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On A Characterization of the Grasshann Manifold Representing the Planes in A Projective Space
Annals of Discrete Mathematics 14 (1982) 129-150 0 North-Holland Publishing Company
ON A CHARACTERIZATION OF THE GRASSHANN MANIFOLD REPRESENTIRG THE PLANES I N A PROJECTIVE SPACE
Alessandro Bichara and Giuseppe Tal l i n i I s t i t u t o Matematico "G. Castelnuovo", U n i v e r s i t i i d i Roma, I t a l y 1.
INTRODUCTION The Grassmann manifold representing the planes i n a p r o j e c t i v e space w i l l
be characterized as a p a r t i a l l i n e space (G,
F ) whose maximal subspaces s a t i s f y
some s u i t a b l e conditions (see s e c t i o n 2 ) . The p r o o f uses the f o l l o w i n g : from
F ) i t i s possible t o construct - i n a natural way - another p a r t i a l l i n e space, say (S, R), which turns o u t t o be isomorphic t o the Grassmann manifold (G,
representing the l i n e s i n a p r o j e c t i v e space [ 3 1 . The r e s u l t s t h a t w i l l be proved here can be generalized t o t h e Grassmann manifold representing the d - f l a t s i n a p r o j e c t i v e space. The p r o o f o f such a general theorem, which goes by i n d u c t i o n ( s t a r t i n g from d = 2 , since the step from d
1 t o d = 2 i s q u i t e special), i s the subject o f another paper by the
authors.
2.
PRELIMINARIES L e t G be a non empty set, whose elements w i l l be c a l l e d p o i n t s , and
F a
(proper) non empty c o l l e c t i o n o f (proper) subsets o f G, which w i l l be c a l l e d
l i n e s . The p a i r (G,
F ) i s s a i d t o be a p a r t i a l l i n e space (PLS) i f the f o l l o w -
ing hold ( [ Z I ) : ( i ) Any two d i s t i n c t p o i n t s i n G belong t o a t most one l i n e i n F. ( i i ) Any l i n e i n F contains a t l e a s t two p o i n t s o f G. ( i i i ) F i s a covering o f G. Two d i s t i n c t p o i n t s p and p ' i n G w i l l be s a i d t o be c o l l i n e a r i f they belong 129
A . Bichara and G. Tallini
130
t o a l i n e , denoted by (p,p'),
i n F; by p
- p'
i t i s meant t h a t p and p '
(p# p ' ) are c o l l i n e a r ; otherwise, p and p ' are non-collinear: p f p ' . (G, F)
w i l l be c a l l e d a proper p a r t i a l l i n e space (PPLS) i f t h e r e
e x i s t two non-collinear p o i n t s i n G. A p o i n t subset H o f a PPLS (G, F) i s c a l l e d a subspace i f any two o f i t s p o i n t s a r e c o l l i n e a r and the l i n e j o i n i n g them i s completely contained i n H; a subspace H o f (G, F) i s a maximal subspace i f i t i s n o t p r o p e r l y contained i n any subspace o f (G, F ) . A PLS (G, F) w i l l be c a l l e d i r r e d u c i b l e i f t h e f o l l o w i n g holds: Any l i n e i n F c o n t a i n s a t l e a s t t h r e e p o i n t s o f G.
(ii')
F i n a l l y , a PPLS (G, F) i s s a i d t o be connected i f For any two d i s t i n c t p o i n t s p, p ' i n G t h e r e e x i s t s a polygonal
(iv)
path j o i n i n g them. L e t P(r,K)
(r
> 4)
be an r-dimensional p r o j e c t i v e space over a f i e l d K.
