C3 geometries arising from the Klein quadric

S A R A H R E E S

C3 G E O M E T R I E S A R I S I N G F R O M T H E

K L E I N Q U A D R I C

1. P R E L I M I N A R I E S

1.1 Introduction

In this paper, I am concerned with Tits geometries of type C a . Due to a

fundamental theorem of Tits in [13], any classification of Tits geometries by

their diagrams rests on a classification of geometries of types C 3 and H a . We do not discuss geometries of type H 3 here. At any rate, these may perhaps be viewed as less interesting; only two finite examples exist, one a

(thin) building, the other a quotient of the first by a group of auto-

morphisms of order 2. Only one thick finite C3 geometry, the 'z¢ 7 geometry', is known which is

not a building or a quotient of such; it is widely believed to be the unique

example of this kind. By a theorem of Aschbacher [1], it is certainly the only such geometry with classical residues and a flag transitive group of automorphisms. In this paper, I exhibit a construction (in (2.1)) which pro-

duces this geometry from the well-known classical geometry, the Klein

quadric, defined over the field of two elements. The same construction yields similar examples over many fields, though all other known examples

arise as quotients of buildings. No thick finite examples are known apart from the d 7 geometry; the evidence suggests that there are no more. I

show in (2.3) that the existence of further examples of geometries belonging q q q

to o c~ c~ and arising from the Klein quadric is equivalent to the existence of a set of qZ + q + 1 points in the projective space PG(5, q),

exterior to a Klein quadric in that space, any two such points being joined by a line exterior to the quadric. Such sets of points cannot exist if q is

equal to 3 or is a power of 2 greater than 2; for other values of q the question remains open.

On the other hand, it ought to be possible to show that any geometry S q q q

that is described by o ~ ,, and is not a building must arise via my construction of (2.1). In the light of a result of Ott [9], in (3.2) I

X X X

associate with any geometry S described by o -c~ c~ and not a building a partial linear space (that is, a geometry of points and lines in

which any two points are incident with at most one common line)

Geometriae Dedicata 18 (1985) 67-85. 0046-5755/85.15. © 1985 by D. Reidel Publishing Company.

6 8 S A R A H R E E S

17 = (~, L~°). We verify that the geometry II has several properties in common with the Klein quadric of order x. If II is indeed the Klein quadric, then S must arise as in (2.1).

The results of (2.3) and (3.2) seem to suggest a possible classification of C 3 x x x

geometries described by o ~ ~. In other words, we may hope to classify those geometries once we have the answers to the following two questions:

(i) For which prime powers q is it possible to find a set of q2 + q + 1 points in PG(5, q) exterior to a Klein quadric in that space, any two points being joined by a line exterior to the Klein quadric?

(ii) Is every partial linear space 17 -- (~, 5¢), such that (a) every line is incident with x + 1 points, (b) the collinearity graph of II is strongly regular with the parameters

of the Klein quadric, (c) II contains a set 6 a of maximal singular subspaces each of which

is a projective plane of order x; every line of 17 is contained in precisely one subspace in 5 e,

(d) the lines of II can be coloured with a set of x 2 + x + 1 colours, in such a way that the set of lines in an element of 5e uses x 2 + x + 1 colours, the set of lines through a point uses x + 1 colours, and there exist no monochrome triangles of lines,

indeed the geometry of points and lines of the Klein quadric?

As yet, I can answer neither question completely.

In the final section of this paper, I leave geometries of type C3 and gener- alise, in (4.1), the construction of (2.1) to produce geometries of arbitrarily

high rank described by the diagram

0 0 0 . . . . . . . 0 ~') ~ .

The only examples given are either thin or defined over infinite fields. I show that no such geometries exist over GF(2) of rank greater than 3.

1.2. Introduction to Tits geometries

Virtually all of this paper is concerned with Tits geometries of type C 3. I am reluctant, therefore, to inflict on the reader unfamiliar with Tits geometries a general introduction to the subject. There are already innumerable

C 3 G E O M E T R I E S A R I S I N G F R O M T H E K L E I N Q U A D R I C 69

introductions available (e.g. in [3], [12], and [13]) which can be consulted if

necessary. I shall try here to introduce only the technical knowledge necessary to understand the main body of this paper.

Essentially, a Tits geometry is a collection of objects (varieties) of assorted types on which we impose a binary, symmetric, and reflexive incidence relation which can relate two distinct varieties x and y only if they are of

different types. We call a set of pairwise incident varieties aflag, and the set

of varieties not in a given flag but incident with every variety in it the residue of the flag. The structure of the geometry is described by a diagram

constructed in such a way that subdiagrams tell us about the structure of the residues of flags.

A diagram consists of a set of nodes - one for each type of varieties - joined by labelled edges. Subdiagrams consisting of two nodes joined by a

labelled edge describe the residues of flags of varieties of all but two types. In theory, the diagram tells us about the structure of all these rank 2 residues and no more; in practice, the structure of these residues may com-

pletely determine the structure of the geometry as a whole.

