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A Class of Buekenhout Unitals in the Hall Plane

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A Class of Buekenhout Unitals in the Hall Plane A class of Buekenhout unitals in the Hall plane S. G. Barwick Abstract Let U be the classical unital in PG(2;q2) secant to `1. By deriving PG(2;q2) with respect to a derivation set disjoint from U we obtain a new unital U 0 in the Hall plane of order q2. We show that this unital contains O'Nan configurations and is not isomorphic to the known unitals of the Hall plane, hence it forms a new class of unitals in the Hall plane. 1 Introduction A unital is a 2-(q3 +1;q+1; 1) design. A unital embedded in a projective plane of order q2 is a set U of q3 + 1 points such that every line of the plane meets U in 1 or q + 1 points. A line is a tangent line or a secant line of U if it contains 1 or q +1pointsofU respectively. A point of U lies on 1 tangent and q2 secant lines of U. A point not in U lies on q + 1 tangent lines and q2 − q secant lines of U. An example of a unital in PG(2;q2), the Desarguesian projective plane of order q2,istheclassical unital which consists of the absolute points and non-absolute lines of a unitary polarity. It is well known that the classical unital contains no O'Nan configurations, a configuration of four distinct lines meeting in six distinct points (a quadrilateral). In 1976 Buekenhout [4] proved the existence of unitals in all translation planes of dimension at most 2 over their kernel. 2 2 Let U be the classical unital in PG(2;q ) secant to `1.WederivePG(2;q )with respect to a derivation set disjoint from U.LetU 0 be the set of points of H(q2), Received by the editors November 1994 Communicated by J. Thas AMS Mathematics Subject Classification : 51E20. Keywords : Unitals, derivation, Hall planes. Bull. Belg. Math. Soc. 3 (1996), 113{124 114 S. G. Barwick the Hall plane of order q2, that corresponds to the point set of U. We prove the following results about U 0. A class of Buekenhout unitals in the Hall plane 115 0 0 2 Theorem 1 The set U forms a Buekenhout unital with respect to `1 in H(q ). Theorem 2 The unital U 0 contains no O'Nan configurations with two or three 0 0 vertices on `1.IfU contains an O'Nan configuration l1;l2;l3;l4 with one vertex 0 0 0 T = l1 \ l2 on `1, then the lines l1 = l1 \ U and l2 = l2 \ U are disjoint in PG(4;q). Theorem 3 If q>5, the unital U 0 contains O'Nan configurations. If H is a 0 0 0 point of U n `1 and l a secant of U through H that meets the classical derivation set, then there is an O'Nan configuration of U 0 that contains H and l. The only unitals previously investigated in the Hall plane is a class of Buekenhout 2 unitals found by Gr¨uning [5] by deriving PG(2;q ) with respect to U \`1.Weshow that the class of unitals U 0 is not isomorphic to Gr¨uning's unital and so forms a new class of unitals in H(q2). In [2] the Buekenhout and Buekenhout-Metz unitals of the Hall plane are studied. 2 Background Results We will make use of the Andr´e [1] and Bruck and Bose [3] representation of a translation plane P of dimension 2 over its kernel in PG(4;q). The results of this section are discussed in [6, Section 17.7]. Let Σ1 be a hyperplane of PG(4;q)and S aspreadofΣ1. The affine plane Pn`1 can be represented by the affine space PG(4;q) nΣ1 as follows: the points of Pn`1 are the points of PG(4;q) nΣ1,the lines of Pn`1 are the planes of PG(4;q)thatmeetΣ1 in a line of S and incidence is the natural inclusion. We complete the representation to a projective space by letting points of `1 correspond to lines of the spread S.NotethatP is Desarguesian if and only if the spread S is regular. We use the phrase a subspace of PG(4;q)nΣ1 to mean a subspace of PG(4;q) that is not contained in Σ1. Under this representation, Baer subplanes of P secant to `1 (that is, meeting `1 in q + 1 points) correspond to planes of PG(4;q) that are not contained in Σ1 and do not contain a line of the spread S. Baer sublines of P meeting `1 in a point T correspond to lines of PG(4;q) that meet Σ1 in a point of t, the line of S that corresponds to T . A Baer subplane tangent to `1 at T corresponds to a ruled cubic surface of PG(4;q)thatconsistsofq + 1 lines of PG(4;q)nΣ1, each incident with the line t of S and such that no two are contained in a plane about t,[9]. 