L e t G be t h e c o l l e c t i o n o f a l l t h e planes i n P(r,K) and F c o n s i s t s o f p e n c i l s o f planes i n P(r,K)
(a p e n c i l o f planes being the s e t o f a l l t h e planes
through a l i n e contained i n a 3 - f l a t P(3,K) i n P(r,K)),
Then (G,
F) i s a
proper i r r e d u c i b l e PLS which i s isomorphic t o Grassmann manifold G
rY2,K (As i t i s w e l l known, Gr,2,K i s an i r r e rtl ducible algebraic (3(r-2))-dimensional manifold i n an(( ) 1)-dimensional representing the planes i n P(r,K).
-
p r o j e c t i v e space; moreover, t h i s manifold i s an i n t e r s e c t i o n o f quadrics.) Any n o n - t r i v i a l ( i . e . p r o p e r l y containing some element o f F) subspace in
(G, F ) i s e i t h e r
( a ) the c o l l e c t i o n o f a l l the planes through a l i n e L belonging t o an h - f l a t (through L), h 3 4 (such a c o l l e c t i o n w i l l a l s o be c a l l e d an L - s t a r ) or (b) the c o l l e c t i o n o f a l l planes belonging t o a 3 - f l a t ; o r ( c ) the c o l l e c t i o n o f a l l the planes through a p o i n t i n a 3 - f l a t ;
such
a s e t o f planes w i l l be c a l l e d a s t a r o f planes. Therefore, a maximal subspace i n (G, F) i s e i t h e r an il-star, and the c o l l e c t i o n o f such maximal subspaces w i l l be denoted by S, o r the s e t o f a l l planes i n a 3 - f l a t i n P(r,K),
and the c o l l e c t i o n o f these maximal sub-
spaces w i l l be denoted by T. The PPLS (G, F) i s connected and s a t i s f i e s : Ale
If t h r e e p o i n t s i n G are pairwise c o l l i n e a r , then t h e r e e x i s t s a
The Grassmann manifold representing the planes in a projective space
131
subspace i n (G, F) through them.
No l i n e i n F i s a maximal subspace. Furthermore, t h e r e e x i s t two A2. c o l l e c t i o n s , say S and T, o f maximal subspaces i n (G, F ) and any maximal subspace belongs e i t h e r t o S o r t o T. Moreover: (1)
SES,TET*eitherSnT=fl,
(11)
VfEF*3!SES93!
(111)
I f S,
SO, S"
element i n S, d i s t i n c t from
E
TET:
or
SnTEF.
f c S , f C -T .
S p a i r w i s e meet a t d i s t i n c t p o i n t s , then any
S" and meeting both S and S' a t d i s t i n c t p o i n t s ,
meet S" too.
T
E
There e x i s t three subspaces f, n,
A3. T and
T
covers f and i s covered by
Axioms A1,
Remark 1.
7
i n (G, F) such t h a t f
E
F,
T.
A2, A3 and t h e connectedness hypothesis, besides being
s a t i s f i e d by Grassmann manifold representing t h e planes i n P(r,K),
are a l s o
s a t i s f i e d by the generalized Grassmann manifold representing the planes i n an e i t h e r i r r e d u c i b l e (and p o s s i b l y Pascalian) o r r e d u c i b l e p r o j e c t i v e space P even o f i n f i n i t e dimension. Remark 2.
The s e t t h e o r e t i c union o f Grassmann manifolds representing t h e
planes i n skew p r o j e c t i v e spaces i s a non-connected PLS s a t i s f y i n g axioms A1, A2, and A3. I n order t o characterize the (generalized) Grassmann manifold representi n g the planes i n one o r more p r o j e c t i v e spaces (see Remark 2 ) , PPLS's (G, F) s a t i s f y i n g A1, A2, and A
w i 11 be proved:
3
w i l l be studied. Namely, the f o l l o w i n g r e s u l t s
I f (G, F) i s a connected PPLS s a t i s f y i n g axioms A1, A2, and A3,
Theorem 1.