The classical examples of Tits geometries are buildings, very regular

geometries many of which are associated with groups of Lie type. If we

restrict attention to geometries described by diagrams whose edges admit (n)

only labels of the form o -o (where, by convention, we abbreviate (2) (3) (4)

o o to c~ ~, o- o to o - - o, o o to ~ ¢)), and thus look only at geometries described by the well-known 'Coxeter' dia-

grams, very few geometries apart from buildings are known. We see some sporadic examples, in particular some associated with sporadic simple

groups, but many other sporadic examples are excluded by this restriction

of the diagram. Further, many of the examples which we see that are not buildings are not strictly sporadiC but arise from buildings as quotients of such; that is, as geometries whose varieties are orbits of varieties of a

building under appropriate groups of automorphisms. (The precise details of how such quotients are formed can be found in [13].) A fundamental theorem of Tits in [13] tells us that every geometry described by a Coxeter diagram arises as a quotient of a building precisely if the same is true of residues which are described by those subdiagrams of type o a u

(5)

(called C3) and o - 0 o (called H3). Thus, in searching for a classification of Tits geometries described by Coxeter diagrams, we look first

70 S A R A H R E E S

for a classification of those described by o (5)

o o o. Here I look at o - -o u.

cr ~ and

1.3. Geometries of type C3

I shall now describe in detail what I mean by a geometry of type C3 (i.e. a geometry described by o- ~ _ _ o ) .

A geometry, S, of type C 3 is a collection of points, lines, and planes. (These three types index the nodes of the diagram, as we read from left to right.) A binary, symmetric, and reflexive relation I relates some of the varieties of S. We require that no two distinct points, no two distinct lines and no two distinct planes are related by I. We also require that the incidence graph (the graph whose vertex set is the set of all points, lines and planes of S, with two such joined by an edge if related by I) of S is connected, that every variety is incident with some other variety, and that every pair of incident varieties is incident with at least two other varieties. If every pair of incident varieties is incident with at least three other varieties we call S thick, whereas if every pair of incident varieties is incident with precisely two other varieties we call S thin.

The diagram imposes on S the following structure. From the stroke o -o, we deduce that, for every plane P of S, the set

of points and lines incident with P (the residue of P) has the structure of a generalized projective plane; thus any two points incident with P are also incident with a unique line incident with P, any two lines incident with P are also incident with a unique point incident with P.

From o_ o, we deduce that, for any point p of S, the set of lines and planes of S incident with p (the residue of p) has the structure of a generalized quadrangle; that is, a geometry of 'points' (the lines of the residue) and 'lines' (the planes of the residue) whose incidence graph has girth 8 and

diameter 4. Finally, the absence of a stroke between the two nodes which represent

points and lines of S tells us that every point incident with a given line is incident also with every plane incident with that line, a condition made to seem vacuous simply because of the names we have chosen to give our

varieties. Frequently we write parameters over the nodes of our diagram, viz.

X x y

o ~ ~. This indicates that any line of S is incident with x + 1 points, that any point is incident with x + 1 lines of a given plane in its residue (so projective planes are of order x), and that any line is incident with y + 1 planes. In any thick C3 geometry, parameters of this kind must exist.


1.4. Buildings o f type C 3

The buildings of type C 3 are precisely the geometries of points, lines, and singular planes of rank 3 polar spaces.

The points and lines of a polar space of rank at most r are characterized by the following axioms due to Buekenhout and Shult [4] (where we define

lines simply to be sets of points): (i) No point is collinear with all others.

(ii) I f p is a point and m a line non-incident with p, then p is collinear

either with a unique point o f m or with every point o f re.

(iii) I f A1 < Az "'" < Ak is a chain of non-empty singular subspaces, then

k ~ r .

(We define, as is usual, a singular subspace, U, to be a set of points such that, for all p, q 6 U, p and q are incident with a line all of whose points are

in U. All singular subspaces of a polar space are, in fact, projective spaces.) The simplest example of a rank 3 polar space is the geometry whose

points, lines, and planes are the vertices, edges, and faces of the octahedron, or, dually, the faces, edges, and vertices of the 3-dimensional cube. This

1 1 1

gives us a geometry described by 0 o c~. A slightly more complicated example arises from the 3-dimensional

Hamming cube whose vertices are elements of a set A~ x A 2 x A3, whose edges are subsets of elements of this set with two coordinates specified, and whose faces are subsets of elements of this set with just one coordinate specified. In the case where A ~ = A 2 = A 3 = { 0 , 1, . . . , k - l } , such a

1 1 k - 1

geometry is described by o ~ ~ . Every finite polar space of rank at least 3 in which every line is incident

with at least three points arises as the geometry of singular subspaces of a quadratic or pseudoquadratic form (see [-11], and also [5] for more specific details of the forms which are involved). Here I am mainly concerned with the O~, geometries, geometries which arise from quadratic forms over even- dimensional vector spaces as follows.

