4-Gonal configurations with parameters r=q2+1 and k=q+1

J.A. THAS

4-GONAL CONFIGURATIONS WITH PARAMETERS

r =q2 + 1 A N D k = q +

1

Part H 1. I N T R O D U C T I O N

A finite 4-gonal configuration [2] is an incidence structure S = ( P , B, I), with an incidence relation satisfying the following axioms (i) each point is incident with r lines (r>~2) and two distinct points are incident with at most one line; (ii) each line is incident with k points (k >_-2) and two distinct lines are incident with at most one point; (iii) if x is a point and L is a line not incident with x, then there are a unique point x' and a unique line L' such that xIL' Ix' IL. If [P[=v and [Bl=b, then v = k ( k r - k - r + 2 ) and b = r ( k r - k - r + 2 ) . In [5] D.G.Higman proves that the positive integer k + r - 2 divides kr(k-1)(r-1). Moreover, under the assumption that k > 2 and r>2, he shows that r ~<( k - 1)2 + 1 and k <~( r - 1)2 + 1.

2.

4-GONAL CONFIGURATIONS

WITH PARAMETERS

r=q2+l

AND

k=q+I

Let S=(P,B,I) be a 4-gonal configuration with parameters r=q2+l, k=q+ 1 (q> 1). If the points x, y (resp. lines L, M) of S are collinear (resp. concurrent), then we write x,,,y (resp. L ~ M ) ; otherwise we write x,~y (resp. L,~ M). The trace of three points x, y, z, for which x,~y, y,~z, z,~x, is defined to be the set tr(x,y,z)={uePII u ~ x , u ~ y , u ~ z } . In [1] R.C.Bose proves that ]tr(x,y, z ) ] = q + l . The triple (x,y, z), where x,,~y, y*,z, z,,,x, is said to be regular provided each point collinear with at least three points of tr(x, y, z) is actually collinear with all points of tr(x, y, z). When the triple (x, y, z) is regular, the span of x, y, z is defined unambigously as the set sp (x, y, z)= tr(u~, u2, u3) for any three distinct points u~, u2, u3 in tr(x,y, z) (remark that Isp(x, y, z)[ = q + 1). From the regularity of the triple (x, y, z) follows immediately the regularity of the triple (u~, u2, u3), where ua, u2, u3 are any distinct points of sp(x, y, z) or tr(x, y, z). If for a point x each triple (x, y, z), with x,~y, yn, z, z,~x, is regular, x is said to be regular. The trace of two non-concurrent lines L, M is defined to be the set tr (L,M) ={NeB II N ~ L , N ~ M } (remark that [tr(L, M ) l = q + l ) . The pair (L, M). Geometriae Dedicata 4 (1975) 51-59. All Rights Reserved Copyright © 1975 by D. Reidel Publishing Company, Dordrecht-Holland

52

J.A. TIlAS

L,~M, is said to be regular provided each line concurrent with at least two lines of tr(L, M) is actually concurrent with all lines of tr(L, M). When the pair (L, M), L,.~M, is regular, the span of L and M is defined unambigously as the set sp(L, M ) = t r ( N l , N2) for any two distinct lines N1, N2 in tr(L,M) (remark that Isp(L,M)l=q+l). From the regularity of the pair (L,M) follows immediately the regularity of the pair (Nx, N2), where Nt, Nz are any distinct lines of tr(L, M) or sp(L, M). If for a line L each pair (L, M), L,~M, is regular, L is said to be regular. A regular point x is called biregular provided each line incident with it is regular. Let x be a regular point and let y be a point which is not collinear with x. The set of the q2 + 1 points which are collinear with x and y is denoted by Ox~,. IfBx,={sp(za, z2, za) II zl, z2, z3 are distinct points of Ox,}, then the incidence structure ~(x,y)=(Oxy, Bxy, e) is a 3 - ( q Z + l , q + l , 1) design (this follows immediately from the regularity of the point x). Consequently 0~(x, y) is a finite inversive plane of order q. Next let O~ be the set {La, L2,..., L~2+a), where LI, L2, .... L~+I are the q2+ 1 lines which are incident with x. If B*y={{L,~,L,~, ...,L,o+~}, be{l, 2 . . . . . q2+l} II Z,~, L~ . . . . . L,~+~ are incident with the resp. points of a circle of the inversive plane n(x, y)}, then the incidence structure x,(x) =(Ox, B'y, e) evidently is a finite inversive plane of order q which is isomorphic to n(x, y). In [11] we have proved that the set B*,, and consequently the inversive plane ~y(x), is independent of the choice of the point y. The inversive plane zcy(x)(resp. the set B'y), which is independent of the choice of y, will be denoted by z~(x) (resp. B*). 3. TIIE 4-GONAL CONFIGURATIONS