then t h e r e e x i s t a p r o j e c t i v e space (1, L ) and a mapping F from the c o l l e c t i o n %
P o f a l l planes i n (1, L ) i n t o G such t h a t : (i)
F i s one-to-one and onto;
( i i ) F consists o f e x a c t l y t h e images under F o f the p e n c i l s o f planes in
?; ( i i i ) S consists o f e x a c t l y the images under F o f L - s t a r s i n (1, L),
L E L; and ( i v ) T consists o f e x a c t l y t h e images under F o f the c o l l e c t i o n s o f planes
132 in
A . Bichara and G. Tallini
?,
each o f them being formed by a l l t h e planes i n a 3 - f l a t i n (G, F ) . Thus, (C,
L ) i s i r r e d u c i b l e i f f (G, F ) i s i r r e d u c i b l e . F i n a l l y ,
if
(I, L ) i s f i n i t e l y generated and Pascalian, then (G, F ) i s isomorphic t o Grassmann m a n i f o l d r e p r e s e n t i n g t h e planes i n (C, Theorem 2 .
L).
L e t (G, F ) be a PPLS s a t i s f y i n g axioms A
A2, and A3. Then each 1' connected component o f (G, F ) i s a PPLS s a t i s f y i n g axioms A1 and A2. I f f o r each connected component o f
(G, F ) A3 holds, then (G,
F) i s the set
t h e o r e t i c union o f g e n e r a l i z e d Grassmann m a n i f o l d s r e p r e s e n t i n g t h e planes i n p a i r w i s e skew p r o j e c t i v e spaces.
3. SOME PROPERTIES OF A CONNECTED PPLS Let
(G, F )
be a connected PPLS s a t i s f y i n g axioms A1, A 2 , and A
sect. 2 . Then: P r o p o s i t i o n I. The c o l l e c t i o n s S and T a r e skew. L e t M and
3
in
PI' ( w i t h M # M I )
be two maximal subspaces having two d i s t i n c t common p o i n t s p and p ' ; then
M and M ' belong t o d i f f e r e n t c o l l e c t i o n s o f maximal subspaces. Thus, two d i s t i n c t maximal subspaces belonging t o t h e same c o l l e c t i o n have a t most one common p o i n t . Proof.
I f a maximal subspace M" were contained i n S n T, then
M" n M"
MI';
a c o n t r a d i c t i o n t o A21, as MI' p r o p e r l y c o n t a i n s a l i n e . Since p and p ' belong t o M y they a r e c o l l i n e a r ; l e t f be t h e l i n e j o i n i n g them; o b v i o u s l y , f
5M
and f c
M' , t h e two maximal subspaces belong
t o d i f f e r e n t c o l l e c t i o n s ' a n d t h e statement i s proved. P r o p o s i t i o n 11.
L e t S and S ' be two d i s t i n c t maximal subspaces i n S, having
a common p o i n t p. I f T i s a maximal subspace i n T , meeting-S and S ' a t t h e l i n e s f and f ' , r e s p e c t i v e l y , then f and f ' meet a t p, so T passes through p. Proof.
f and f ' are d i s t i n c t l i n e s (otherwise, two d i s t i n c t maximal subspaces
133
The Grassrnann manifold representing the planes in a projective space S and S' i n S would meet a t t h e l i n e Since any l i n e i n
f = f ' , which i s impossible by prop. I ) .
F has a t l e a s t two points, t h e r e e x i s t p o i n t s p '
E f and
p" E f ' such t h a t p, p ' and p" are p a i f w i s e d i s t i n c t . These p o i n t s are p a i r wise c o l l i n e a r (indeed, p, p '
E
s,
p, p"
€
s',
and p ' , p"
T);
€
thus, by A1,
there e x i s t s a subspace H through them and N i s contained i n a maximal subspace M. I f 11 belonged t o
M
s,
then M = S (by prop. I, M n
2 Hence M
s i m i l a r l y , M = S ' ( s i n c e t1 n S '
= S);
t o S, then S = S' a c o n t r a d i c t i o n .