Given a 2n-dimensional vector space V = (el , e2 . . . . . e2,), we define a quadratic form Q by Q(Z xiei)=XIXn+ I ~-X2X,+ 2 + "'"-JvXnX2n. A bilinear form, B, is given by B(x, y) = Q(x + y) - Q(x) - Q(y). We call a subspace U of V a singular subspace if for all u, w ~ U, Q(u) = Q(w) = 0 and B(u, w) = O.

The singular subspaces of dimensions 1 and 2 of V form the points and lines, while the singular subspaces of higher dimensions form the remaining singular subspaces, of a polar space of rank n. In this particular geometry,

7 2 S A R A H R E E S

the maximal singular subspaces (of projective dimension n - 1) can be par-

titioned into two classes by a relation which relates two subspaces whose intersection has even codimension in each. The geometry can be described either by the diagram

or by the diagram

0 0 0 . . . . . . . o o sing. sing. sing. sing. sing. pts lines pins n - 2 n - 1

spaces spaces

one c]ass of n -- 1 spaces

0 0 0 . . . . . . . -

sing. sing. sing. s i n g ' ~ ' ~ second class of pts lines pins n - 3 " O

spaces n - 1 spaces

(where for the second diagram we ignore singular n - 2 spaces, and define

two n - 1 spaces in opposite classes to be incident if they intersect in a hyperplane of each).

For the sake of brevity, I exclude details. A more exact account of this

geometry can be found in [11]. We observe here that the geometry is uniquely recognizable amongst rank n polar spaces with at least three

points per line as that in which every singular n - 2 space is incident with precisely two singular n - 1 spaces. For n = 3, we have a geometry of

points, lines, and planes, a well-known rank 3 polar space, usually viewed

embedded in projective 5 space, and known as the Klein quadric.

1.5. Quotients of C 3 geometries

Suppose that S is a C3 geometry and A a group of automorphisms of S

(that is, a group permuting the varieties of S in such a way as to preserve

the sets of points, lines, and planes and the incidence relation). The action of A can be factored out to give a quotient geometry also of type C3 (and with residues isomorphic to those of S) precisely if

(i) A acts semiregularly on the points of S, and (ii) no two points in the same orbit of A are collinear.

This follows from the more general conditions on such a group given by Tits in [13].


1.6. The d 7 geometry

Only one thick finite geometry of type C3 is known which is not a building. First discovered by Neumaier (see [8]), the geometry is usually known as

the d 7 geometry since it may be described as a geometry on cosets of certain subgroups of the group t iT . The following description of this

geometry is due to Ronan and Stroth ([10]). We describe the d 7 geometry S to have as points the elements of the set

{1, 2 . . . . . 7}, as lines all the 3-sets of that set, and as planes one orbit of 15 projective planes of order 2 defined over the point set { 1, 2 . . . . . 7} under the

action of d 7 on the seven points. Incidence is defined by set inclusion (and

reverse inclusion). 2 2 2

S is thus a geometry defined by o o o in which every point points lines planes

and every plane are incident. It is not a quotient of a building.

2. THE CONSTRUCTION OF C 3 GEOMETRIES FROM THE KLEIN QUADRIC

(2.1) THEORE M. Suppose that Fl is an O~(K) geometry (Klein quadric) embedded in PG(5, K), and suppose that there exists a set ~ of points of PG(5, K) exterior to YI and such that every line of II is perpendicular to precisely one point of ~. Then we can define a geometry S as follows:

We define the point set of S to be the set of points of ~, the line set of S to be the set of points of H and the plane set of S to be one class of planes of Fl. Further, we define a point and line of S to be incident if, as points of PG(5, K),

they are perpendicular with respect to the defining form of the quadric. We define a line and plane of S to be incident if they are incident as subspaces of H. Finally, we define every point and every plane of S to be incident.

S is a geometry defined by o ~ ~. The residue of a plane of S points lines planes

is a classical projective plane over K, and the residue of a point an Os(K) generalized quadrangle (that is, the generalized quadrangle in PG(4, K)

associated with the form xlx2 + x3 x4 + x2). Proof. It is clear that we only need to verify the diagram for S, and in

fact only need to check that the residue of a plane is indeed a classical projective plane and that the residue of a point is also the appropriate generalized quadrangle.

Now if P is a plane of S, P is also a plane of 1-I. As the lines of the residue of P we see those points of YI contained in P. As the points of the residue we see all the points of ~ , each of which defines, by the intersection of its


perpendicular with P, a unique line of P in 17. Thus the residue of P must clearly have the structure of (the dual of) the projective plane which is P itself in 17.

As for the residue of a point; by considering the restriction of the defining form of 17 to the hyperplane of PG(5, K) perpendicular to an exterior point, we see that, for all p e I-I, p± intersects r l in a set of points and lines which forms a generalized quadrangle of type 05(K). Since each line of II determines a unique plane of 17 in each class, we see that the points and lines of 17 c~ p± correspond precisely to the lines and planes of Res(p).

(2.2) EXAMPLES. (i) If we take K to be GF(2), Theorem (2.1) gives us an alternative description of the d 7 geometry mentioned in Section 1.

