Journal of Geometry. Vol. 5/2 1974. Birkh~user Verlag Basel

ON THE GRASSMANIAN OF LINES IN PG(4,q)

AND R(I,2) REGULI

J. W. Freeman*

An R(I,2) regulus is a collection of qel mutually skew planes in PG(5,q) with the property that a line meeting three of the planes must meet all the planes. An (Z,~)-configuration is the collection of lines in PG(4,q) meeting a line s and a plane ~ skew to i. A correspondence between (s in PG(4,q) and R(I,2) reguli in the associated Grassmanian space G(I,4) is examined. Bose has shown that R(I,2) reguli represent Baer subplanes of a Desarguesian projective plane in a linear representation of the plane. With the purpose of examining the relations between two Baer sub- planes of PG(2,q2), the author examines the possible intersections of a 3-flat with an R(I,2) regulus.

1. Motivation and Main Results.

Let PG(n,q) or S(n) denote the n-dimensional pro- k

jective space over the Galois field GF(q), q = p , p odd

prime. The lines of S(3) correspond in a i-i manner onto

the points of a nondegenerate hyperbolic quadric K = G(I,3)

in S(5) under the Pl~cker coordinate map ~, [17, p.327].

If p and q are two skew lines of S(3) with ~(p) = P,

~(q) = Q, then P and Q are not conjugate with respect to

(w.r.t.) K. The lines of S(3) meeting both p and q

correspond under ~ to the points of a nondegenerate

*This work was supported (in part) by National Science Foundation Grant No. GP-40583.

159

2 Freeman

hyperbolic quadric section of K, Q(3). In fact Q(3) is

the intersection of K = G(I,3) with the intersection of

the tangent spaces of K at P and Q. The quadric Q(3) is

ruled by two collections of lines, termed reguli. Each

such collection is an R(l,t-l) regulus (see this section

below) of S(2t-l) in the simplest case t = 2. Hence the

configuration of lines meeting two skew lines in S(3)

corresponds under ~ to the points of an R(I,I) regulus.

Let ~ be a line skew to a plane ~ in S(4). The

collection of lines of S(4) which meet ~ and ~ in each a

point is called an (~,~)-configuration. The elements of

an (~,~)-configuration correspond under ~ to the points

of an R(I,2) regulus (see below) lying on the associated

Grassmanian space, G(I,4) = L(9), (proposition 6). -i

Conversely, let Q be a point on L(9) with q = ~ (Q)

a line of S(4). Let T(Q) be the tangent space to L(9) at

Q. Then T(Q) is a 6-dimensional projective subspace

(6-flat) and there is a i-i correspondence between R(I,2)

reguli in T(Q)~ L(9) and (q,~)-configurations of S(4),

(proposition 7, corollary 3).

The L(9) locus may be viewed as the intersection of

five degenerate quadrics in S(9). The polar properties

for a locus given by more than one quadric together with

the group of projective transformations of S(9) induced

by the projective group of S(5) are the basic tools for

the analysis of L(9).

The polar space [w.r.t. L(9)] of a point Q is a 6-flat

[4-flat] when Q is [is not] on L(9). Unlike the polar

space at a point w.r.t, a single quadric not all lines of

S(9) meeting L(9) in a single point Q are in the polar

space of L(9) at Q, (section S, remark i). From this point

160

Freeman 3

of view we discus~ a partition of the points of S(9).

The partition follows ~rom a property of the points of

S(9)'~L(9) given by H. R. Brahana [6, p.543]. Connections

with some properties of L(~) to the work of J. A. Todd [16]

are given in section 4.

Let C be a collection of mutually disjoint projective

subspaces (flats) of a projective space S. A transversal

to the collection is a (projective) line which has exactly

one point in common with each member of C. An R(l,t-l)

regulus is a collection of mutually skew (t-l)-dimensional

projective subspaces of a space, S(2t-l), with the following

properties: i) R(l,t-l) has at least 3 distinct members,

ii) every transversal to 3 distinct members of R(l,t-l) is

a transversal to R(l,t-l), iii) if L is a transversal to

R(l,t-l), each point of L is on a member of R(l,t-l).

Some basic properties of R(l,t-l) reguli are discussed

in [8, p.162]. Dembowski [9, p.220] calls an R(l,t-l)

regulus a (t-l)-regulus. Let E be a simple extension of

a field F with [E:F] = t. The points of the projective

line over E may be represented by the pairs [i,~] and [0,i]

with ~ e E, see [5]. The points [l,m] and [0,i] with m ~ F

give rise to an R(l,t-l) regulus in the (2t-l)-dimensional

projective space over F. The connection between R(l,t-l)

reguli, regular spreads and the linear representation of

finite Desarguesian projective planes is discussed in [8]

(see corollary to Thm. 12.1, p.164).

Bose and Barlotti [4] have given a representation of

a class of projective planes (delta-planes) in a 4-

dimensional projective space in which the definition of

incidence is not simply the inclusion relation. Incidence

in the representation of the delta-planes is given by how

161

4 Freeman

the objects which represent the 'delta-points' and 'delta-

lines' intersect. The representation of the delta plane

was originally developed in a 5-dimensional projective

space. With this point in view the present paper concludes

with an examination of how a 3-flat and an R(I,2) regulus

may intersect (proposition i0).