T(O) OF

J. TITS

Consider an ovoid O of the threedimensional projective space PG(3, q) over the Galois field GF(q). Let PG(3, q) be embedded as a hyperplane H in PG(4, q)=P. Define points as (i) the points of P-tI, (ii) the hyperplanes X o f P for which [Xc~O[= 1, and (iii) one new symbol x®. Lines are (a) the lines of P which are not contained in H and meet O (necessarily in a unique poin0, and (b) the points of O. Incidence is defined as follows: Points of type (i) are incident only with lines of type (a); here the incidence is that ofP. A point X o f type (ii) is incident with all lines ~ X o f type (a) and with precisely one line of type (b), namely the one represented by the unique point of O in X. Finally, the unique point x® of type (iii) is incident with no line of type (a) and all lines of type (b). This incidence structure T(O), which is due to J. Tits [2], is a 4-gonal configuration with parameters k = q + 1, r=q 2+ t, v = ( q + l)(q 3 + 1), b=(q 2 + 1) x (q3 + 1). It is easy to check that x~ is a biregular point of the 4-gonal

4-GONAL CONFIGURATIONS

53

configuration T(O). If Q(d, q), de{3, 4, 5}, is the 4-gonal configuration arising from a non-singular hyperquadrie Q of index 2 of the projective space PG(d, q), then I(0) is isomorphic to Q(5, q) if and only if O is an elliptic quadric of PG(3, q). In this case each point and each line of I"(0) are regular. We also remark that it is easy to show that each line of T(O) is regular if and only if O is an eUiptie quadric. In [11 ] we have proved the following: (a) Up to an isomorphism there is only one 4-gonal configuration with parameters r = 5 and k = 3. (b) If the 4-gonal configuration S = (P, B, I) with parameters r = q2 + 1 and k=q+ 1, where q is even and q>2, possesses a biregular point, then S is isomorphic to a 4-gonal configuration T(O) of J. Tits. (c) If the 4-gonal configuration S = (P, B, I) with parameters r = q2 + 1 and k = q + 1, where q is odd, possesses a biregular point x and if the inversive plane n(x, y) is egglike for each y,*,x (i.e. if the inversive plane n(x) is egglike), then S is isomorphic to a 4-gonal configuration I"(0) and consequently to Q(5, q) (taking account of the fact that each ovoid O of PG(3, q), q odd, is an elliptic quadric). Finally we remark that each known 4-gonal configuration with parameters r = q 2 + 1, k=q+ 1 (q> 1) is isomorphic to a configuration T(O) of J.Tits. 4. THEOREM

If the 4-gonal configuration S=(P, B, I) with parameters r=q2+ 1 and k=q+ l, where q is even and q>2, possesses a regular point, then S is isomorphic to a 4-gonal configuration T( O) of J. Tits. Proof Let x be a regular point of the 4-gonal configuration S. Taking account of the main theorem in [11] we have only to prove that x is also biregular. For that purpose we consider a line L~ which is incident with x and also a line L2 with L~ -~L2. We shall show that the pair (L1, L2) is regular. Let L'I be the line which is incident with x and concurrent with L2. We also consider a line L~, L'I #L~, which is concurrent with L1 and L2. Let y, z, u be defined by LI IylL2IzlL2IulL'~. The q+ 1 lines which are concurrent with L~, L~ (resp. Lt, L2) are denoted by Lt, L2 . . . . . Lq+l (resp. L~, L2 . . . . . Lq+ 0. Call vz, w~ (resp. vi, w't) the points for which L'~Iv, ILt Iw~IL'2 (resp. L~Iv'~IL'tIw~IL2), i=3, 4 .... , q + l . If v~, w~, x , , x~2. . . . , x~.q_t (resp. v~, w~, x',, x'~2. . . . . x~.q-x) are the q + l points, incident with L, (resp. L'~), then tr(x, z,x~j)=C,j (resp. tr(x, z,x~j)=C;j) is a circle of the t

t

!