We find the point set of the geometry as a set of seven points in PG(5, 2) exterior to the quadric, any two such points being joined by an exterior line. Eight possible such sets exist; their existence is used in a 1910 paper by Conwell [6] to demonstrate the now well-known isomorphism between d 8 and f~(2), a group of automorphisms of the Klein quadric, s¢ 8 permutes the eight sets transitively in its natural action, while preserving the structure of the quadric. Thus t iT , as the stabiliser of one such set of seven points, appears naturally as the group of automorphisms of our geometry.

(ii) Suppose that we can find a (projective) plane in PG(5, K) exterior to the Klein quadric in that space (we can find such planes precisely when K is an ordered field; thus we see that we can get no finite examples in this way). The points of such a plane form a suitable set ~.

Observe that any point of the Klein quadric is perpendicular to a unique line of ~. Thus two lines of S have either one or all of their points in common, according as they correspond to distinct or to the same lines of ~. Note also that any collineation of PG(5, K) which preserves I-I, its two classes of singular planes and the exterior plane ~ induces an automorphism of the geometry S. Thus S admits a group G of automorphisms isomorphic to the direct product of two copies of the group of special orthogonal 3 × 3 matrices over K. G acts flag transitively on S.

A geometry S which arises in this way is a quotient of the 07(K) building (the rank 3 polar space defined by the form Q(Z xlel) = XxX2 + x 3 x ~ + x5 X 6 + X 2 , or equivalently by Q(E yi fl) = y~ + y22 + y2 + y] _ y2 _ y2

_ y2), which we shall call N, by a group of automorphisms, A. A consists of those automorphisms induced by matrices of the shape

(0

C 3 GEOMETRIES ARISING FROM THE KLEIN QUADRIC

where

75

X = p - - S

S p - -

- r q

for p2 -t- q2 + r 2 + S 2 = 1.

In order to see this, observe first that the action of the group H of 4 x 4

matrices of shape

p q r s

p --s r s p - -

- - r q

w i t h p2 + q2 _+_ r 2 -t- s 2 = 1 o n t h e v e c t o r s p a c e ( Y l , 2 2 , 2 3 , Y4) is equivalent to the action of the group ~ 1 of unit quaternions on itself by right multiplication. Since this second action is clearly regular, we see that H acts

regularly on (yl , Y2, Y3, Y4). From this we can deduce that the group A of automorphisms of ~ acts semiregularly on the set of points of ~ , with no two points in the same orbit of A being collinear. Thus by Section 1, A does

indeed induce a quotient of ~ of type C 3 . Further, since H acts transitively

on (Yl, Y2, Y3, Y4), we see that each plane of ~ contains a point of every orbit of A. Thus in ~/A every point and every plane are incident.

A further inspection of A shows that for any line x and any plane P of :~,

either x has an image under A incident with P, and hence xA and PA are incident in ~/A or P is incident with a line y all of whose points are images

of the points of x. In this case xA and yA are distinct lines of ~/A with the same point set.

To verify that the quotient geometry ~/A is indeed the geometry arising

from the Klein quadric as in (2.1) it is enough to verify that we can recover the Klein quadric within ~/A. I sketch without proof how this may be done; the details of the proof are simply algebraic.

We find the Klein quadric within YJ/A as the geometry I I ' which has as its points the lines of ~/A and as its lines the residues of point, plane flags

of ~/A. Thus two lines xA, yA of YJ/A correspond to collinear points of H' if they are coplanar in ~/A.

First we observe that if x is any line of ~ and y a line as above such that, in ~/A, xA and yA are distinct lines with the same point set, then xA and yA are clearly not coplanar in ~/A. Thus xA and yA are not collinear in II ' .


Next we note that if P is a plane of N and x a line of ~ with no image in P, then the lines of P coplanar with some image of x all pass through a single point of P.

Finally, we observe that the orbits of a set of lines in a plane of N form a singular subspace (a plane) of H'. Such a subspace is maximal as a singular subspace. For, as we observed above, for any line x outside P with no image in P there exists a line y of P such that x A and yA are non-coplanar in N/A.

This is enough to verify the Buekenhout-Shult axioms as in Section 1, and show that H' is a rank 3 polar space.

Finally, to show that 17' is actually the Klein quadric it is enough (essentially by Tits [11]) to verify that any line of 17' is incident with exactly two singular planes. An unpleasant algebraic calculation verifies this.

(iii) From the 3-dimensional cube and, more generally, the 3-dimensional Hamming cube, we also obtain examples. Since we do not find these polar spaces as quadrics in projective space the construction must clearly be slightly different. Instead of identifying the points of S as points of projective space exterior to the quadric, we find them as sets of points of the polar space intersecting each plane in a line, or even as the sets of such lines.

Thus from the ordinary 3-dimensional cube we may define a geometry S 1 1 1

belonging to o c~ ~ whose point set is the set of parallel classes of edges of the cube, whose line set is the set of faces of the cube, and whose plane set is the set of vertices of the cube which have even weight when described by vectors in GF(2) 3. Incidence between points and lines is defined such that a parallel class of edges is defined incident with those faces containing edges in that direction, and also with all vertices of even weight. A face is defined to be incident with precisely those vertices it

contains. Similarly, we may define a geometry from the k x k x k Hamming cube,

choosing in general as our plane set the set of vertices of the cube with weight congruent to zero mod k.