The facts of proposition i0 are used in the examination 2

of the relations between two Baer subplanes of PG(2,q ).

The intersection relations are discussed in a forthcoming

paper by R. C. Bose and the author.

2. A Representation of the Lines of S(4) and S(9).

A line s of S(4) may be determined by two linearly

independent points on s having vector representations, ~,

~, which correspond to a 2 x 5 matrix:

i) = 0 <i<4. Yi -- --

A line represented as in i) may be associated with the

vector (P(ij)), 0 < i < j < 4 where

ii) P(ij) =

Xo X, l ]

Yi Yj

It is well known [13] that the association (denoted as ~)

is well-defined on the lines of S(4) and i-i onto a locus

L(9) of points in S(9) defined by five (dependent) quadratic

relations labeled Qi in iii) below.

iii) Qi: P(jk)P(Zm) - P(js + P(jm)P(ks = 0

with {i,j,k,s = {0,1,2,3,4}, j < k < s < m.

162

Freeman 5

These quadrics are related by:

iv)

P(04)Q 0 - P(14)Q I + P(24)Q 2

p(03)Q0 - P(13)Q 1

p(02)Q0 - P(12)Q 1

P(OI)Q 0 - P(12)Q2

+ P(23)Q 2

+ P(23)Q 3

- P(S4)Q3 = 0

- P(34)Q 4 = 0

- P(24)Q 4 = 0

+ P(13)Q 3 - P(14)Q4 = 0

The five basis vectors (ei) , 0 _< i _< 4,

Ze. in the u n d e r l y i n g v e c t o r space g ive 1

for S(4). The ten lines of S(4), < el,

to a basis {< e . . >} f o r t h e u n d e r l y i n g 13

P(Ol)Q I - P(02)Q 2 + P(03)Q 3 - P(04)Q4 : 0.

and the vector

rise to a frame

ej >, map under

vector space of

S(9). To avoid unnecessary double subscripts the point

(P(01),P(02),P(03),P(04),P(12),P(13),P(14),P(23),P(24),P(34))

of S(9) is also denoted by (P0,PI,P2,P3,P4,P5,P6,P7,P8,Pg).

All calculations and matrices are taken w.r.t, these bases.

Since any line of S(9) meets a quadric in either 0,1,2 or

q+l points, it follows that any line of S(9) meets L(9)

in either 0,1,2, or q+l points.

3. Conjugacy w.r.t. L(9) and Polar Properties.

The equations iii) may be written in the form:

v) Qi: xA'xt = 0,

where A. is an upper triangular matrix and x = (x0,...,x 9) 1

is a coordinate vector of X e S(9). Let R be a point of

S(9) with coordinates r. A point S with coordinates s is

said to be conjugate to R w.r.t, the locus L(9) if

vi) [(Ai+At)f = 0

163

6 Freeman

for each i : 0,1,2,3,4. Thus, conjugacy of a point w.r.t.

L(9) is equivalent to conjugacy w.r.t, each of the five

quadrics, Qi" Consequently, the conjugacy relation w.r.t.

L(9) is independent of coordinate system, symmetric, and

if R is conjugate to {Yk } w.r.t. L(9), then R is conjugate

to every point dependent on {Yk }. The set P(R) of points

conjugate to any point R E S(9) w.r.t. L(9) is a subspace

called the polar space of L(9) at R. If R ~ L(9) the

polar space has some properties of a tangent space for

one quadric (propositions 1,2). So, we also denote P(R)

by T(R) when R e L(9). For the remainder of this paper

the term conjugacy will only be used to mean conjugacy

w.r.t. L(9) as defined above. If the polar space of R

passes through S, then the polar space of S passes through

R.

LEMMA i. If R e L(9), then the polar space, T(R), is a

6-flat.

Pf: The polar flat w.r.t, each quadric Qi is a hyperplane

so the five relations iii) determine a system five linear

equations. It is sufficient to show the system of equations

has rank 3 when R s L(9). At least one coordinate, Ri, of

R is nonzero and by inspection of the associated 5 x i0

matrix a 3 x 3 subdeterminant containing R i may be found

which is not zero. Hence, the rank is at least three.

By using the identities iv) it follows that any one of the

linear equations is a linear combination of three of the

remaining four equations. Hence, the rank of the matrix

is at most three and the lemma is established.

PROPOSITION i. Let i.) R e L(9); 2.) r = e-l(R). Then

for each point Q e T(R)~ L(9), Q # R, Q corresponds under -i

to a line in S(4) meeting the line r. For each line m

164

Freeman 7

of S(4) meeting the line r, m corresponds under ~ to a

point M e T(R)~L(9).

Pf. Let Q ~ T(R)~ L($) with q = -I(Q). The coordinates

of Q then satisfy all five linear relations vi) of the

polar of R and also relations iii). Points of S(4) having

row vectors ~i' ~' [el' ~] may be chosen which span the

line r[q]. The five 4 • 4 subdeterminants of the 5 x 4 t t t)

matrix (r~, r2, ql' q2 are precisely the relations vi)

which Q satisfies. Hence, the rank of the matrix is at

most three. The rank of the matrix is at least three

since the two lines are distinct. Hence, the two lines,

r,q, meet in a point.

