Uniqueness of the inversive plane of order sixty-four

An inversive plane is an incidence structure of points and circles such that: When the inversive plane \(\mathbf{I}\) is finite, there is an integer \(n \ge 2\), called the order of \(\mathbf{I}\) such that \(\mathbf{I}\) has\(n^2+1\) points, \(\mathbf{I}\) has \(n^3+n\) circles and every circle of \(\mathbf{I}\) is incident with \(n+1\) points of \(\mathbf{I}\). 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 \(\mathrm{PG}(3,q)\) is a set \(\Omega \) 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 \(\Omega \) is a plane meeting \(\Omega \) in more than one point. (A tangent plane is a plane meeting \(\Omega \) in a unique point.) The incidence structure \(\mathbf{I}(\Omega )\) of points of \(\Omega \) 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 \(\mathbf{I}(\Omega )\), for some ovoid \(\Omega \) of \(\mathrm{PG}(3,q)\). All known finite inversive planes are egglike. Moreover, given ovoids \(\Omega _1, \Omega _2\) of \(\mathrm{PG}(3,q)\). \(\mathbf{I}(\Omega _1)\) is isomorphic to \(\mathbf{I}(\Omega _2)\) if and only if there is a collineation g of \(\mathrm{PG}(3,q)\) with \(\Omega _1^g=\Omega _2\).

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: \(\mathbf {I}\) \((\Omega )\), where \(\Omega \) is an elliptic quadric of \(\mathrm{PG}(3,q)\), and \(\mathbf {I}\) \((\Omega )\), where \(\Omega \) is a Tits ovoid of \(\mathrm{PG}(3,2^{2e+1})\), \(e \ge 1\) [39] (and these are not isomorphic). The inversive planes \(\mathbf {I}\) \((\Omega )\), where \(\Omega \) is an elliptic quadric of \(\mathrm{PG}(3,q)\), 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].

Here we show that every inversive plane of order sixty-four is Miquelian (Corollary 2 in Sect. 4). Previously known results along these lines are : uniqueness of the inversive planes of orders 2, 3, 4 [42], 5 [8, 13, 37], 7 [14, 37], 9 [35] and 16 [1, 21, 22] and classification of the inversive planes of orders 8 [1, 15, 29] and 32 [23]. (For order 4, see also [34].)

Ovoids have been used to construct maximal arcs (and thereby partial geometries) [36], unitals [7, 19] and generalised quadrangles [12, p. 304], [25, 3.1.2]. Thus our results have consequences for enumerating Buekenhout–Metz unitals, Thas maximal arcs and Tits generalised quadrangles, which we will not dwell on here.

An oval of \(\mathrm{PG}(2,q)\) is a set of \(q+1\) points, no three collinear. A line l is external, tangent, secant to O accordingly as \(|l \cap O|\) 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.

Using this result, the same year, Barlotti and Panella independently classified ovoids of \(\mathrm{PG}(3,q)\), q odd.

Barlotti proved a little more (and Segre proved a slightly stronger result four years later, and gave a far more explicit statement).

The best result in this direction is that of Brown from 2000, although we will not need it here.

Earlier, in 1963, Dembowski had shown that inversive planes of even order arise from ovoids.

So the union of a conic of \(\mathrm{PG}(2,q)\), 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 \(\mathrm{PG}(2,2)\), \(\mathrm{PG}(2,4)\) and \(\mathrm{PG}(2,8)\) are regular, and irregular hyperovals of \(\mathrm{PG}(2,q)\) had been constructed for \(q=2^h\), \(h=5\) and \(h \ge 7\). Segre raised the question of existence of irregular hyperovals in \(\mathrm{PG}(2,16)\) and \(\mathrm{PG}(2,64)\). The next year, Lunelli and Sce [18] constructed irregular hyperovals in \(\mathrm{PG}(2,16)\). Nearly four decades passed before the other question was settled [28] by the construction of two irregular hyperovals in \(\mathrm{PG}(2,64)\), one with a group of order 60; the other with a group of order 15. The last of the hyperovals in \(\mathrm{PG}(2,64)\) 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 \(\mathrm{PG}(2,64)\) 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.