Each such an example is a quotient of a building, in fact a quotient of the 3-dimensional Hamming cube from which it arises by the cyclic group of automorphisms generated by the automorphism which maps each vertex (x, y, z) to the vertex (x + 1, y + 1, z + 1) (addition taken mod k).

The question now is: are there any more examples? In particular, are there any thick, finite examples other than the d 7 geometry? We see that

(2.3) LEMMA. The construction of (2.1) yields a 9eometry described by q q q

o ~ ~ precisely if we can f ind in PG(5, q) a set o f q2 -Jr- q + 1


points exterior to a Klein quadric in that space, any two such points bein9

joined by a line exterior to the quadric. Proof First observe that if p and r are two points in PG(5, q) exterior to a

Klein quadric, H, then p± c~ r ± contains no lines of I I precisely if (p, r ) is

external to the quadric.

For if (p, r ) meets I I in a point s, p± c~ r ± = p± c~ s ± contains a line of any plane of I I containing s. Conversely, if p~- c~ r ± contains a line m of H,

then the 3-space m ± contains the line (p, r ) and also some singular planes

of H. In a projective 3-space, any line and plane are forced to intersect. To finish, note that, since for each point p ~ H, p± must contain a distinct

line of each plane of H, there must be as many points in 17 as there are lines in a plane of H; hence q2 + q + 1. Thus we need to know, in order to know

if other thick, finite geometries exist, for which values of q such sets of q2 + q + 1 points exist.

Of course, for q = 2 we know that such sets exist. For q = 3, the answer is negative.

(2.4) LEMMA. I t is not possible to f ind a set of 13 points in PG(5, 3),

exterior to a Klein quadric over GF(3), any two joined by an external line. Proof Suppose that V = {v 1, v 2 . . . . , v13 } is a set of vectors in GF(3) 6

giving rise to such a set. Suppose that Q and B are the quadratic and

bilinear forms which define the Klein quadric. Let V + be the set of vi s V such that Q(vi) = 1 and V - be the set of v~ e V such that Q(v~) = - 1.

Now, for all distinct v~, vj E V +, we must have B(vi, vj )= 0, since we

require Q(vl +_ v~), equal to Q(vi) + Q(vj) +_ B(vi, vj), to be non-zero. Simi- larly, for distinct v~, vj e V- we must have B(vi, v j) = O.

From this, it is elementary to deduce that each of V + and V- is a

linearly independent set. Hence I V + I ~ 6, I V - I ~< 6, and thus I V[ ~< 12.

For q even, the d 7 geometry is the unique example of a geometry as in (2.1) over GF(q). This result is due to J. A. Thas, and relies on the following.

L E M M A (Fisher and Thas [7, Theorem (3.1)]). Suppose that F is a 9rid

quadrangle (that is, the 9eometry of singular subspaces of a form x l x 2 + x a x4) in PG(3, q), where q is an even power of 2. I f ~- is a set of q + 1 points ofPG(3, q), exterior to F, any two points of g" being joined by a line of PG(3, q) exterior to F, then 3- is the set of points on a line.

Hence we have


(2.5) LEMMA (due to J. A. Thas). Suppose that a set ~ o f qZ + q + 1

points exists in PG(5, q), exterior to a Klein quadric FI in that space, any two

points of N ~ being joined by a line exterior to H. I f q is even, then q = 2.

Proof Let p be a point of rI and consider ~tp, the set of points of ~t in the (projective) 4-space pa. As the set of points on a line of the C 3 geometry of (2.1), ~p has size q + 1. Let rl, r z . . . . . rq+a be its elements.

Now consider a (projective) 3-space U of p± not containing p. U intersects 17 in a grid quadrangle F. Each line (p, r~) joining p to an element of Np meets U in a single point sg. Since each line (ri , rj-) is exterior to H we see that each line (s~, s j ) must also be exterior to F. Now, by the above

lemma, we see that the points Sl, s2 . . . . . sq+ 1 must lie on a line of U. Thus the points p, Sx, . . . , Sq+~ span a (projective) plane in U. Since (p, s~) = (p, ri), the same must be true of the points p, r I . . . . . rq+ 1.

Now consider the line ( r 1, r2). Suppose it contains fl points of Np. Observe that the 3-space (rx, r2) ± intersects FI in a set of q2 + 1 points

containing no lines. Thus no two singular points in (ra, r2) ± are perpendicular with respect to rI.