Conversely, suppose two distinct lines r,q of S(4)

meet in a point. Thus, the associated 5 x 4 matrix obtained

from any four points spanning the lines (as in the above

paragraph) has rank 3. Hence, the five 4 x 4 subdeterminants

are zero which are precisely the relations vi) evaluated at

R and Q. Thus, Q e T(R)~L(9). We note that for R e L(9),

R e T(R)~L(9).

PROPOSITION 2. Let i.) C e L(9); 2.) D ~ L(9); 3.) C and

conjugate w.r.t. L(9). Then the line CD meets L(9) in

exactly C.

Pf. Let c, d be coordinate vectors representing C, D,

respectively. Any point W on the line CD other than D has

a coordinate vector,

W: w = c + 6d, 6 e GF(q).

For each i, i = 0,...,4, cA.c t = O, and for at least one l

k, k ~ {0,1,2,,3,4},

d_~d_ t # O.

165

8 Freeman

If W is to be on L(9) then in particular

However C and D are conjugate, hence the only solution

is 6 = 0. Thus, C is the only point of the line CD

meeting L(9).

Remark i. The above development shows that some properties

of the conjugacy relation for a single quadric extend to

conjugacy w.r.t. L(9). However~ not all properties extend.

For if R is on L(9) every line in the polar space of R

passing through R and not contained in L(9) meets L(9) in

only the point R. However, there are lines meeting L(9)

in only the point R and not in the polar space of R. With

C = < e01 > = < (i 0 0 0 0 0 0 0 0 0) >,

D = < (0 1 0 0 1 0 1 1 0 i) > it is easily checked that

C e L(9), D s L(9), the line CD meets L(9) in only C, and

yet C and D are not conjugate.

4. Generating Flats of L(9).

The correspondence between lines of S(3) and the

points of nondegenerate hyperbolic quadric in S(5), often

termed the Klein quadric is well known [17~ p. 327]. The

Grassmanian space of the lines of S(4) has analogous

properties. The facts developed in this section are

indicated in [16, p.521,526]. They are included here

only for continuity of exposition and to emphasize proper-

ties of ~.

A nondegenerate hyperbolic quadric Q(5) of S(5) is

ruled by two collections of plsnes (generating planes).

For, if P is a point of S(3) the lines of S(3) passing

through P correspond under e to the points of a plane,

166

Freeman 9

z(P), on Q(5) and dually, the lines in a plane A of S(3)

correspond under e to the points of a plane z(A) on Q(5).

Two distinct planes of the same collection meet in exactly

a point. If P,Q are points of S(3) then ~(P)~z(Q)

corresponds under -i to the line PQ of S(3). If A,B are

planes of S(3), then ~(A)~z(B) corresponds under -i to

the line A/~B of S(3). Two planes, one from each class,

either meet in a line or are skew depending whether the

associated point and plane in S(3) are incident or not.

The analogous situation occurs in the L(9) case.

There are at least two types of planes of S(9) which are

generating planes of L(9). The lines of a 3-flat passing

through a given point correspond to the points of a

generating plane of L(9). Label such planes type I planes.

The lines of S(4) belonging to a plane of S(4) correspond to

a generating plane of L(9) also. Label such a plane type

If.

PROPOSITION 3. Any generating ' plane of L(9) is either a

plane of type I or II.

Pf. Let z be a generating plane of L(9) with 3 noncollinear

points of ~ labeled P,Q,R. Since zCL(9) these 3 points

are mutually conjugate and correspond to 3 lines p,q,r of

S(4) no two of which are skew. Let p,q meet in the point 0.

Either r passes through 0 or not. The plane ~ is of type I

or type II if r passes through 0 or not.

A similar argument shows:

PROPOSITION 4. The lines of S(4) passing through~ point

of S(4) correspond under ~ to the points of a 3-flat of

S(9) lying in L(9). Conversely , if G(3) is a generating

3-flat of L(9), then the points of G(3) must correspond t o_o

the lines of S(4) passing through ~ point. Two generating

3-flats of L(9) meet in exactly ~ point.

167

i0 Freeman

COROLLARY I. If a flat of dimension n of S(9) is a

generating flat of L(9), then n ~ 3; generating t-flats

occur for t = 0,1,2,3.

5. A Group of PFojective Transformations , a Partition of

s(9), and the polar Space of a Point n~ ~ L(9 )'

The group of projective transformations of S(4)

induces a group G' of projective transformations on S(9)

which leave L(9) invariant. In this section we show that

the group G' acts transitively on the points of L(9) and

on the points of S(9)~L(9). We also examine the polar

flat of a point of S(9)~L(9). See [i, p.92] for defini-

tions.

We first obtain a partition of the points of S(9).

Let F(3) be a 3-flat of S(4). Since PGL(5,q) is transitive

on the 3-flats of S(4)~ F(3) may be taken to be

< eo,el,e2,e3 >. The six lines < ei,e j > map under ~ to

six independent points < e.. >, 0 < i < j < 3 which span i] -- --

a 5-flat, G(5), of S(9). This 5-flat sections L(9) in

exactly a nondegenerate hyperbolic quadric Q(5) since

is i-i. Consider another 3-flat F'(3) of S(4). Hence

F(3)~F'(3) is a plane F(2). Similarly the lines of

F'(3) correspond to the points of a quadric Q'(5) in a

5-flat G'(5). The lines of F(2) correspond to the points

of a plane G(2) in L(9) so G'(5)~G(5) is at least a plane.

