On deriving unitals

DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 197/198 (1999) 137 141

On deriving unitals Aart Blokhuis a'b, Christine M. O ' K e e f e ~,* a Department of Mathematics and Computer Science, Free Unit,ersity q/' Amster&mT, De Boelelacm 1081a, 1081 HV Amsterdam, Netherlands t, Department o/ Mathematics and Computer Science, Technical University q/ Eimthoven, P.O. Boy 513. 5600 MB, Eindhoven, A'etherlands c Department q~ Pure Mathematics, The University Of Adelaide, Adelaide 5005. Australia Received 9 July 1997; revised 6 January 1998; accepted 3 August 1998

Abstract We provide short proofs that suitable unitals in derivable projective planes give rise to unitals in the derived planes, Some known constructions of unitals in Hall planes are immediate corollaries. @ 1999 Elsevier Science B.V. All rights reserved Keywords. Unital; Projective plane; Affine points, Baer subplane

1. Introduction Throughout this note ~z will denote a projective plane o f square order q. A blockin9 set in 7z is a set o f points which meets every line o f ~z, and a blocking set ,N is minimal if ~ \ { P } is not a blocking set for any point P ~ ~ . A tam.lent o1" a blocking set ~ is a line which meets ,~ in exactly one point; so a blocking set is m i n i m a l if and only if each o f its points lies on a tangent. A unital in 7z is a set o1" q x / ~ + 1 points which meets every line o f 7z in either 1 or x / ~ + 1 points. A line which meets a unital q/ in 1 point is called a tangent while a line which meets 4/ in V ~ + I points is called a secant. Thus a unital is a blocking set, and in fact: T h e o r e m 1 (Bruen and Thas [6]). Let ~ be a minimal bloc'kin# set in a prq/ective plane o f order q. Then

IBI ~
I with equality iJ and only (/" ,~ is a unital.

A Baer subplane o f rr is a subplane o f order x/q' Each line o f zr meets a Baer subplane in either 1 or x/q + 1 points, and a set o f ~ + 1 points which is the intersection o f a Baer subplane with a line is a Baer subline. * Corresponding author. E-mail: [email protected]. 0012-365X/99/$ see front matter (~) 1999 Elsevier Science B.V. All rights reserved PII: S001 2 - 3 6 5 X ( 9 8 ) 0 0 2 2 8 - 3

138

A. Blokhuis, CM. O'Keefe/Discrete Mathematics 197/198 (1999) 137-141

Let (,,~ be a line of n and let s~' be the affine plane n \ ( ~ . A set 9 of V~ + 1 points of ( ~ is called a derivation set if for each pair of points X, Y ~ ~d such that the line AT meets ( ~ in a point of 9 , there is a Baer subplane of n containing X, Y and ~. (This Baer subplane is necessarily unique.) Given a derivation set ~ for n, there is an affine plane ~¢~ of order q with points the points of ~¢, and lines, each set of points of ~d on a line of n meeting { ~ \ ~ together with each set of points of .d on a Baer subplane of rc containing 9 . The unique completion of this affine plane to a projective plane by the addition of the points on a line ?~ is the plane n ~ derived from n with respect to 9 . In the following, we will regard n and n ~ as having the same affine points and we will also identify the points of ( ~ \ ~ in n with the corresponding points of #~ in n ~. (For a full discussion of derivation see [12].)

2. Deriving unitals, I A first approach to constructing unitals in derived planes relies on the characterisation of unitals among blocking sets (Theorem 1).

Theorem 2. L e t q be a square and let n be a projective plane o f order q. L e t ~ be a derivation set in n. I f there is a unital ~l[ in n such that o~l N @ = ~ and each Baer subplane o f n containing ~ meets vii in 1 or at least v ~ + 1 points, then there exists a unital in the derived plane 7r~ with the same points as ~ll.

Proof. Let , d = n \ f ~ , where E~ is the line containing 9 , and let ~ " = @ ' \ ~ . Let ~ " be the set of points P ' E #~ which correspond to the points P E f ~ N ~' (so I ~ ' ] = 1 or V ~ + 1 ) and let ~ ' ~ = ok"U ~/". Then ~2Z~y is a set of q v ~ + 1 points in n ~ which meets every line of n ~ in 1 or at least x/~+ 1 points. Thus ~Z/~ is a blocking set. It is minimal, for otherwise there is a point P E ~¢~ on no tangent, implying ]o~'e]~>(q + 1)x/~ + l. By Lemma l, o~,~,' is a unital in n ~. []

Theorem 3. L e t q be a square and let n be a projective plane o f order q. L e t ~ be a derivation set in n. I f there is a unital oil in n such that all N 9 = ~ and each Baer subplane o f n containing 9 meets dig in V ~ + 1 or at least 2 v ~ + 1 points, then there exists a unital in the derived plane n ~ with the same affine points as ~ll.

Proof. Let d = 7r\#~ where #~ is the line containing 9 , and let °~"= ~ \ ~ . By assumption ~" has 1 or ~ + 1 points on each line of n, other than ~ , meeting f ~ \ 9 and 0 or at least v ~ points on each Baer subplane of n on 9 . The set of Baer subplanes of n containing c~ is a set of x/q + 1 parallel classes of lines in the affine plane ~ ' ~ derived from ~d; so let 9 ' be the set of v ~ + 1 points of f ~ which lie on these lines in the projective completion n ~. It follows that v~'~= ~ " U 9 ' is a blocking set of q x / ~ + 1 points in n ~ which meets every line in 1 or at least v / ~ + 1 points. As in the proof of Theorem 2, :ht~ is a unital. []

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3. Deriving unitals, II An alternative approach to constructing unitals in derived planes verifies the proper-ties of a unital directly.