r

54

J.A. THAS

inversive plane z~(x, z) (i= 3, 4,..., q+ 1; j = 1, 2,..., q-1). Evidently the circles Csl, Cl2 . . . . . Cl,q_: (resp. C[1, C[~ ..... C[,q-x) constitute a flock F~ (resp. F~) of the inversive plane z~(x, z) (i=3, 4 .... , q+l). The carders of this flock Ft (resp. F't) are the points y and u. As q is even the inversive plane ~(x, z) is egglike, and so each flock of ~(x, z) is uniquely determined by its carriers ([6], [I0]). Consequently Fa=F, . . . . . F~+I=F~=Ff~ . . . . . F~+I. Next we consider a point x~, ie{3, 4, ..., q+l}, je{1, 2, ..., q - l } . From the preceding there follows that each line L'z, •=3, 4 .... , q+ 1, is incident with a point of sp(x, z,x~j)-{x, z}. There follows immediately that the point x~j is incident with one of the lines L't. So we have proved that the pair (L1, L2) is regular. As (L~, L2) is regular for each L2 which is not concurrent with L~, the line Lx is regular. Hence the point x is biregular, and so the theorem is completely proved.

Remark. In the same way it is possible to prove the following result: I f the 4-gonal configuration S=(P, B, I) with parameters r = q 2 + 1 and k = q + 1, with q odd, possesses a regular point x and if the inversive plane ~(x,y) is egglike for each y,~x (i.e. if the inversive plane ~(x) is egglike}, then S is isomorphic to a 4-gonal configuration T(O) and consequently to 0(5, q). 5. CHARACTERIZATION OF THE 4-GONAL CONFIGURATION 0 ( 5 , q )

If each point of the 4-gonal configuration S=(P, B, I) with parameters r = q2 + 1 and k = q + 1 (q > 2) is i'egular, then S is isomorphic to the 4-gonal configuration Q(5, q) arising from a non-singular hyperquadric Q of index 2 of the projective space PG(5, q).

Proof We have to distinguish two cases. (a) q even. From 4. follows immediately that each line of S is regular and that S is isomorphic to a 4-gonal configuration T(O) of J. Tits. As each line of S is regular the ovoid O is an elliptic quadric. Consequently T(O) is isomorphic to Q(5, q), and so S is isomorphic to Q(5, q). (b) q odd. Taking account of the remark in 4. it is sufficient to prove that the inversive plane zt(x, y) is egglike for each pair (x, y) with x,~y. Let x, y be two non-collinear points of S and let C be a circle of the inversive plane zt(x, y) of order q. Now we consider a point z of tr(ul, u2, u3) {x, y}, where ul, u2, ua are any three distinct points of C. Next a bijection -

4-GONAL CONFIGURATIONS

55

0 of z~(x, y) onto itself is defined in the following way: if u~z~(x, y), if L~ is the line incident with u and x, if L~ I wl I MIz, if MIw2 IL2 Iy, then O(u) is the point of zc(x, y) which is incident with the line L2. We remark that the fixed points of 0 are precisely those of C. We shall prove that 0 is an inversion of the inversive plane x(x, y). For that purpose we have to show that 0 is an automorphism of~z(x, y). Consider an arbitrary circle D of z~(x, y). If we project D from the point x there arises a circle E of the inversive plane ~(x) (see 2.). This circle E is also the projection from x of a circle D' of z~(x, z). If we project D' from z there arises a circle E' of the inversive plane z~(z). This circle E' is also the projection from z of a circle D" of z~(y, z). If we project D" from y there arises a circle E" of the inversive plane z~(y). This circle E" is also the projection from y of a circle D" of ~(x, y). As O(D)=D" the bijection 0 is an automorphism of z~(x, y). Consequently 0 is an inversion of the inversive plane z~(x, y). Hence every circle of the finite inversive plane z~(x, y) is the axis of an inversion, and so ~(x, y) is miquelian [3]. There follows immediately that z~(x, y) is egglike, and so the theorem is completely proved. 6. LEMMA

Suppose that (x, y, z), with x ,v y, y,~ z, z ,~x, is a regular triple of points of the 4-gonal configuration S = (P, B, I) with parameters r = q2 + 1 and k = q+ 1 (q> 1). I f the point u is collinear with no point of sp(x, y, z), then u is collinear with exactly two points of tr(x, y, z).