Since PGL(5,q) is transitive on pairs of 3-flats of S(4)

a routine coordinate calculation shows:

LEMMA 2. For any pai r f distinct 3-flats F(3), F'(3) of

S(4), the associated 5-flats G(5), G'(5) under ~ span an

8-flat of S(9).

168

Freeman ii

COROLLARY 2. If G(5), G'(5) is a pair of 5-flats of S(9)

obtained as in lemma 2, then G(5)/~G'(5)~L(9).

The (q5-1)/(q-l) 3-flats of S(4) determine a corre-

sponding set of 5-flats,

vii) {G(5)}

of S(9) each meeting L(9) in a quadric Q(5). The number

of points in one of the 5-flats not on the corresponding

Q(5) is q2(q3-1). Any two 5-flats in vii) have no point

in common in S(9)~L(9). The total number of points

either in one of the 5-flats and not in L(9), or on L(9)

is seen to be the number of points of S(9). This proves:

LEMMA 3. The points of S(9) may be partitioned into the

following sets of points: i.) th e points of L(9),

2.) the points not on L(9) and belonging to the 5-flats

associated with the 3-flats of S(4) given in vii). Each

point of S(9)~L(9) is in exactly one such 5-flat.

COROLLARY 3. Let G' < PGL(10,q) be the Kroup of ppojective

transformations T' of S(9) induced from the projective

transformations T of S(4). Then G' acts transitively on

the points of L(9) and acts tpansitively on the points of

S(9) not in L(9).

Pf. Since PGL(5,q) acts transitively on the lines of S(4),

the induced group G' leaves L(9) invariant and acts tran-

sitively on the points of L(9). It suffices to show that

G' is transitive on the points of S(9)~L(9).

Let R be a point of S(9)~L(9). There is a unique

5-flat G(5) of the collection in vii) passing through R

(lemma 3). This flat meets L(9) in a quadric Q(5). There

is a secant line of Q(5) passing through R. This is also

a secant line of L(9) since G(5) meets L(9) in Q(5). The

169

12 Freeman

two points of Q(5) on a secant line correspond to two skew

lines of S(4). Since PGL(5,q) is transitive on pairs of

skew lines in S(4), G' is transitive on secant lines of

L(9). By Witt's theorem [14, p.98] the group of projective

transformations of G(5) leaving invariant Q(5) and a

secant line is transitive on the points of the line not

in Q(5). G' is transitive on the collection vii) since

G is transitive as 3-flats of S(4). Hence G' is transitive

on the points of S(9)~L(9).

COROLLARY 4. The polar flat of any point R not on L(9) is

Pf.

be

a 4-flat.

By Corollary 3 the coordinates of R may be taken to

viii) R: p(l 0 0 0 0 0 0 1 0 0) = < e01 + e23 >.

By vi) the polar flat is given by five linearly independent

equations. Since the polar relation is preserved under

nonsingular linear transformations, the statement follows

from Corollary 3.

REMARK 2. Let P(R) be the polar 4-flat of R w.r.t. L(9).

Then P(R) is the same 4-flat as the polar space of R w.r.t.

Q(5) = GR(5)~L(9) if the associated 5-flat GR(5) in vii)

is considered as the entire space.

By proposition 2 every line RQ with Q e L(9) ~P(R)

is a tangent line to L(9). By Remark I, there is a line

passing through R meeting L(9) in a single point Q where

Q ~ P(R). A line of S(9) passing through R meeting L(9)

in exactly one point Q where Q ~ P(R) is said to be a

pseudo-tangent. Through points not on L(9) there pass

secants, tangents and pseudo-tangents to L(9).

170

Freeman 13

PROPOSITION 5. Let R { L(9) and consider the associated

5-flat GR(5) given in vii).

rt angent, if S e P(R)

A line RS with S ~ L(9) is a Jsecant, if S ~ GR(5)~P(R)

Lpseudo-tangent, if S s GR(5).

Pf. By Corollary 3, R may be taken to have coordinates

as in viii). Let S e L(9). Then S satisfies all 5

equations in ill). A point W # R on the line RS has

coordinates

W = S + 8R = (S0+8 S I S 2 S 3 S 4 S 5 S 6 $7+8 S 8 $9) ,

for some 8 e GF(q). Using relation iii) and the hypothesis

that S e L(9) it follows that W e L(9) if and only if

ix) 368=0, S36=0,$9i8=0 ,S88=0; 6(6+(S0+$7) )=0. By remark 2, P(R)~ GR(5) and the equations of G5(R) are

X 9 = X 8 = X 6 = X 3 = 0. There are three cases to consider

for S e L(9).

Case i: If S e P(R), then S 6 = S 3 = S 9 = S 8 = 0 and

S O + S 7 = 0. Hence, by ix), W e L(9) if and only if

6 = O. Hence, as expected, if S ~ P(R)~L(9), then RS

is a tangent line.

