Uniqueness of the inversive plane of order sixty-four

The uniqueness of the inversive plane of order sixty-four, up to isomorphism, is established. Equivalently, it is shown that every ovoid of PG(3,64)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{PG}(3,64)$$\end{document} is an elliptic quadric.


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 D incident with both P and Q whose only common point with C is P; (iii) (iii) there are at least four points; (iv) (iv) there is a non-incident point,circle pair; and (v) (v) every circle is incident with a non-empty set of points. (See [12, pp. 252-253].) When the inversive plane I is finite, there is an integer n ≥ 2, called the order of I such that I has n 2 + 1 points, I has n 3 + n circles and every circle of I is incident with n + 1 points of I. In fact, a finite inversive plane of order n is exactly a 3 − (n 2 + 1, n + 1, 1)-design ( [12, pp. 252-254]).
An ovoid of PG(3, q) is a set of q 2 + 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]). A finite inversive plane is egglike if it is isomorphic to I( ), for some ovoid of PG (3, q). All known finite inversive planes are egglike. Moreover, given ovoids In 1963, Dembowski proved that every inversive plane of even order is egglike [11], and hence has order a power of two.
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, 2 2e+1 ), e ≥ 1 [39] (and these are not isomorphic). The inversive planes I( ), where is an elliptic quadric of PG (3, q), are called Miquelian for, by a result of van der Waerden and Smid [41], they are characterised by the Theorem of Miquel [12, 6.1.5]. The bundle theorem is a similar configuration to Miquel's Theorem [12, pp. 255-256]. It is satisfied by every egglike inversive plane [12, 6.1.4]. In 1980, Kahn proved the converse, so: an inversive plane is egglike if and only if it satisfies the bundle theorem [17].

Background results
An oval of PG(2, q) is a set of q + 1 points, no three collinear. A line l is external, tangent, secant to O accordingly as |l ∩ O| is 0, 1 or 2. An example of an oval is a nondegenerate conic. Many other ovals are known in characteristic two; see [27] for the most recent survey.
A 1955 result of Segre shows that the situation in odd characteristic is in strong contrast to that in characteristic two.
Barlotti proved a little more (and Segre proved a slightly stronger result four years later, and gave a far more explicit statement).

Theorem 3 [2, Sect. 3] [33, Theorem V] An ovoid of PG(3, q), q even, is an elliptic quadric if and only if every secant plane section is a conic.
The best result in this direction is that of Brown from 2000, although we will not need it here.

Theorem 4 [4] An ovoid of PG(3, q), q even. is an elliptic quadric if and only if some secant plane section is a conic.
Earlier, in 1963, Dembowski had shown that inversive planes of even order arise from ovoids.
Theorem 5 [11] An inversive plane of even order is egglike, and so has order a power of two. Mitchell 1910 [20] The tangent lines to a conic in PG(2, q), q even, are concurrent.

Hyperovals in PG(2, 64)
So the union of a conic of PG(2, q), q even, and the point of concurrency of its tangent lines is a hyperoval, which is called a regular hyperoval.
By 1957 [32], it had been shown that all hyperovals of PG(2, 2), PG(2, 4) and PG(2, 8) are regular, and irregular hyperovals of PG(2, q) had been constructed for q = 2 h , h = 5 and h ≥ 7. Segre raised the question of existence of irregular hyperovals in PG (2,16) and PG(2, 64). The next year, Lunelli and Sce [18] constructed irregular hyperovals in PG (2,16). Nearly four decades passed before the other question was settled [28] by the construction of two irregular hyperovals in PG(2, 64), one with a group of order 60; the other with a group of order 15. The last of the hyperovals in PG(2, 64) was constructed the following year [30]; it has a group of order 12. The hyperovals with groups of orders 60 and 15 were generalised to the infinite families of Subiaco hyperovals in [10] in 1996. The hyperoval with a group of order 12 was generalised to the infinite family of Adelaide hyperovals in [9] in 2003. In 2019 [40], hyperovals of PG(2, 64) were classified by Vandendriessche.
The regular hyperoval gives rise to two ovals, the conic and the point conic. The Subiaco hyperoval with a group of order 60 gives rise to 3 ovals; the Subiaco hyperoval with a group of order 15 gives rise to 6 ovals and the Adelaide hyperoval gives rise to 8 ovals.
In more detail, the regular hyperoval contains representatives of two isomorphism classes of ovals-the conic and the pointed conic; the Subiaco hyperoval with a group of order 60 contains representatives of three isomorphism classes of ovals, with groups of orders 60, 12 and 1; the Subiaco hyperoval with a group of order 15 contains representatives of six isomorphism classes of ovals, one with a group of order 15, one with a group of order 3 and four with a group of order 1; and the Adelaide hyperoval contains representatives of eight isomorphism classes of ovals, two with a group of order 12, one with a group of order 3 and five with a group of order 1.