{p,p"l).
s ' 3 {p,p')
implies
Therefore, i f I1 belonged
$ S and, by A*,
H E T . Since !1
belongs t o T and contains the d i s t i n c t p o i n t s p ' and p" i n T E T , by prop. I,M = T. Moreover, p E M and 14 = T imply t o S; thus,
{p)
5T n S
f
and
p E T. The p o i n t p i n T belongs
p E f; s i m i l a r l y , p E f ' ; therefore, t h e
d i s t i n c t l i n e s f and f ' meet a t t h e p o i n t p and t h e statement i s proved. Proposition 111. Any T i n T i s a p r o j e c t i v e space. Proof. I t i s enough t o prove t h a t i n the l i n e space T Veblen-Nedderburn axiom holds : L e t fl and f
be two l i n e s i n T meeting a t the p o i n t p3; i f f3 and f4 2 are d i s t i n c t l i n e s i n T, each o f them meeting both fl and f2 a t p o i n t s d i s t i n c t from p3, then f3 and f4 meet a t a p o i n t . = f2 n f4; then
Through
..,4)
t h e r e i s e x a c t l y one maximal subspace S E S (see i A211). Such maximal subspaces a r e p a i r w i s e d i s t i n c t ( i f i # j and Si = sj, fi ( i = 1,.
then the maximal subspace Si would share w i t h T E T the p o i n t s i n fi which i s impossible by A21). Now, fi c Si,
i = 1,2
and {p
l 3
= fl
U
f
j'
n f2 imply
I p3 } -C 1S n S2; since S1 # S2, by prop. I, {p31 = S1 n S2. By the same argument, {pl} = S2 n Sg and {pz} = S1 n S3; moreover, S meets S1 and
4 S2 a t t h e p o i n t s q1 and q2, respectively. The t h r e e maximal subspaces S1, S2, and S pairwise meet a t d i s t i n c t p o i n t s (see (3.1)) and S4 meets S1 and S
2
3 a t d i s t i n c t points; therefore, by A2111, S4 meets S3 a t a p o i n t
134
A . Blchara and C. Tallini
q: {q} = S3 n S4. Since T meets l y , and S3 n
S4
S3 and S4 a t the l i n e s f3 and f 4' respective-
= {q}, by prop. 11, f 3 and f 4 meet a t the p o i n t q and the
statement i s proved. The p r o j e c t i v e space which a r e members o f T , c o n t a i n p r o j e c t i v e planes forming a c o l l e c t i o n 11 o f subsets o f G. Clearly, any element i n 11 i s a subspace i n (G,
F ) which i s contained i n a maximal subspace belonging t o
Proposition I V .
T.
L e t T and T ' be two elements i n T through a p o i n t p i n G.
, and S3 a r e three p a i r w i s e d i s t i n c t elements i n S through p 2 f { E F. I f t h e l i n e s fi, and such t h a t T n Si = fi E F , and T ' n S i i = 1,2,3, belong t o t h e same plane a i n T, then a l s o the l i n e s f; belong
Assume S1, S
t o a unique plane i n T I .
Proof.
I t i s enough the prove
L e t a' be the plane i n T' through f ' and f;. 1 t h a t f ' belongs t o a ' .
3
L e t f and f ' be two l i n e s n o t through p, the former i n a , t h e l a t t e r i n a ' . Since a and a' a r e p r o j e c t i v e planes, f meets fl,
meets
fi and f;.
Set {qi}
= f n fi,
i = 1,2,3,
I t i s easy t o check t h a t the f i v e p o i n t s qi,
and
'
f2 and f3, and f '
{q!} = f ' n f!, j = 1,2.