2 Let U be the classical unital in PG(2;q ) secant to `1. Buekenhout [4] showed that the set of points U in PG(4;q) corresponding to U forms a non-singular quadric that meets the underlying spread in a regulus. If U is a unital of a translation plane P of dimension at most 2 over its kernel and U corresponds to a non-singular quadric of PG(4;q) that contains a regulus of the underlying spread, then U is called a Buekenhout unital with respect to `1. Note that the classical unital is Buekenhout with respect to any secant line. 2 2 Let PG(2;q ) be the Desarguesian plane of order q and let D = fT0;:::Tqg be 2 a derivation set of `1. Deriving PG(2;q ) with respect to D gives the Hall plane of order q2, H(q2)(see[8]).TheaffinepointsofH(q2) are the affine points of PG(2;q2). TheaffinelinesofH(q2) are the lines of PG(2;q2) not meeting D together with the 116 S. G. Barwick Baer subplanes of PG(2;q2)thatcontainD. The line at infinity of H(q2) consists 0 2 of the points of `1nD and q +1newpoints,D = fR0;:::;Rqg.IfwederiveH(q ) with respect to D0,wegetPG(2;q2). The Hall plane contains other derivation sets 0 0 2 of `1,wecallD the classical derivation set of H(q ). Let Σ1 be a hyperplane of PG(4;q)andletS be the regular spread of Σ1 that 2 generates PG(2;q ). Let R = ft0;:::;tqg be the regulus of S that corresponds to D. The spread S0 obtained from S by replacing the regulus R with its complementary 0 0 0 2 regulus R = fr0;:::;rqg (that is, S = SnR[R) generates the Hall plane H(q ). We use the following notation throughout this paper: we denote the lines at 2 2 0 infinity of PG(2;q )andH(q )by`1 and `1 respectively; if D is the derivation set used to derive PG(2;q2)togiveH(q2), then we denote by D0 the classical derivation 0 2 set of `1;inPG(4;q), we denote the spreads of Σ1 that generate PG(2;q )and 2 0 H(q )byS and S respectively. If T is a point of `1, we denote the corresponding line of the spread in PG(4;q)byt.LetU be the non-singular quadric of PG(4;q) that corresponds to U and U 0. 3 The Buekenhout unital 2 Let U be the classical unital in PG(2;q ) that is secant to `1. Derive the plane using a derivation set D that is disjoint from U.LetU 0 be the set of points in H(q2) that corresponds to the point set of U. Theorem 1 The set U 0 forms a unital in H(q2). Proof The set U 0 contains q3 +1 pointsof H(q2). We show that every line of H(q2) meets U 0 in either 1 or q +1 points, from which it follows that U 0 is a unital. Clearly 0 0 `1 meets U in q +1pointssince`1 meets U in q +1points. 2 0 Let l be a line of H(q )thatmeets`1 in the point T .IfT is not in the derivation set D0, then the points of l lie on a line of PG(2;q2)andsol contains 1 or q +1 points of U. Hence l contains 1 or q +1pointsofU 0. Suppose T is in the derivation set D0,thenT=2 U 0.LetU be the non-singular quadric of PG(4;q) that corresponds to U,thenU also corresponds to U 0.Letα be the plane that corresponds to the line l,soα \ Σ1 = t, the line of the spread corresponding to T .Nowα meets U in either a point, a line, a conic or two lines. If α \U contains a line, then t contains a point of U which is a contradiction as T=2 U 0.Thusα meets U in either 1 or q +1 pointsandsol meets U 0 in either 1 or q +1points.Notethatifl is secant to U 0, then the q +1pointsl \ U 0 are not collinear in PG(2;q2); they form a conic in the Baer subplane that corresponds to l and a (q +1)-arcofPG(2;q2). 2 The proof of this theorem shows in fact that U 0 is a Buekenhout unital with 0 2 0 respect to `1 in H(q ).
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