Proof If V = { u e P II u is coUinear with no point of sp(x, y, z)}, then iV[ = ( q + 1) x (q3 + 1) - q - 1 - (q + 1)(q 2 + 1) q + (q + 1) q (q2 _ q)/2 + q(q + 1) = q2(q2 _ 1)/2 =d. Let V={ul, u2 . . . . . u~}. The number of points w of tr(x,y, z) which are collinear with ut is denoted by ts. If we count the number of ordered pairs (u~, w), where uf~V and wetr(x, y, z) is coUinear with u~, we obtain: t, = ( q +

1)(q2-q)q=q2(q2-

1).

(1)

f

If we count the number of ordered triples (us, w, w'), where u ~ V and where the distinct points w, w'~tr(x, y, z) are collinear with us, we obtain: t,(t,-

1) = ( q +

1) q ( q 2 -

q) = q 2 ( q 2 _

1).

(2)

t

From (1) and (2) follows that ~,,t~=2q2(q2-1). As -(y~,t,)2)/d=O, we have t,=r=(E,t,)/d=2 v i e { l , 2 . . . . , d}. Consequently each element of V is collinear with exactly two points of tr(x, y. z).

56

J.A. THAS

7. THEOREM

Suppose that the 4-gonal configuration S=(P, B, I), with parameters r=q 2 + 1 and k = q + 1 (q > 1), has a 4-gonal subconfiguration S' = (P', B', I'), with parameters k' = r' = k, for which the following condition is satisfied: if x, y, zeP', with x,~,y, y,,,z, z,,~x, then the triple (x, y, z) is regular and moreover sp(x, y, z)=P'. Then we have (i) S has an involution 0 whichfixes P' pointwise (ii) S' is isomorphic to the 4-gonal configuration Q(4, q). Proof (i) If x ~ P - P ' , then the q2 + 1 points of P' which are collinear with x constitute an ovoid Or of S' [12]. Now we consider three distinct points y, z, ueOr. As sp(y, z, u)=P', it follows that sp(y, z, u)=Ox. Let weOx - s p ( y , z, u). Since w is collinear with no point of sp(y, z, u), the point w is coUinear with exactly two points x, x' of tr(y, z, u) (remark that x'~P'). Consequently sp (y, z, u) u {w} c Or c~ Ox, Next let w' e sp (y, z, u) and consider sp(w, w', 0, where t is a variable point of sp(y,z, u)-{w'}. As sp(w, w', t ) c O x n O ~ , , we have lU,sp(w, w', t ) l = q 2 - q + 2<<.lO~c~O~,l. Suppose a moment that s c O t - Or,. The point s is collinear with q + 1 points of Ox. which do not belong to Orr~Or,. Hence IO~c~Or,l+q+l<<.lOx,I = qZ + 1 or q2 + 3 ~
4-GONAL CONFIGURATIONS

57

Hence P' contains at least one point which is collinear with x, y, z. From [8] there follows immediately that S' is isomorphic to the 4-gonal configuration

Q(4,q). (b) q odd. Let x, y, zEP' with x,~ y, y,~ z, z ~ x. A reasoning analogous to that in (a) leads to [tr (x, y, z)nP'l is even. As Itr(x, y, z)c~P'l >2=~tr(x,y,z) c P ' , we have Itr(x, y, z)r~p'le{0, 2, q + l } . If tr(x,y, z)c~P'=(w, w'}, we say that (w, w') is a nice pair of P'. We shall prove that each pair (w, w'), w,~w', of P' is nice, i.e. that [tr(x, y, z)c~P'l~{0, 2} Yx, y, z~P' with x,~y, y ' ~ z , Z'uX.