J
J
are pairwise d i s t i n c t ;
qj moreover, i f S and S' are the maximal subspaces i n
9 through
f and f ' ,
, S 2 , S 3 are p a i r w i s e d i s t i n c t 1 and {qi) = S n S i , Iq!) S' n Sj ' Since S1, S2, and S3 pairwise meet a t J d i s t i n c t p o i n t s and S' ( # S) meets S, and S2 a t d i s t i n c t p o i n t s , S and S' r e s p e c t i v e l y , then the f i v e subspaces S, S', S
have a common p o i n t q, which i s obviously d i s t i n c t from q1 and q ' Thus, 1' t h e three maximal subspaces S, S1 and S' pairwise meet a t d i s t i n c t p o i n t s
3 ( # S') meets S and S1 a t t h e d i s t i n c t p o i n t s q3 and p ( r e s p e c t i v e l y ) . Therefore, S3 and S' have a common p o i n t q ' . Since Sg n T ' = f i , S' n T ' = f '
and S
and I q ' ) = S3 n S', by prop. 11, f ' and f ' meet a t 4. Hence, f ' and f i a r e 3 coplanar. The plane through them contains f ' and t h e p o i n t p on f i ; thus, i t i s a'
and f i belongs t o a ' .
The Grassmann manifold representing the planes in a pmjective space
F)
4. THE PARTIAL LINE SPACE (S, R ) ASSOCIATED WITH (G, Take p E G, a E
n,
with p
135
o f S, c o n s i s t i n g P ,a o f those max'imal subspaces i n S meeting a a t l i n e s i n F through p, i . e .
r = {SES: P ,a i s uniquely defined. Proposition V. I f a , a'
(4.1
E
n
E
a; then the subset r
and S n a E F 1
S 3 p
and p
E
,
a , p ' E a ' , then
1
Proof. Since a i s a p r o j e c t i v e plane,
through t h e p o i n t p i n a there are a t
l e a s t two d i s t i n c t l i n e s fl and f 2 o f
F.
The maximal subspace S1 and S2 i n
r and now i t w i l l be shown P ,a t h a t S1 # S2. I f S1 = S2, then t h i s member o f S would share w i t h a maximal
S through fl and f 2 ( r e s p e c t i v e l y ) belong t o
subspace T i n T a s e t I containing the d i s t i n c t l i n e s
fl and f2, which
and S1 # S2; (4.1) f o l l o w s . i s impossible by A21. Therefore, S1, S2 E r P ,a L e t S1 and S2 be two d i s t i n c t elements i n r n r S1 and S2 p,a p',a'. meet a a t l i n e s i n F through p and a' a t l i n e s i n F through p'; thus p and p ' belong t o
S, n S2. By prop. I, p = p ' .
Mhen a
a ' , (4.2) i s obviously true. Assume a
#
a ' . L e t T and T ' be
the maximal subspaces i n T through a and a ' respectively; then (otherwise, S1 would meet T a t two l i n e s i n
T # T'
F through p, one belonging t o
a, t h e other t o a ' , which c o n t r a d i c t s A21) and T n T ' =
{PI.
Furthermore,
by prop. I V , any element Sg i n r , d i s t i n c t from S1 and S2, meets T' a t P ,a being coplanar w i t h S1 n T ' and S2 n T ' belongs t o a l i n e i n F, which
-
r
P,a"
hence, rp,a
5
-
rpl,al. By the same argument,
(4.2) follows.
r p',a'
C r
-
p,a
and
{r : a E ll , p ~ a o} f subsets o f S i s defined; P ,a since i t i s n o t a proper c o l l e c t i o n (see prop. V ) , l e t R be the proper Thus, the c o l l e c t i o n
c o l l e c t i o n associated w i t h it.
136
A . Bichara and G. Tallini
The pair (S, R ) is a PLS. Moreover, two d i s t i n c t elements
Proposition VI.
in S a r e collinear i n (S, R ) i f f they have a common point i n G. Proof. Let -
S and S ' be two d i s t i n c t elements in S. If S n S '
{p}, then l e t
f be a line in S through p; through f there i s a maximal subspace T E T meeting
s'
a t one point a t l e a s t ; thus, i t meets
s'
a t a line f '
F. In the
E
projective space T , the d i s t i n c t l i n e s f and f ' a r e joined by a plane therefore, S and S ' belong to r
.