Choose r e ~ \ ( r 1, r 2 ) . Since every plane of PG(5, q) intersects H, ( r l , r2, r ) must contain at least one point p' of II. Suppose that p' # p. As above, the plane (p', r~, r2) contains r, but the plane (p, rl, r2) does not; for if so

(p', rl, r2) = (r, r 1, r2) = (p, r l, r2), and hence p' E (p, rx, r2) _ p±. But we know that p' and p, as two points of (r~, r2) ±, cannot be perpendicular. Thus we see that each point of ~ not on (rx, r2) is perpendicular to exactly one point of (ra, r2) ± n H. Hence, counting, we see that i.~[ _-- q2 -t- q + 1 = fl + (q2 -I- 1)(q + 1 -- fl), and hence fl = q.

Thus any two points of N span a line of PG(5, q) intersecting N in exactly q points; that is, the points of N form a 2-(q 2 + 1, q, 1) design. The equality r ( k - 1)= 2 ( v - 1) for a 2-(v, k, 2) design with r blocks through each point gives us q 2 + q = r ( q - 1). For q even, this has no solution unless q = 2.

It seems at this stage unlikely that any further examples of geometries arise as in (2.1) over finite fields. Note that if any other such geometries did exist they would be extremely interesting, as by [21 they would not be quotients of buildings.

3. RECOVERING THE KLEIN QUADRIC F R O M C 3 GEOMETRIES

It seems that our results of Section 2 might help us to classify geometries of type C3. If we could show that all thick, finite C 3 geometries which were


not buildings arose as in (2.1), then the existence or non-existence of such geometries would depend only on the combinatorial condition of (2.3).

x X x

We look here at geometries described by o c~ ~ (only these parameter types arise from the construction of (2.1)) and display some results which suggest that all such geometries, if not buildings, do indeed arise from the Klein quadric.

Observe first a result of Ott. X x x

T H E O R E M (Ott [9]). I f S is a geometry described by o ~; ~,

where x is finite, and x > 1, then either

(i) S is a building, or

(ii) in S every point is incident with every plane.

The same result is trivially true for x = 1. Further: x x x

(3.1) LEMMA. Suppose that S is a geometry described by o c~ ~

in which every point and every plane is incident. Then any two planes of S are

incident with a unique common line.

Proof Certainly no two planes of S can be incident with more than one common line. For suppose that P, Q are two planes, both incident with two lines m and n. Then m and n, as two lines in the projective plane Res(P), are incident also with a point p. Further, p is incident also with both P and Q. Now in the generalized quadrangle Res(p) we have two lines (P and Q) which intersect in two points (m and n); an impossible situation.

Now in S we have (x + 1)(x 2 + 1) planes in the residue of any point, and thus (x + 1)(X 2 -+- 1) planes in all. If P is any one plane, x(x 2 + x + 1) other

planes are incident with some line of Res(P). Thus, since 1 + x(x 2 + x + 1) = (x + 1)(x 2 + 1), we see that every plane of S distinct

from P is incident with some line of Res(P).

Finally, we deduce x X X

(3.2) THEOREM. Let S be a geometry described by o 0 ,,

(x < oo) which is not a building. Define a geometry II = (~, 2 ' ) o f points and

lines as follows: ~ is the set o f lines of S; £P is the set o f residues of point,

pIane f lags of S; incidence is defined by set inclusion. Then

(i) H is a partial linear space in which any line is incident with x + 1

points;

(ii) the collinearity graph o f FI is strongly regular, described by

@ x ( x + 1) 2 1 ~ x 3 ( x + 1)2 @ , ~

~22xa + x - 1 -') (~x- 1)(x + 1) z)

80 S A R A H REES

(iii) I I contains a set 5P o f maximal singular subspaces each o f which is a

projective plane o f order x; every line o f H is contained in precisely one

subspace in 5P ;

(iv) the lines o f I I can be coloured with a set o f x 2 + x + 1 colours, in such

a way that the set o f lines in an element o f 5~ uses x 2 + x + 1 colours,

the set o f lines through a point uses x + 1 colours, and there exist no

monochrome triangles o f lines.

Proof By the theorem of Ott, every point of S is incident with every plane. We see that [~[ = (x 2 + x + 1)(X 2 -b 1), and that every line is inci-

dent with x + 1 points. From (3.1), we see that H is a partial linear space. Each point of H is incident with (x + 1) 2 lines, and thus is collinear with x(x + 1) 2 other points.

Now choose m, n, two lines of S (so points of H).

Suppose first that m and n are coplanar (corresponding to collinear points of 17). Then x 2 + x - 1 lines in the unique plane P incident with

both m and n are coplanar with both. Furthermore, if Q is a plane distinct

from P and incident with m, and R is a plane distinct from P and incident with n, by (3.1) a unique line o is incident with both Q and R, and thus

coplanar with both m and n. Again by (3.1), o determines uniquely Q and R,

and is not incident with P. Thus we see that m and n are coplanar with precisely x 2 -[- x - 1 -[- x 2 ~-- 2x 2 + x - i common lines.

Now suppose that m and n are not coplanar. Then again for any plane Q

incident with m and any plane R incident with n, by (3.1), a unique line o is incident with both Q and R. Thus (x + 1) z lines are coplanar with both m

and n.

So we see that the collinearity graph of H is strongly regular with the

required parameters.