Theorem 4. L e t q be a square and let 7r be a projective plane o f order q. Let ~ be a derivation set on a line { ~ in ~z. L e t o2/ be a unital in 7z such that ~2/N ~J = O, each Baer subplane o f r~ containing @ meets ~ in at least 1 point and (i) t f f , ~ is a secant o f ~2/ then at least ( x / ~ + 1) 2 Baer subplanes on ~ meet ¢/ itl exactly one point, (ii) tf/~_ is a tangent o f q/ then at least x / ~ ( x / ~ + 1) Baer subplanes on ~ meet q/ in exactly one point. Then there exists a unital in the derived plane 7rv with the same points as 4/.

Proof. Let o2/~ be defined as in the proof of Theorem 2. In order to prove that ~k':~ i,,; a unital in rc~, it suffices to show that each Baer subplane on c~ meets ~2/ in either t or x / ~ + 1 points. Let 5~ = {~)l . . . . . :~'s}, where s = q ( x f q + 1), be the Baer subplanes on ~ and let ti=i~iN°2/[~l. (i) Counting the ordered pairs (P, #2), where P E °2/, ~ E 5~ and P E ~ , yields - v / q ) ( v ~ + 1)

~ti=(qv~ i=l

and counting the ordered triples (P, Q, ;~3) where P, Q E ~2/,M E 5,~ and P, Q E ,~ yields s

~ti(t,-

1) =-(x/~ + 1)(q - x/q-- l)(v/q + 1)v/q

i--I

(this is the number of secants distinct from {-~ on a point of ~ multiplied by the number of ordered pairs of points of o2/ on such a secant). Thus we obtain s

~--~(t,.- ( V ~ + 1))2 = q(x/~ + 1) 2. i=1

Now, there are at least ( v ~ + 1) 2 Baer subplanes ~ E 5 P with t~ = 1, and ( , ~ + 1 )2 of. them contribute q (x/q + 1 )2 to the sum. The remaining Baer subplanes in ,c/' therefore contribute zero to the sum, and hence intersect a2/ in exactly x/q + 1 points. (ii) The argument in this case is analogous, with ~ t , = q v ~ ( v / - q + 1)

and

~ti(ti-

1)=(v~+

1)(q- v~)(v~+

1)~.

[]

t=|

Theorem 5. L e t q be a square and let r~ be a projective plane o f order q. L e t ~ be a derivation set in ~. I f there is a unital 02/ in rt such that ~ N ~ = 9 , each Baer

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subplane o f n containing ~ meets °ll in at least v/~ + 1 points and there are at least V ~ + 1 Baer subplanes on ~ that meet ell in exactly xfq + 1 points, then there exists a unital in the derived plane n ~ with the same affine points as ~ll. Proof. Let ~ be as in the proof o f Theorem 3. It suffices to show that each Baer subplane on ~ meets ~ \ ( ~ in 0 or v/~ points o f n \ ( ~ . Our argument is again analogous to that given in Theorem 4(i), but with ti = ]~i n (~k'\f~)]/>0. We find that

~ti=(qv~-

V ~ ) ( V ~ + 1)

i=1

and

~-~ti(ti-

1 ) = ( x / ~ + 1 ) ( q - 1)v/q(v/~ - 1).

[]

i=1

4. Examples Example 1 (Buekenhout [7]; Gr/ining [11]). Let ~/Z be a classical unital in a Desarguesian plane n = P G ( 2 , q ) . Let ~ be a Baer subline comprising the points o f d// on a secant line to ~//. Theorem 3 or 5 guarantees the existence of a unital in the Hall plane PG(2, q)~, with the translation line a secant and with the same affine points as ~ .

Example 2 (Barwick [4]; see also Rinaldi [13]). Let ~ be a classical unital in a Desarguesian plane n = PG(2, q). Let ~ be a Baer subline comprising points on a secant line to q/, disjoint from ~#. Theorem 2 or 4 guarantees the existence o f a unital in the Hall plane PG(2, q)~, with the translation line a secant and with the same affine points as ~N.

Example 3 (Barwick [5]; see also Buekenhout [7]; Rinaldi [14]). Let ~/ be a Buekenhout-Metz unital (that is, a unital constructed as in [7, Remark 4]) in a Desarguesian plane n = PG(2, q), with respect to the tangent ( ~ , and let c~ be a Baer subline on ( ~ , disjoint from ~#. Theorem 2 or 4 guarantees the existence o f a unital in the Hall plane PG(2, q)~, with the translation line a tangent and with the same affine points as 9/. Example 4 (Barlotti and Lunardon [2], see also Dover [9]). Let ~' be a BuekenhoutMetz unital in a Desarguesian plane n = PG(2,q), with respect to the tangent ( ~ . Let T = ~ n og and let ~ be a Baer subline comprising the points o f ~// on a secant line through T. Theorem 2 or 4 guarantees the existence of a unital in the Hall plane PG(2,q) ~, with the translation line a tangent and with the same affine points as d//. We remark that every known unital in PG(2, q2) can be shown to be o f BuekenhoutMetz type [ 1,3, 8, 10], and that Buekenhout-Metz unitals are also known as Buekenhout unitals o f parabolic type.

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Acknowledgements W e t h a n k P r o f e s s o r J e f T h a s f o r s u g g e s t i n g t h e u s e o f t h e v a r i a n c e trick. T h i s w o r k was supported by the Australian Research Council.

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