Case 2: If S ~ G5(R)'~P(R) , then S 9 = S 8 = S 6 = S 3 = 0

and S O + S 7 # 0. Thus 6 = 0 and 6 = -(S O + S 7) ~ 0 give

two distinct points on the line RS and on L(9). Hence, if

S e G5(R)~-P(R) , then RS is a secant line.

Case 3: If S { Gs(R) and on L(9), then at least one

of $9,$8,$3,S 6 is nonzero. Hence from ix), if W is to be

on L(9), then 6 = 0 is the only solution. This establishes

the proposition.

171

14 Freeman

The properties given in lemma 3 and proposition 5 are

indicated in Brahana's work on metabelian groups [6].

However Brahana's statement on tangency [6, p.548] seems

unnecessarily vague and proposition 5 regarding lines

meeting L(9) and passing through a point R not on L(9) is

much more pPecise.

6. R(I,2) Reguli and (w,~)-configu~ations.

Let w be a line and z be a plane of S(4). Suppose

w and ~ are skew. The collection of lines of S(4) meeting

both w and ~ is called a (w,~)-configuration. Recall

(section i) that an R(I,2) regulus is a collection of

q+l skew planes in S(5) such that a line meeting three of

the planes meets all q+l planes. We call these planes the

axis-plangs of the R(I,2) regulus.

PROPOSITION 6. ~ (w~)Tconfiguration corresponds under

to the points of an R(I,2) regulus on L(9).

Pf__. Label the points of w, P0,...,Pq and let W = ~(w).

The lines of S(4) meeting w in P. correspond under e to i

the points of a generating 3-flat, St(3) , in the tangent

space, T(W), (propositions 1 and 4). Any two of the q+l

3-flats S.(3) meet in exactly W since w is the only line 1

of S(4) passing through all the Pi" Since < P''~l > is a

3-flat of S(4) the lines of S(4) passing through P. and l

meeting ~ correspond under ~ to the points of a plane,

~i' of St(3). Any two of these planes are skew since no

line of S(4) meets ~ in a point and s in two points.

Consider the collection of {~i }, of planes in

T(W)~L(9) obtained from an (w,~)-configuration. Let S

be a point on a plane ~k of the collection with s = ~-I(S).

The line s of S(4) meets w in the point Pk and ~ in a

172

Freeman 15

point, Q(s). The planar pencil < w,Q(S) > with vertex

Q(S) corresponds under ~ to a generating line of L(9).

This line meets each element of {~.} in a point. Hence, 1

the line is a transversal to the collection through S.

Thus, at each point of the collection of skew planes

there passes a transversal to the collection. Any two

planes of {~.} span a 5-flat in the tangent 6-flat, T(W). m

A dimension argument using the transversals to the collection

{~.} shows that the collection spans a 5-flat. Hence, the i

collection {~k } is an R(I,2) regulus.

REMARK 3. The R(I,2) regulus on L(9) determined by a

(w,~)-configuration spans a 5-flat of T(W) not passing

through W.

Pf. As in the proof of proposition 6, the axis planes do

span a 5-flat S(5) in T(W). Suppose S(5) passes through

W. Then the two generating 3-flats of L(9), < W,~. > and ]

< W,~ k > with j # k are in a 5-flat. Hence, they meet in

at a line which contradicts proposition 4.

PROPOSITION 7. Let i.) W be a point of L(9), 2.) ~ = e-!(W),

3.) T(W) be the tangent 6-flat of L(9) at W, 4.) S(5) be

any 5-flat not passing through W. Then , S(5) ~L(9) is an

R(I~2) regulus and the points of R(l~2) correspond under -i

to a (w~)-configuration for a unique plane ~ skew to

W,

Pf. As before, label the points of w, P0,...,Pq.

L(9) AT(W) is a collection of q+l 3-flats S.(3) pairwise 1

meeting in W, ( p r o p o s i t i o n s 1 and 4). Since S(5) does

not pass through W, (q+l) skew planes ~. are determined in z S(5) where

x) ~. : S ( 5 ) ~ S . ( 3 ) . 1 1

173

16 Freeman

-! The points of T 0 correspond under ~ to the lines of S(4)

meeting w in the point P0 and which are in a 3-flat~ F0(3) ,

of S(4). This 3-flat F0(3) does not contain w since W { T 0

on L(9). Hence, F0(3)~w = P0' That is, the lines of

F0(3) through P0 correspond under ~ to the points of the

plane T 0 .

For any plane ~' of F0(3) with P0 ~ ~'' the image

under e of the (w,~')-configuration is an R(I,2) regulus

on L(9) having T 0 as an axis-plane (proposition 6). We

now need to find a unique plane ~, of F0(3) so that the

(w,~)-configuration corresponds under ~ to the points of

the planes {~.} of (x). To determine this plane in S(4) i

consider another plane 71 # T 0 of the collection in (x).

