Uniqueness of the inversive plane of order sixty-four

Designs, Codes and Cryptography https://doi.org/10.1007/s10623-022-01014-6 Uniqueness of the inversive plane of order sixty-four Tim Penttila1 Received: 29 May 2021 / Revised: 18 November 2021 / Accepted: 25 January 2022 © The Author(s) 2022 Abstract 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) is an elliptic quadric. Keywords Inversive planes · Ovoids · Ovals Mathematics Subject Classification 51E20 · 51E21 1 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 1 , 2 of PG(3, q). I(1 ) is isomorphic to I(2 ) if and only if there is a collineation g of g PG(3, q) with 1 = 2 . Communicated by M. Lavrauw. B Tim Penttila 1 School of Mathematical Sciences, The University of Adelaide, Adelaide, Australia 123 T. Penttila 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, 22e+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]. 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. 2 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. Theorem 1 [31] An oval of PG(2, q), q odd, is a conic. Using this result, the same year, Barlotti and Panella independently classified ovoids of PG(3, q), q odd. Theorem 2 [2,24] An ovoid of PG(3, q), q odd, is an elliptic quadric. 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. 123 Inversive planes of order 64 3 Hyperovals in PG(2, 64) Theorem 6 U.G. Mitchell 1910 [20] The tangent lines to a conic in PG(2, q), q even, are concurrent. 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 = 2h , 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. Theorem 7 There are four isomorphism classes of hyperovals in PG(2, 64). 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. Corollary 1 There are nineteen isomorphism classes of ovals in PG(2, 64). 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 hyperov (...truncated)