For that purpose we consider an ovoid O~ of S' (u~P-P'). Let ~ (resp. fl) denote the number of sets sp (x, y, z), x, y, zE O~, for which Itr (x,y, z) c~P'l = 0 (resp. Itr(x, y, z)c~e'l=2). We remark that u, O(u)~tr(x, y, z). Now we have ~x+fl=q 3 +q (remark that (O,, {sp(x,y, z) II x, y, z~Ou}, e) is an inversive plane of order q) and oc(q - 1) + fl ( q - 3) = number of points of P - P' which are not collinear with u or O(u)= (q + 1) (q3 + 1) - 2 (q-' + 1) (q - 1) - 2 - (q + 1) x(q2+l). Consequently ~x=fl=(q3+q)[2. So with Ou there correspond (q3 + q)/2 nice pairs of P'. Next we count the number of pairs (Ou, nice pair corresponding with O~) in two different ways. If ~, is the number of nice pairs of P', we obtain ((q + 1)(q3 + 1) - ( q + 1)(q2 + 1))fl]2 = ~,( q - 1)]2. Hence ~=qa(q+l)(q2+l)[2. But the number of pairs (w, w'), w,~w', of P' also equals q3(q+l)(q2+l)[2, and so each pair (w, w'), w,~w', of P' is nice. There results that Itr(x, y, z) c~P'le{0, 2} Vx, y, zeP' with x,~y, yo.,z, z,.,.,x. Now we consider an ovoid O,, of S' ( u e P - P ' ) and a point w e P ' - O ~ . The q + 1 points of O~ which are collinear with w are denoted by Xo, xt, ..., x~. Since u, w etr (xo, x~, x2), we have Itr (xo, x~, x2)n P ' I = 2. Let tr (Xo, x~, x2)c~P'={w, w'}. Then the elements of sp(xo, xx, x2) are exactly the q+ 1 points of P' which are collinear with w and w'. As sp(xo, xl, x2)cO~, it follows that sp(xo, xt, x2)={xo, xl, ..., xq}. Consequently the mapping 0~ determined by O.,(x)=x VxeO~ and Ou(w)=w' V w e P ' - O ~ is an involutorial permutation of P' fixing O~ pointwise. We prove that 0~ is an involution of the 4-gonal configuration S'. If w e P ' - O ~ , x~O., x ~ w , then O.(x)~Ou(w). Next we consider distinct points wl, w 2 ~ P ' - Ou with wl ,,o W2. The line of S' which is incident with wt and w2 is denoted by L, and the unique point of Ou which is incident with L is denoted by x. If wIL, w # x , then the q + 1 points of O. which are collinear with w constitute a circle of the inversive plane I~ =(O~, {sp(xo, xl, x2)[I Xo, x~, x2~O~}, ~). With the q points w there correspond q circles of I~ which have two by two only the point x in common. These circles are denoted by C~, C2 .... , Cq, where (7, corresponds with wt, i = 1, 2. IlL' is the line defined by O.(wl) and x, then with the points w'IL', w' ~x, there also correspond q circles Ct, C2, ..., C~ of I~ which have two by two only the point x in corn-

