Uniqueness of the inversive plane of order sixty-four

Tim Penttila

doi:10.1007/s10623-022-01014-6

Abstract

The uniqueness of the inversive plane of order sixty-four, up to isomorphism, is established. Equivalently, it is shown that every ovoid of mathrm{PG}(3,64) is an elliptic quadric.

Highlights

An ovoid of PG(3, q) is a set of q2 + 1 points, no 3 collinear, if q > 2; if q = 2, it is a set of 5 points, no 4 coplanar
A secant plane to an ovoid is a plane meeting in more than one point. (A tangent plane is a plane meeting in a unique point.) The incidence structure I( ) of points of and plane sections by secant planes is an inversive plane of order q ( [12, 6.1.2])
All known finite inversive planes are egglike

Summary

Introduction

An inversive plane is an incidence structure of points and circles such that:. (i) (i) every three distinct points are incident with a unique circle; (ii) (ii) given two points P, Q and a circle C on P (but not on Q), there is a unique circle. When the inversive plane I is finite, there is an integer n ≥ 2, called the order of I such that I has n2 + 1 points, I has n3 + n circles and every circle of I is incident with n + 1 points of I. (A tangent plane is a plane meeting in a unique point.) The incidence structure I( ) of points of and plane sections by secant planes is an inversive plane of order q ( [12, 6.1.2]). There are two known families of finite inversive planes: I( ), where is an elliptic quadric of PG(3, q), and I( ), where is a Tits ovoid of PG(3, 22e+1), e ≥ 1 [39] (and these are not isomorphic). Our results have consequences for enumerating Buekenhout–Metz unitals, Thas maximal arcs and Tits generalised quadrangles, which we will not dwell on here

Background results

The computational results

Consequences