The points of ~i correspond under -i to lines of S(4)

meeting w in P1 # P0 and lying in some 3-flat Fl(3) not

containing w. The 3-flats F0(3) and FI(3) are distinct

since T 0 and ~i are distinct generating 3-flats S0(3) ,

SI(3) of L(9). Hence, F0(3) and FI(3) meet in a plane,

7. Since P0 and P1 are not in F0(3)~FI(3)=~ is skew to

w,

We now show that for this choice of ~, the (w,~)-

configuration corresponds under ~ to the points of the

planes in (x), so the collection (x) is an R(I,2) regulus

(proposition 6). Let Q be any point of ~. The planar

pencil < w,Q > with vertex Q corresponds to a generating

line, g, of L(9). In particular, the two lines QP0' QPI

of the pencil correspond to two points Q0' QI' on 70, ~l'

respectively. Hence, the line QoQI = g is in S(5). The

line g must meet each plane 7. in a point, since S(5) 1

meets L(9) in exactly the skew planes of (x) and g, a

generating line, is in S(5). Hence, g is a transversal

174

Freeman 17

to the collection {~.}. The planar pencil < w,Q > with 1

vertex Q corresponds under ~ to a transversal of the

collection. Thus, the collection {~.} is an R(I,2) i

regulus.

COROLLARY 5. Let i.) F(3) b_~e ~ 3-flat of S(4) meeting

line w in exactly ~ oip_qi~ , 2.) ~ be the generating

plane of type ~ o__nn L(9) cprresponding under ~ to the lines

of F(3) passing through PO , 3.) ~(w) = W e L(9). Then

there is a i-i correspondence between (w~)-confi~urations

in S(4) and R(I~2)reguli of T(W) on L(9) where ~ is any

plane of F(3) not passing through P~ and ~ is an axis

plane of each of the R(I~2) reguli.

REMARK 4. Given two R(I,2) reguli in S(5) two frames for

S(5) may be chosen showing that PGL(6,q) is transitive on

R(I,2) reguli of S(5). By the natural embedding of S(5)

in S(9) it follows that any R(I,2) regulus may be viewed

as the image of an (w,~)-configuration in S(4).

7. R(I,2) Reguli and 3-flats.

Bose has shown that R(I,2) reguli represent Baer

subplanes in a linear representation of PG(2,q2), [3,11].

With this fact in mind and with the purpose of examining

intersection relations of Baer subplanes of PG(2,q 2) we

now investigate how a 3-flat of S(5) may intersect an

R(I,2) regulus.

An elementary coordinate argument following [5] may

be used to show that any three skew planes of S(5) deter-

mine a unique R(I,2) regulus. Let R(I,2) be an R(I,2)

regulus with axis-planes labeled ~.. Since the axis-planes l

are skew each point P of R(I,2) is on a unique transversal

line t(P), in R(I,2). It follows that:

175

18 Freeman

REMARK 5. A. Consider three transversals of R(I,2) that

are not in the same 3-flat. Then any two of the three

3-flats spanned by pairs of the 3 transversals meet in

exactly the common transversal.

B. Let P, Q e ~i' P # Q" Then the two 3-flats

< t(P), ~i >' < t(Q), Z'l > are distinct.

PROPOSITION 8. Let t(P), t(Q) be distinct transversals

of R(I,2) meeting the axis planes -m ~" . . . . in ~i' Q-i' respectively.

Then, ! transversal of R(I,2) passinK through ! point of

the line PiQ i is in the 3-flat < t(P), t(Q) > and is a

transversal to the R(I~I) regulus determined b_flany three

distinct lines PjQj, PkQk , PmQ m.

Pf. The 3-flat < t(P), t(Q) > does not contain any plane

~'i (B, remark 5) and thus meets ~.l in the line PiQ i. The

3 lines PiQi, i = 0,1,2 determine a unique R(I,I) regulus,

R(I,I). Since a line of the opposite regulus meets each

of these three lines in a point, it meets each of the

planes T0, ~i' ~2 in a point. Hence, for each point R

on PoQ0 , the line of this opposite regulus through R must

be a transversal to R(I,2) and may be properly labeled t(R).

Now t(R) must meet each of the remaining axis planes.

Since t(R) is in < t(P), t(Q) > the remaining lines of

R(I,I) are precisely the lines PiQi , i = 3,...,q.

This shows that two transversal lines of R(I,2)

determine a 3-flat which meets R(I,2) in exactly a non-

degenerate hyperbolic quadric.

PROPOSITION 9. Let i.) two lines si, ~ lie in two

distinct axis planes, ~i' ~ of R(I~2), 2.) F(3) = < ~i' ~ >"

Then either F(3) meets R(I,2) in a nonde~enerat e hyperboli c

quadric, or F(3) meets R(I,2) exactly i__n_n ~i , ~ and a

transversal t.

176

Freeman 19

Pf. The transversals of R(I,2) through s. meet 7. in a

line (proposition 8) which meets s. in at least one point.

Thus, F(3) contains at least one transversal line, t.

If F(3) meets R(I,2) in one further point, R of 7., l

then R and s. span z.. Hence, ~. is in F(3). This implies 1 l l

that s. meets 7. in a point which contradicts the fact that

the axis planes are skew. Hence, if F(3) meets R(I,2) in

a further point R, then R must be in a plane, ~k' with

i ~ k ~ j. Now t meets ~k in a point, Tk, whence the line

TkR is in F(3). The three lines, si, s. and TkR are skew,

share a common 3-flat, and thus determine an R(I,I) regulus.