CI E
II ;
P 9Q If S n S ' = 0, no element in R through S and S' exists.
Let f be a l i n e in S E S. Through f there i s a maximal subspace T E T . If p E f , l e t f ' be a l i n e in T through p , d i s t i n c t from f . Through f ' there i s a maximal subspace S'
E
S , which i s obviously d i s t i n c t from S and meets
S a t p ; thus, through S there i s an element of R (joining S and S'). Hence, R i s a cover of S. Moreover, f o r any r E R ,
I rl > 2
two elements i n R have a t most one common p o i n t
(s, R )
(see (4.1) and any (by (4.2 ); i t follows t h a t
i s a PLS.
Let p be a point in G . The collection S
of a l l elements n S through P p i s a subspace of (S, R ) . (Indeed, any two d i s t i n c t elements in S are P
collinear i n (S, R )
and the l i n e through them i s completely contained in S ) . P Let p be a point i n G and T an element i n T through p.
Proposition VII.
s in
(S, R ) i s isomorphic t o the s t a r F consisting P P ,T of the lines in F through p and belonging to the projective space T. Thus,
Then the subspace
s i s a projective space and i s of f i n i t e dimension h i f f T i s of f i n i t e P dimension
h
t
1.
Proof. -
Any element in S meets T a t a l i n e i n F (see A21). P P,T Let cp be the mapping defined by
q:SES+SnTEF Clearly,
cp
P,T'
i s one-to-one and onto (see A211). Moreover,
cp
-1
maps pencils
onto lines, i n R , belonging t o S (Indeed, any pencil i n F conP ,T P' P,T s i s t s of a l l lines in T through p , b e l o n g i n g t o a plane n; such a pencil in F
i s the image under
cp
of the l i n e r
).
P Now, i t will be proved t h a t cp maps lines in S onto pencils i n F P P,T' ,Q
137
The Grassmann manifold representing the planes in a projective space be a l i n e i n S and S and S ' two d i s t i n c t p o i n t s on it; they p,a' P meet T a t ( d i s t i n c t ) l i n e s f and f ' , which are j o i n e d by a plane a through Let r
= r (see prop. V); furthermore, v(rp,a) = v ( r p ) Psa' p,a ,,I i s the p e n c i l c o n s i s t i n g of the l i n e s i n F through p and belonging t o a . P,T It f o l l o w s t h a t cp i s an isomorphism between S and the s t a r o f l i n e s through P p i n t h e p r o j e c t i v e space T.
p i n T; then, r
From prop. Proposition
VII,
VIII.
i t f o l l o w s immediately:
L e t T and T ' be any two d i s t i n c t elements i n T through
a p o i n t p i n G. Then T i s o f f i n i t e dimension h t 1 i f f TI i s o f f i n i t e dimension
h t 1.
Next, we prove: Proposition
IX. Any T i n
T i s a 3-dimensional p r o j e c t i v e space. Furthermore,
the c o l l e c t i o n S o f a l l elements i n S through a p o i n t p i n G i s a p r o j e c t P i v e plane, which i s a subspace o f (S, R ) .
Proof. I f T Assume T #
i s the space
r, and
7
l e t q1
i n A3,
then T i s a 3-dimensional p r o j e c t i v e space.
-
be a p o i n t i n T
and q
2 (G, F ) i s connected, t h e r e e x i s t both a f i n i t e subset
p o i n t s i n G and a f i n i t e subset P1 = 41 3 Pn = q2 and l i n e fi
fi
3
{fl,
{pi,
..., f n - l 1
i = 1, 2,
pitll,
a p o i n t i n T. Ip
l,..., p n l
o f lines i n
. . ., n - 1.
have t h e common p o i n t p1 = 9,;
since
T
of
F such t h a t Through any
there i s e x a c t l y one T . i n T (see A211). The subspaces 1
Since
and T1
has f i n i t e dimension equal t o three,
V I I I , T1 i s a 3-dimensional p r o j e c t i v e space. Since TinTitl 2 [pitll ( i = 1, ...,n - 2 ) and Tn-l n T2 {pnl, a l l Tiis ( i = 1, n-1) and T
by prop.
