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A Characterisation of Translation Ovals in Finite Even Order Planes

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A Characterisation of Translation Ovals in Finite Even Order Planes A characterisation of translation ovals in finite even order planes S.G. Barwick and Wen-Ai Jackson School of Mathematics, University of Adelaide Adelaide 5005, Australia Abstract In this article we consider a set C of points in PG(4,q), q even, satisfying cer- tain combinatorial properties with respect to the planes of PG(4,q). We show that there is a regular spread in the hyperplane at infinity, such that in the correspond- ing Bruck-Bose plane PG(2,q2), the points corresponding to C form a translation hyperoval, and conversely. 1 Introduction In this article we first consider a non-degenerate conic in PG(2, q2), q even. We look at the corresponding point set in the Bruck-Bose representation in PG(4, q), and study its combinatorial properties (details of the Bruck-Bose representation are given in Section 2). Some properties of this set were investigated in [4]. In this article we are interested in combinatorial properties relating to planes of PG(4, q). We consider a set of points in PG(4, q) satisfying certain of these combinatorial properties and find that the points 2 arXiv:1305.6673v1 [math.CO] 29 May 2013 correspond to a translation oval in the Bruck-Bose plane PG(2, q ). In [3], the case when q is odd is considered, and we show that given a set of points in PG(4, q) satisfying the following combinatorial properties, we can reconstruct the conic in PG(2, q2). We use the following terminology in PG(4, q): if the hyperplane at infinity is denoted Σ∞, then we call the points of PG(4, q) \ Σ∞ affine points. Theorem 1.1 [3] Let Σ∞ be the hyperplane at infinity in PG(4, q), q ≥ 7, q odd. Let C be a set of q2 affine points, called C-points, and suppose there exists a set of planes called C-planes satisfying the following properties: 1. Each C-plane meets C in a q-arc. 1 2. Any two distinct C-points lie in a unique C-plane. 3. The affine points of PG(4, q) are of three types: points of C; points on no C-plane; and points on exactly two C-planes. 4. If a plane meets C in more than four points, it is a C-plane. Then there exists a unique spread S in Σ∞ so that in the Bruck-Bose translation plane P(S), the C-points form a q2-arc of P(S). Moreover, the spread S is regular, and so P(S) ∼= PG(2, q2), and the q2-arc can be completed to a conic of PG(2, q2). The case when q is even is more complex. The combinatorial properties only allow us to reconstruct a translation oval in PG(2, q2). The main result of this article is the following theorem. Theorem 1.2 Consider PG(4, q), q even, q > 2, with the hyperplane at infinity denoted 2 by Σ∞. Let C be a set of q affine points, called C-points and consider a set of planes called C-planes which satisfies the following: (A1) Each C-plane meets C in a q-arc. (A2) Any two distinct C-points lie in a unique C-plane. (A3) The affine points that are not in C lie on exactly one C-plane. (A4) Every plane which meets C in at least three points either meets C in exactly four points or is a C-plane. Then there exists a regular spread S in Σ∞ such that in the Bruck-Bose plane P(S) ∼= 2 PG(2, q ), the C-points, together with two extra points on ℓ∞, form a translation hyperoval of PG(2, q2). We begin in Section 2 with the necessary background material on the Bruck-Bose repre- sentation. In Section 3 we investigate combinatorial properties of conics and translation ovals in PG(2, q2), q even, and show that they satisfy properties (A1-4) of Theorem 1.2. The rest of the article is devoted to proving Theorem 1.2. In Section 5 we discuss further aspects of the problem, and avenues for further work. 2 2 The Bruck-Bose representation We use the Bruck-Bose representation of PG(2, q2) in PG(4, q) introduced in [1, 5, 6]. See [2] for more details of this representation. Let Σ∞ be the hyperplane at infinity of PG(4, q) and let S be a spread of