Ovoids in PG(3, 64)
In order to prove this result, we follow the strategy established in [23].
The set of secant plane sections of an ovoid of PG(3, q) on a tangent line is a called a pencil with carrier .
Let O 1 and O 2 be ovals of PG(2, q), q even and let P be a point not on either oval, nor equal to either nucleus. Then O 1 and O 2 are compatible at P if they have the same nucleus, they have a point Q in common, the line P Q is a tangent line to each oval and every secant line to O 1 on P is external to O 2 (and hence every external line to O 1 on P is secant to O 2 ).
In order to keep the treatment as synthetic as possible, we lift some of the ideas of [5]. An augmented fan of ovals of PG(2, q) is a set F of ovals with common nucleus N and common point Q indexed by the points (other than P) of a conic C on P with nucleus N so that F = {O X : X ∈ C, X = P} such that O X and O Y are compatible at XY ∩ P N, for all X = Y ∈ C \ {P}. C is called the augmenting conic of F.
Using this terminology, we can restate the Plane Equivalent Theorem without using coordinates. PG(3, q), q even, is equivalent to an augmented fan {O X : X ∈ C, X = Q} of ovals of PG(2, q). Moreover, for every tangent line each pencil with carrier gives rise to such a set and for each plane π of the pencil there is a parameterisation of the planes π s of the pencil, s ∈ G F(q) such that π 0 = π, a parameterisation of the points P s of the augmenting conic, other than the common point, and there is a homography M s taking π s ∩ to O P s and the carrier line of the pencil to the line Q N .

Theorem 8 [Plane Equivalent Theorem] [23, Theorem 2.1] An ovoid in
We note that the Plane Equivalent Theorem [23, Theorem 2.1] was proved in [26] and [16] (independently), and first published in [21]. (As it was applied, in [5,6,22,23], it was (slightly) modified into a more useful form, and it is in that form that it is stated here, rather than the original form of [26].)

Lemma 1 [23, Lemma 3.2]
Let (O 1 , P 1 ) and (O 2 , P 2 ) be two oval-point pairs. If there is a collineation g such that g P 2 = P 1 and gO 2 is compatible with O 1 at P 1 , then g maps the lines through P 2 secant to O 2 to the lines through P 1 external to O 1 , and the one line through P 2 tangent to O 2 to the one line through P 1 tangent to O 1 . Now the lines through any point P can be parameterised by a single non-homogeneous parameter in the following fashion: select two lines, say [a 1 , b 1 , c 1 ] and [a 2 , b 2 , c 2 ] and give them parameters ∞ and 0, respectively. Then the line [λa 1 +a 2 , λb 1 +b 2 , λc 1 +c 2 ] also passes through P and will be labelled λ. The lines through any point are thus parameterised with G F(q) ∪ ∞, in the same way as PG(1, q) is parameterised when (x, y) is given parameter x/y (with 1/0 = ∞).
Any collineation g such that g P 2 = P 1 maps the lines through P 2 to the lines through P 1 and induces an element g ∈ P L(2, q) on the corresponding sets of parameters. Returning to the situation of the two oval-point pairs (O 1 , P 1 ) and (O 2 , P 2 ) we can insist that the two tangent lines involved both be assigned the parameter ∞. The stabiliser of ∞ in P L(2, q) is the group A L(1, q). Therefore a collineation g such that g P 2 = P 1 and gO 2 is compatible with O 1 at P 1 induces an element g ∈ A L(1, q) that maps the set of parameters of the external lines to O 2 through P 2 to the set of parameters of the secant lines to O 1 through P 1 . The next theorem provides a converse to this result.
Theorem 10 [23, Theorem 3.3] Let (O 1 , P 1 ) and (O 2 , P 2 ) be two oval-point pairs, with the lines through P 1 and P 2 parameterised in such a way that the tangent lines to the two ovals receive the parameter ∞. Let S 1 and E 2 be the sets of parameters of the secant lines to O 1 through P 1 and the external lines to O 2 through P 2 , respectively. Then there exists a collineation g such that such that g P 2 = P 1 and gO 2 is compatible with O 1 at P 1 if and only if there is an element g ∈ A L(1, q) such that g E 2 = S 1 . Now it is straightforward to see which ovals can appear together in an ovoid. For every ovalpoint pair (O, P), we parameterise the lines through P as described above and determine the local secant parameter set L as the set of parameters of the secant lines. Then two oval-point pairs (O 1 , P 1 ) and (O 2 , P 2 ) with local secant parameter sets L 1 and L 2 are said to match provided that there is some element g ∈ A L(1, q) such that g L 1 = {x ∈ G F(q) : x / ∈ L 2 }. Two ovals can only appear together in the same pencil of an ovoid if they both have points at which the corresponding two oval-point pairs match.

The computational results
For each of the 19 ovals of PG(2, 64), the group stabilising each oval was computed, along with the orbits of this stabiliser on non-nucleus points off the ovals, and a list of all possible oval-point pairs was made.
The conic has a group of order 1,572,480 which has one orbit on points not on the hyperoval containing the conic; the pointed conic has a group of order 24,192  Of the 19 ovals, 18 had points at which they do not match with anything at all; hence, none of these 18 ovals can be a secant plane section of an ovoid. The remaining oval is a conic. Hence, every plane section of an ovoid of PG(3, 64) is a conic.

Theorem 11 Every ovoid of PG(3, 64) is an elliptic quadric.
Proof This follows from the results above and Theorem 3.

Consequences
Theorem 12 Every inversive plane of order 64 is Miquelian.
Proof This follows from Theorems 11 and 5.

Corollary 2 There is a unique inversive plane of order 64.
Proof This follows from the result of van der Waerden and Smid mentioned in the introduction and the fact that any two elliptic quadrics of PG (3, q) are equivalent under a collineation of PG (3, q).