58

J.A. THAS

mon. There follows that {C1, C~ .... , C~} = {C1, C2 . . . . . Cq}, and so C2~{C2, ..., C~}. Consequently O~(w2)IL', and so O,(wl)NOu(w2). From the preceding there follows immediately that 0, is an involution of the 4-gonal configuration S' which fixes O, pointwise (remark that the fixed points of 0, are precisely those of O~). Now we shall prove that each pair of lines of S' is regular. For that purpose we consider two non-concurrent lines LI, L2 of S' and also three distinct lines L~, L2, L~ which are concurrent with L1 and L2. We have to show that each line which is concurrent with L1 and L2 is also concurrent with L3. Suppose a moment that the line L3 is concurrent with L~ and L~, and is not concurrent with L3. We introduce the following notations: L t I x i I L t , Lalx2ILz, Lzly2IL2, L z l y I I L z , LIlzIIL'3, LalzzlL2, LtlxalLa, Lslys IL'2 , xaIL3IzaIL3 * ' (remark that La4L*). The set of the q+ 1 points of S' which are collinear with Yl and x2 (resp. x2 and z~) is denoted by 6"1 (resp. C2). Through C1 (resp. (72) pass ( q - l ) / 2 ovoids O~, (resp. Or,), i = 1, 2, ..., ( q - 1)/2 (since the pairs (Yl, x2), (zl, x2) are nice). There holds: O.,(xl) = x~ , O~,(y2)= Y2 , 0u,(y~) =x2, O,~(L~) = L'~ , 0~,(L2)=L~. Consequently 0,,(L3) =M~ is concurrent with LI and L2. Remark that L3 and M~ are concurrent. Hence M~ is concurrent with L~, L2, L3. There holds Mt¢{L'I,L;,L'3 } (from Mt =L~ there would follow that L~ is concurrent with La, a contra• diction). In the same way one proves that O,~(Ls)=M~ * * is concurrent with L~, L2, L~ and that M*~{L'I, L'2, L'3}. Next we show that M t 4 M j if i4j. Suppose that M~=Mj, i ~ j and let MtlwILa. Then 0~, and O~j contain C1 and w, and so O.,=O~ or i=j, a contradiction. Hence we have (q-1)/2 distinct lines M , and also ( q - 1)/2 distinct lines M*. As there are q + 1 lines which are concurrent with L~ and L2 and as Mr, Ml*~{L1, ' L2, ' L3}, ' at least one of the lines Mt will coincide with one of the lines M*. Suppose that M~=M*, s, te{1, 2, ..., ( q - 1)/2}. Then M~ is concurrent with La and L*, a contradiction since S' contains no triangles. Consequently L3 is concurrent with L~, L2, L~. Hence each pair of lines of S' is regular and so S' is isomorphic to the 4-gonal configuration Q(4, q) [8]. t

t

t

!

t

t

t

!

l

l

Remark. Assume that q is odd. In (ii) we have shown that O. is the set of the fixed points of an involution 0u of Q(4, q). Consequently O. is the intersection of Q(4, q) with some subspace of PG(4, q)~Q(4, q) [4]. As IO~1 = q2+ 1 there follows immediately that Ou is the intersection of Q (4, q) with a subspace PG(3, q) ofPG(4, q), and that O, is an elliptic quadric ofPG(3, q).

4-GONAL CONFIGURATIONS

59

BIBLIOGRAPHY 1. Bose, R.C., 'Graphs and Designs', C.LM.E., II ciclo, Bressanone 1972, 1-104. 2. Dembowski, P., Finite Geometries, Springer-Verlag, 1968, 375 pp. 3. Dembowski, P., 'Automorphismen endlicher M/Sbius-Ebenen', Math. Z. 87 (1965), 115-136. 4. Dieudonn6, J., La gdomdtrie des groupes classiques, Springer-Verlag, 1955, 130 pp. 5. Higman, D.G., 'Partial Geometries, Generalized Quadrangles and Strongly Regular Graphs', Atti del convegno di geometria combinatoria e sue applicazioni, Perugia (1971). 6. Orr, W.F., 'The Flock Conjecture', Private communication (1973), 7. Payne, S.E., 'Generalized Quadrangles of Even Order', 35 pp. (to appear). 8. Payne, S.E., 'Finite Generalized Quadrangles: A Survey', Proc. internat. Conf. projective Planes, Washington State Univ. 1973, 219-261. 9. Tallini, G., 'Problemi e Risultati sulle geometrie di Galois', Relazione N. 30, Ist. Mat. Univ. Napoli, (1973), 1-30. 10. Thas, J.A., 'Flocks of Finite Egglike Inversive Planes', C.I,M.E., 1I ciclo, Bressanone 1972, 189-191. 11. Thas, J.A., 'On 4-gonal Configurations with Parameters r=q2q-1 and k = q + l ' , Geometriae Dedicata 3 (1974), 365-375 12, Thas, J.A., '4-gonal Subconfigurations of a Given 4-gonal Configuration', Rend. Accad. Naz. Lincei, fasc. 6, Serie 8, vol. 53 (1972), 520-530.

Author's address:

J. A. Thas, Seminar of Higher Geometry University of Ghent, Krijgslaan 271 - Gebouw S.9., 9000 Gent, Belgium (Received July 29, 1974)