bound on F -pure threshold is much smaller than corresponding bounds on log canonical threshold; see paragraph (1.2).…”