Each line of the opposite regulus meets ~i' ~j' ~k and

hence is a transversal to R(I,2). Hence F(3) must be the

unique 3-flat meeting R(I,2) in the R(I,I) regulus passing

through s.. l

Our calculation of the possible intersections of a

3-flat with an R(I,2) regulus depends on the following

lemma.

LEMMA 4. Let i.) the axis planes of an R(I,2)regulus b__ee

labeled ~., i = 0~i~ ,q-l~ 2.) ~ be a nonruling plane --9- "'" '

of R(l~2) which meets R(I~2) in at least three noncollinear

points, ~, [i' P-~' so that no two are on the same axis

plane of R(l~2), 3.) t(P0) , t(Pl) , t(P ) be the associated

transversal lines passing through the respective points.

Then, i.) If the transversals lie in the same 3-flat, then

meets R(I~2) in a (degenerat e or nondegenerate) conic.

2.) If the transversals are not in the same 3-flat, then

meets R(I~2) i_nnexactly the three points.

Pf. Since the three points are not collinear at least two

distinct transversals are determined. The first statement

then follows from proposition 8.

177

20 Freeman

To establish the second statement we may suppose

that the three points determine three distinct trans-

versals which do not lie in the same 3-flat. Using the

idea given in [5, p.88], coordinates may be introduced

so that,

P0 = < e0 > e T0: < e0, el, e 2 >

PI = < el+e4 > ~ ~i: < e0+e3' el+e4' e2+e5 >

P = < e 5 > e ~: < e 3, e 4, e 5 >

Thus t(P 0) = < e0, e 3 >, t(P I) = <el, e 4 >,

t(P ) = < e2, e 5 >, no transversal of these three is in

the 3-flat determined by the remaining two, and

= < e 0, el+e 4, e 5 >,

~k = < (i 0 0 k 0 0), (0 1 0 0 k 0), (0 0 1 0 0 k) >,

for ~k distinct from ~ . By linear algebra it follows

that there is no point distinct from P0' PI' P~ which on

and ~k for any k, k = 0,1,. .. ,q-l,~.

Since any 3-flat meets each axis plane of R(I,2) in

at least one point, the following proposition gives the

possible intersections of a 3-flat with an R(I,2) regulus.

PROPOSITION i0. A 3-flat 7 of S(5) is in exactly one of

the followin~ classes w.r.t, an R(!,2)" regulus.

I. ) 7. contains one axis plane ~. of R(I,2) and meets

the remainin ~ planes i__nn each a point. The points are on

a transversal to R(I,2).

II.) i) ~ R(I,2) is a nonde~enerate hyperbolic quadric.

ii) Z__A R(I,2) is 3 lines, si, ~, t__, with s{,

belonging to two distinct axis planes ~i' ~j' respectively'

The line t is a transversal of R(I,2) and si, s.. --j

178

Freeman 21

iii) Z meets R(I~2) in a plane tangent to an R(I~I)

regulus o__nnR(l~2).

iv) ~ meets one axis plane ~. in a line s with I

the remaining ~ points of intersection coplanar add lying

on a nondegenerate conic with s meeting the plane of the

conic in exactly the '(q+l)-st' point of the conic.

IIl.) i) Z meets R(I~2) in exactly !transversal line.

ii) Z meets R(I~2) in a set of q+l points, no 4

coplanar. 2

Pf. Since there are q +q+l 3-flats containing a given

plane of S(5), if Z contains an axis plane of R(I,2) then

Z must be in class I (B, remark 5).

If Z does not contain an axis-plane and meets R(I,2)

in at least 3 lines, then the proof of proposition 8 implies

that Z meets R(I,2) in a nondegenerate hyperbolic quadric.

Hence Z is of class ll(1). If Z meets two axis planes in

lines, then Z is either class ll(i) or l!(iii) (proposition

9).

Suppose Z meets exactly one axis plane, n0, in exactly

a line s. Hence Z meets each remaining axis plane in

exactly a point. Thus q points are determined. If 3 of

these q points are collinear (q odd), then the common line

must be a transversal. Since this transversal is in Z,

then the q points of intersection must be on this trans-

versal, t. If t and s are skew, then Z would contain T 0

which contradicts the hypothesis of this paragraph. Hence,

t and s span a plane which is tangent to the nondegenerate

hyperbolic quadric determined by the transversals to s as

in proposition 8. Hence, if Z meets exactly one axis plane

in a line, s, and if three of the q remaining points are

collinear, then ~ is of class ll(iii).

179

22 Freeman

Assume now that Z meets one axis plane in exactly

a line, s, and that no three of the q remaining points

of intersection with the axis planes are collinear. Let

be the plane determined by any three, R0, RI, R 2. If

contained s, then the line R0R I would meet s in a point.

Hence R0R I meets R(I,2) in three points and would then be

a transversal. Hence, Z is of class II(iii) and the q

points are collinear. Since we are assuming that no three

of the q points are collinear, z must not contain the line

s. Hence ~ meets s in a point, distinct from R0, RI, R 2.

Hence ~ meets R(I,2) in a nondegenerate conic (lemma 4)

and the line s must meet ~ in a point of the conic. Thus,

is of class il(iv).