Σ∞. Call the points of PG(4, q) \ Σ∞ affine points and the points in Σ∞ infinite points. The lines and planes of PG(4, q) that are not contained in Σ∞ are called affine lines and affine planes respectively. Consider the incidence structure whose points are the affine points of PG(4, q), whose lines are the affine planes of PG(4, q) which contain a line of the spread S, and where incidence is inclusion. This incidence structure is an affine plane, and can be completed to a translation plane denoted P(S) by adjoining the line at infinity ℓ∞ whose points are the elements of the spread S. The translation plane P(S) is the Desarguesian plane PG(2, q2) if and only if the spread S is regular. Note that the affine planes of PG(4, q) 2 that do not contain a line of S correspond to Baer subplanes of PG(2, q ) secant to ℓ∞. We introduce the coordinate notation that we will use in the Bruck-Bose representation of PG(2, q2) in PG(4, q) (so S is a regular spread). See [2] for more details of this coordi- natisation. In PG(2, q2), points have coordinates (x, y, z), x, y, z ∈ GF(q2) and the line at infinity has equation z = 0. In PG(4, q), points have coordinates (x0,...,x4), xi ∈ GF(q), and we let the hyperplane at infinity Σ∞ have equation x4 = 0. Let τ be a primitive 2 2 2 element in GF(q ) with primitive polynomial x −t1x−t0 over GF(q). Let α, β ∈ GF(q ), then we can uniquely write α = a0 + a1τ, β = b0 + b1τ for ai, bi ∈ GF(q). The Bruck- 2 Bose map takes a point P =(α, β, 1) ∈ PG(2, q ) \ ℓ∞ to the point P =(a0, a1, b0, b1, 1) ∈ 2 2 PG(4, q)\Σ∞. In PG(2, q ), ℓ∞ has points {P∞ = (0, 1, 0)}∪{Pδ = (1, δ, 0) : δ ∈ GF(q )}. If δ = d0 +d1τ, for d0,d1 ∈ GF(q), then in PG(4, q), these points correspond to lines p∞,pδ of the regular spread S where p∞ = h(0, 0, 1, 0, 0), (0, 0, 0, 1, 0)i, pδ = h(1, 0,d0,d1, 0), (0, 1, t0d1,d0 + t1d1, 0)i. 3 Properties of conics and translation ovals 2 Let C be a non-degenerate conic in PG(2, q ), q even, that meets ℓ∞ in a point P∞, so C has nucleus N ∈ ℓ∞. Let C = C\{P∞}. We use the term C-points for (affine) points 2 of PG(2, q ) in C. If α is a Baer subplane secant to ℓ∞, then α meets C in either a subconic, or in at most four points; so α meets C in either a q-arc or in at most four points. Baer subplanes secant to ℓ∞ that meet C in a q-arc are called C-planes. The following properties about C-planes and C-points is straightforward to prove, and is also a special case of Theorem 3.2. 3 2 Lemma 3.1 Let C = C∪{P∞} be a conic in PG(2, q ), q even, that meets ℓ∞ in a point P∞. Define C-points to be points of C, and C-planes to be Baer subplanes secant to ℓ∞ that meet C in a q-arc. Then the following hold. 1. Two distinct C-points lie in a unique C-plane. 2. Every affine point not in C lies in a unique C-plane. 3. If a Baer subplane secant to ℓ∞ contains at least three C-points, then it either con- tains exactly four C-points, or is a C-plane. Our aim in this article is to work in the Bruck-Bose representation in PG(4, q), and consider a set of C-points and C-planes that satisfy the properties given in Lemma 3.1, and see if we can reconstruct the conic. In fact, it turns out that these geometric properties do not uniquely reconstruct the conic, rather, they determine a translation oval. So we first show that a translation oval satisfies the required geometric properties. n Recall that a translation oval in PG(2, 2h) is projectively equivalent to O(2n)= {(t2 , t, 1) : t ∈ GF(q) ∪ {∞}} where (n, h) = 1. Note that O(2n) has nucleus N = (0, 1, 0), and O∪{N} is a translation hyperoval. Further, if n = 1, then O(2) is a non-degenerate conic. n Theorem 3.2 Let O(2n) = {(t2 , t, 1) : t ∈ GF(q) ∪ {∞}} be a translation oval in 2 h PG(2, q ), q =2 , (n, h)=1, where ℓ∞ has equation z =0. Define C-points to be points n of O = O(2 ) \{(1, 0, 0)}. Define C-planes to be Baer subplanes secant to ℓ∞ that contain q points of O. Then the following hold. 1. Each C-plane meets O in a q-arc. 2. Any two distinct C-points lie in a unique C-plane. 3.
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