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
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
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