Now the set of lines incident with any plane of S certainly forms a singular subspace of I I which is a projective plane of order x. Such a

subspace must be maximal as a singular subspace; for a line m of S non-

incident with a plane P cannot possibly be coplanar with every line incident

with P, since this would require the impossible situation that every line of Res(P) were incident with a single point of Res(m). Clearly the set of all such subspaces of H forms a suitable set 50.

To verify (iv), let a set of x 2 + x + 1 colours be in correspondence with the points of S, and colour each line of I I according to the point of the point, plane residue of S by which it is defined. It is immediately clear that x 2 + x + 1 colours are needed to colour the lines of a member of 5 ~, and that x + 1 are needed to colour the lines through a point of II. The non-


existence of monochrome triangles of lines follows from the non-existence of triangles in the generalized quadrangles that are the residues of points in S.

NOTE. It is easy to verify that the properties (i)-(iii) of (3.2) hold in any Klein quadric. However, it is also easy to construct other geometries in which they hold. For example, if H' is any Klein quadric, let (9 be one of its two classes of singular planes, and define a second geometry II as follows. Let H have the same point set as H'. Moreover, define the point sets of maximal singular subspaces of H' in a set 5 P to be the same as those of the maximal singular subspaces in (9. Any two subspaces in 5 p then intersect in a point. Now define the lines of H to be the same as those of H' almost everywhere, but replacing the lines of just one element N of (9 by another set of lines which also make ~ into a projective plane. H' cannot be a Klein quadric, for it is clear that we have broken the second axiom of Buekenhout-Shult. However, the conditions (i)-(iii) still hold.

It is not clear to me whether the property (iv) is enough to finally force H to be a Klein quadric. It should certainly be observed, however, that (iv) does not hold in all Klein quadrics. For, if H is a Klein quadric, the property (iv) in II is equivalent to the existence of a set N of exterior points as in (2.1).

Finally, observe the following

(3.3) COROLLARY. Suppose that S is a geometry described by x x X

o o _ ~ (x < ~ ) which is not a building. Let H be the geometry o f

points and lines defined from S as in (3.2). I f H is the geometry o f points and

lines o f the Klein quadric, then S must arise as in (2.1).

4. G E N E R A L I Z A T I O N S I N T O H I G H E R D I M E N S I O N S

We look finally at generalizations of the construction of (2.1) which involve quadrics in higher dimensional projective spaces.

(4.1) THEOREM. Let H be an O~,(K) quadric, where n >1 3, embedded in

P G ( 2 n - 1, K). Suppose further that there exists a set ~ o f points of

PG(2n - 1, K) exterior to H, such that each n - 2 space of H is perpendicu-

lar to a unique point o f ~.

Then we may define a geometry S as follows:

S has as its point set the set o f points o f H, as its line set the set o f lines o f

H . . . . , as its set o f n - 3 spaces the set o f n - 3 spaces o f H, as its set o f

n - 1 spaces one of the two classes o f n - 1 spaces o f H, andfinally as its set

o f 'symps' the set o f points in .~. Incidence in S between varieties o f S which


are also subspaces of H is defined exactly as in El. A syrup is defined to be

incident with a variety of S which is also a subspace of H of dimension at most

n - 3 precisely if it contains it in its perp (with respect to the defining form of

the quadric). Every syrup is defined to be incident with every n - 1 space of S.

S is a geometry defined by 0 0 0 0 . . . . . . . O

points lines planes 3 n - 4 spaces spaces

( ) {.)

n - 3 1 n - 1 space spaces

I symps

The residue of a syrup is a rank n - 1 polar space of type 02, - I (K) (that is,

defined by the form Q(E xiei) = XlX2 + x3x4 + "'" + X 2 n - 3 X Z n - 2 -~- X 2 n - 1 ) ;

the residue of a f lag consisting of a tower of subspaces of projective dimensions

O, 1, 2, . . . , n - 4 is the geometry of(2.1). I omit the proof, which is merely a generalization of the proof of (2.1).

(4.2) EXAMPLES. (i) There are examples for all n greater than or equal to 3 over subfields K of ~. In these cases we can find subspaces of PG(2n - 1, K) of projective dimension n - 1 exterior to the Klein quadric. The points of such a set form a suitable set ~. For such an example C3 residues are of two types: each C3 residue is either an 07(K ) building or the geometry of (2.2(ii)), a quotient of the Or(K) building. Hence, by Tits [13, Theorem 1], we see that all these examples are themselves quotients of buildings.

(ii) As in (2.2(iii)), we can find similar fairly uninteresting examples (also quotients of buildings) connected with the n-dimensional cube and, more generally, the n-dimensional Hamming cube. Since we do not view these within projective space we are obliged to define our symps slightly differ- ently, as parallel classes of edges, incident with subcubes of the cube of dimension at least 2 precisely if some of the edges of the parallel class are contained in the relevant cube. Thus we get geometries belonging to

1 1 1 1 1 1

0 0 0 0 . . . . . . . ~9

C 3 G E O M E T R I E S A R I S I N G F R O M T H E K L E I N Q U A D R I C

associated with the ordinary n-dimensional cube, and belonging to

1

O

associated with

k - 1 } .