Assume now that E meets each axis plane in exactly

a point. Thus, q+l points of ~ are determined. If three

such points were eollinear, then the common line is a

transversal and the remaining points of intersection would

be collinear. Hence, if three points of intersection are

collinear then Z is of class III(i).

Suppose that Z~R(I,2) is a set of q+l points, no

three collinear. If four points of these points were

coplanar then the plane would meet R(I,2) in a nondegenerate

conic (lemma 4). We will show now that all 3-flats containing

a plane, ~, which meets R(I,2) in a nondegenerate conic C

must be of class If(i) or II(iv).

The q+l distinct transversals determined by the points

of the conic C are all in the same 3-flat(lemma 4) and

determine an R(I,I) regulus with R(I,I)(-~ : C. Hence,

is contained in a 3-flat of class II(i), Z(~).

Label the points of the conic Qi where Qi = ~i ~ C.

Through Qi there pass q lines, s , of ~i such that <~,s

180

Freeman 23

is a 3-flat distinct from Z (7). If two such 3-flats were

the same, say Z ', then Z ' would contain ~i" Hence Z ' would

be of class I. Since ~ contains C then Z' could not

contain ~. This contradiction establishes the fact that

the q 3-flats <7 ,s are distinct and distinct from Z(~).

Let s m, of ~i, ~j pass through Qi' Qj' respectively,

with s m, # ~(~). Similarly, the two 3-flats <~,~>

<7, m > are distinct. Hence, for a plane ~ meeting R(I,2)

in a nondegenerate conic, the q2+q+l 3-flats passing

through~ are either of class II(i),l, or of class ll(iv),

q(q+l). Each such 3-flat meets R(I,2) in a line. Thus,

if a 3-flat meets R(I,2) in exactly q+l points, no three

collinear, then it follows that no four are coplanar.

Hence, they lie on a rational cubic if q ~ 5115, p.311].

The number of 3-flats in each class described in

Proposition i0 is given below.

I. (q+l)(q2+q+l)

2 II. i) q +q+l

2 2 it) (q+l) (~)(q2+q+l) 2

iii) (q+l)2(q-l)(q2+q+l)

iv) (q+l)2(q-l)(q2)(q2+q+l)

Ill. i) (q-l)2(q2)(q2+q+l)

4 ,, 4 22 it) (q -q)~q -q )

References

[1]

[2]

ARTIN, E.: Geometric Algebra. New York and London, !nterscience. 1957.

BAKER, H.F. : Principles of Geometry. Vol. 4. New York, Academic Press. 1952.

181

24 Freeman

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

BOSE,R.C.: On a Representation of the^Baer Subplanes of the Desarguesian Plane PG(2,q 2) in a Projective Five Dimensional Space PG(5,q). Proceedings of International Colloquium on Combinatorial Theory. Rome, lialy, September 3-15, 1973.

BOSE,R.C., BARLOTTI,A.: Linear representation of a class of projective planes in a four dimensional projec- tive space. Ann.Mat. Pura Appl. 88(1971) 9-32.

BOSE,R.C., SMITH,K.J.C.: Ternary Rings of a class of Linearly Representable Semi-translation Planes. Atti del convegno de combinatoria e sue applica- zioni. Perugia (1971) 69-101.

BRAHANA,H.R.: Finite Metabelian Groups and the lines of a projective four-space. Amer.Jour.Math. 73(1951) 5 3 9 - 5 5 5 .

BRAHANA,H.R.: Metabelian p-groups with five generators and orders p12 and pll. Ill. J.Math 2(1958) 641-717.

BRUCK,R.H., BOSE,R.C.: Linear Representations of Projec- tive Planes in Projective Spaces. J.Algebra. 4 (1966) 117-172.

CHOW,W.: On the Geometry of Algebraic Homogeneous Spaces. Ann. of Math. 50(1949) 32-67.

DEMBOWSKI,P.: Finite Geometries. Berlin-Heidelberg, Springer-Verlag. 1968.

FREEMAN,J.W.: ~ representation of the Baer subplanes of PG(2,q ~) in PG(5,q) and properties of a regular spread of PG(5,q). Proc. Inter. Conf. Proj. Planes. Wash. St. Univ. Press. (1973) 91-97.

HODGE,W.V.D.,PEDOE,D.: Methods of Algebraic Geometry, vol.2. Cambridge, Cambridge Univ. Press. 1968.

KLEIMAN,S.,LAKSOV,D.: Shubert Calculus. Amer.Math. Monthly. 79(1972) 1061-1082.

O'MEARA,O.T.: Introduction to Quadratic Forms. New York, Academic Press. 1963.

SEGRE,B.: Lectures on Modern Geometry. Mono. Mat. C.N.R. vol. 7. Rome, Edizioni Cremonese. 1961.

182

Freeman 25

[16]

[17]

TODD,J.A.: The locus representing the lines of a four-dimensional space and its application to linear complexes in four dimensions. London Math. Society,series 2. 30(1930) 513-550.

VEBLEN,O., YOUNG,J.W.: Projective Geometry,vol.l. New York, Ginn and Company. 1910.

J.W. FREEMAN Department of Mathematics Colorado State University Fort Collins, Colorado 80521, USA.

(Eingegangen am 31. Juii 1974)

183