In order to prove this result, we follow the strategy established in [23].

The set of secant plane sections of an ovoid of \(\mathrm{PG}(3,q)\) on a tangent line \(\ell \) is a called a pencil with carrier \(\ell \).

Let \(O_1\) and \(O_2\) be ovals of \(\mathrm{PG}(2,q)\), 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 PQ 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 \(\mathrm{PG}(2,q)\) 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 \(F=\{O_X:X \in C, X \ne P\}\) such that \(O_X\) and \(O_Y\) are compatible at \(XY \cap PN\), for all \(X \ne Y \in C \setminus \{ P\}\). C is called the augmenting conic of F.

Using this terminology, we can restate the Plane Equivalent Theorem without using coordinates.

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].)

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 \(\infty \) and 0, respectively. Then the line \([\lambda a_1 + a_2,\lambda b_1 + b_2, \lambda c_1 + c_2]\) also passes through P and will be labelled \(\lambda \). The lines through any point are thus parameterised with \(GF(q) \cup \infty \), in the same way as \(\mathrm{PG}(1,q)\) is parameterised when (x, y) is given parameter x/y (with \(1/0 = \infty \)).

Any collineation g such that \(gP_2 = P_1\) maps the lines through \(P_2\) to the lines through \(P_1\) and induces an element \(g' \in P\Gamma L(2, q)\) 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 \(\infty \). The stabiliser of \(\infty \) in \(P\Gamma L(2,q)\) is the group \(A\Gamma L(1,q)\). Therefore a collineation g such that \(gP_2= P_1\) and \(gO_2\) is compatible with \(O_1\) at \(P_1\) induces an element \(g' \in A\Gamma L(1,q)\) 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.

Now it is straightforward to see which ovals can appear together in an ovoid. For every oval-point 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 \(g' \in A\Gamma L(1, q)\) such that \(g'L_1 = \{x \in GF(q) :x \notin L_2\}\). 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.

For each of the 19 ovals of \(\mathrm{PG}(2,64)\), 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 which has two orbits on points not on the hyperoval containing the pointed conic; the oval O with a group of order 60 has 76 orbits on points not on the hyperoval containing O; the oval O with a group of order 15 has 275 orbits on points not on the hyperoval containing O; the three ovals O with a group of order 12 have 351 orbits on points not on the hyperoval containing O; the two ovals O with groups of order 3 have 1375 orbits on points not on the hyperoval containing O; the ten ovals O with groups of order 1 have 4095 orbits on points not on the hyperoval containing O.

After this computation, there remains a list of 45107 oval-point pairs. For each oval-point pair, the local secant parameter set L was computed, and a canonical representative of the orbit of L under \(A\Gamma L(1,64)\) was stored (the representative was the lexicographically least local secant parameter set in the orbit). For each oval-point pair, a canonical representative of the orbit of the complement in GF(64) of L was also stored. Two oval-point pairs match if and only if the canonical representative of the local secant parameter set of the first is actually equal to the canonical representative of the complement of the local secant parameter set of the second. All possible matches between oval-point pairs were thus determined.

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 \(\mathrm{PG}(3,64)\) is a conic.

A point P of a generalised quadrangle of order (s, t) is 3-regular ifwhenever Q and R are points that are not collinear, and both not collinear with P. The points and lines contained in an elliptic quadric of \(\mathrm{PG}(5,q)\) form a generalised quadrangle of order \((q,q^2)\), called Q(5, q) [25, Sect. 3.1]. Our work yields two characterisations of this generalised quadrangle for \(q=64\).

A generalised quadrangle has Property (G) at a pair \(\{P,P'\}\) of collinear points if, every triple \(\{P,Q,R\}\) of points, with P, Q, R pairwise non-collinear and \(P' \in \{ P,Q,R\}^\perp \), is 3-regular.