83

1 1 1 k - 1

O O . . . . . . . . (~ c~

0 1

the n-dimensional H a m m i n g cube over {0, 1, 2, 3, . . . ,

Finally, we ask if there can be any thick, finite examples of such geometries.

Much as in Section 2, the following becomes clear.

(4.3) L E M M A . There is an example of a rank n 9eometry described by

q q q q q

0 0 0 . . . . . . . ( ) T ) ,

o q

arisin9 over GF(q) as in (4.1), precisely if it is possible to f ind in PG(2n - 1), q) a set of (q" - 1)/(q - 1) points, exterior to an O+,(q) quadric, any two such points being joined by a line exterior to the quadric.

It should be clear that, since lower rank geometries appear as residues in geometr ies of higher rank, a geomet ry of rank n described by

q q q q q

O O 0 . . . . . . . ? ~

/

6 q

cannot arise as in (4.1) unless there exist also such geometries of rank n - 1, and further of rank 3.

We might hope for higher rank geometries of this type over GF(2), but this is not possible, for we have the following.

(4.3) L E M M A . Suppose that 2 is a set of points in PG(2n - 1, 2) exterior to an O~-,(2) quadric embedded in that space, any two points of 2 being joined by an exterior line, then 121 <<. 2n + 1.

Proof Let V = {vl, v 2 . . . . , vr} be a set of vectors in GF(2) 2" leading to such a set 2. Let Q and B b e the quadrat ic and bilinear forms which define the quadric, H. Fo r all vi E V, Q(vi) = 1. Since also Q(vl + v j) = 1, i f / ~ j, we

must have B(vi, v j) = 1 for all i ¢ j (but B(vi, vi) = Q(vi) + Q(vi) = 0 for all i).

84 S A R A H REES

N o w cons ide r the set {v i - v 1 : i = 2 . . . . . r}. It is e l emen ta ry to show tha t

such a set of vectors m u s t be l inear ly i n d e p e n d e n t in GF(2) 2", a n d therefore

c o n t a i n s at m o s t 2n e lements . T h u s 121 ~< 2n + 1.

N o w since for n > 3, 2n + 1 < 2" - 1, we do n o t expect examples of the

geomet ry of (4.1) over GF(2) except in r a n k 3.

A C K N O W L E D G E M E N T S

This pape r is a revised form of a chap te r of m y Ph .D. thesis (Oxford, 1983).

M y t h a n k s are due to m y thesis supervisor , D r Pe te r C a m e r o n , w i thou t

w h o m the thesis w o u l d never have appeared . I also t h a n k A. E. B rouwer for

some helpful d iscuss ions on cons t ruc t ion .

R E F E R E N C E S

1. Aschbacher, M., 'Finite Geometries of Type C a with Flag Transitive Automorphism Groups', Geom. Dedicata 16 (1984), 195-200.

2. Brouwer, A. E., and Cohen, A. M., 'Some Remarks on Tits Geometries', lndag. Math. 45 (1983), 393-402.

3. Buekenhout, F., 'Diagrams for Geometries and Groups', J. Comb. Theory A27 (1979), 121 151.

4. Buekenhout, F., and Shult, E. E., 'The Foundations of Polar Geometry', Geom. Dedicata 3 (1974), 155-170.

5. Carter, R. W., Simple Groups of Lie Type, Wiley, London, New York, Sidney, Toronto, 1972.

6. Conwell, G. M., 'The 3-Space PG(3, 2) and its Group', Ann. Math., Series 2, 11 (1910), 60-76.

7. Fisher, J. C., and Thas, J. A., 'Flocks in PG(3, q)', Math. Z. 69 (1979) 1-12. 8. Neumaier, A., 'Some Sporadic Geometries Related to PG(3, 2)', Archly Math. 42 (1984),

89-96. 9. Ott, U., 'On Finite Geometries of Type B 3' (preprint).

10. Ronan, M. A., and Stroth, G., 'Minimal Parabolic Subgroups for the Sporadic Groups', European J. Combin. 5 (1984), 59-92.

11. Tits, J., 'Buildings of Spherical Type and Finite BN-Pairs', Lecture Notes in Math. 386, Springer-Verlag, Berlin, Heidelberg, New York, 1974.

12. Tits, J., 'Buildings and Buekenhout Geometries', in M. Collins (ed.), Finite Simple Groups, Academic Press, New York, 1981, pp. 309-320.

13. Tits, J., 'A Local Approach to Buildings', in Davis, Grunbaum and Sherk (eds), The Geometric Vein (The Coxeter Festschrift), Springer-Verlag, Berlin, Heidelberg, New York, 1981, 519-547.

C 3 G E O M E T R I E S A R I S I N G F R O M THE K L E I N Q U A D R I C 85

Received August 1, 1984)

Author's address:

D6partement de Math6matique, Service de G6om6trie, C.P. 216, Universit6 Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, B-1050 Bruxelles, Belgium

C3 geometries arising from the Klein quadric

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