...,
are 3-dimensional p r o j e c t i v e spaces. F i n a l l y , t o prove t h a t S i s a p r o j e c t i v e plane, i t i s enough t o r e c a l l P t h a t t h e r e e x i s t s an element T i n T through p (through p t h e r e i s a t l e a s t one l i n e f i n F which i s contained i n a maximal subspace i n T); T i s a 3-dimensional p r o j e c t i v e space and from prop. V I I the statement follows.
138 5.
A . Bichara and G. TaNini
THE SUBSPACES OF (S, R ) F i r s t we prove: Given t h r e e p a i r w i s e c o l l i n e a r p o i n t s i n (S,R), Sly
P r o p o s i t i o n X.
S2, and
S3, t hro ugh t h e same p o i n t p i n G (when c o n sidered as subspaces o f ( F , G ) ) ,
t h e r e e x i s t s a p r o j e c t i v e p l a n e i n (S, R ) through them. Proof. -
By prop. I X , S
and S3.
P
i s a p r o j e c t i v e p l a ne i n (S, R), t hrough Sly
S2,
Given t h r e e independent p a i r w i s e c o l l i n e a r p o i n t s i n (S,
Proposition X I .
n o t t hro ugh t h e same p o i n t i n G (when c o n s i dered as subspaces o f
R),
(G, F ) ) ,
t h e r e e x i s t s , i n (S, R ) , a p r o j e c t i v e p l a n e t hrough them.
P roof .
Under t h e assumptions, t h e t h r e e p o i n t s (which o b v i o u s l y e x i s t )
a r e p a i r w i s e d i s t i n c t . (Indeed,
p1 = p2 would i m p l y p1 E S1
and p1 E S2,
= p ) . Thus, t h e t h r e e l i n e s (pi,p.) i # j, i,j = 1,2,3, J 2 3 i n F are p a i r w i s e d i s t i n c t (otherwise, S , and S 3 would c o n t a i n t h e l i n e 1 ' s2 (pl,p2) = (pl,p3), which c o n t r a d i c t s A 1 1 ) . Consequently, t h e r e e x i s t s a 2 subspace i n (G, F ) c o n t a i n i n g ply p2, and p3 (see A1); t h i s subspace i s a
and t h en
p1 = p
member a o f subspace
T
n.
( S i n c e i t meets S1 a t a l i n e , i t i s cont ained i n a maximal
E T and i s t h e p r o j e c t i v e p l a n e spanned i n
Consider t h e f o l l o w i n g subset o f
:
ab t u r n s o u t t o be a p r o j e c t i v e p l a n e . I n f a c t , i f S i
elements i n lines
~ l h ,t h e n
fl and f
T b y pl, p2, and p3).
and S;
a r e two d i s t i n c t
t h e y meet t h e p r o j e c t i v e p l a n e a a t two ( d i s t i n c t )
o f F and f, and f 2 meet a t a p o i n t p i n a ; thus, S i and S; 2 belong t o t h e l i n e r i n R. L e t r and r , p # p ' , be two Pya Pya P' ,a
The Grassmann manifold representing the planes in a projective space ( d i s t i n c t ) l i n e s i n a s . Since t h e l i n e (p,p')
in
F on
CY
139
e x i s t s , the maximal
subspace S E S through i t i s the o n l y element belonging t o
r P,"
The statement f o l l o w s .
nr PI,"'
As a c o r o l l a r y t o prop. X and X I : Proposition X I I .
Any subspace o f (S, R ) i s a p r o j e c t i v e space.
6. THE COLLECTION
P
OF MAXIMAL SUBSPACES I N (S, R)
L e t p be a p o i n t i n G and S
Proposition X I I I .
(1
F ) i s a PPLS, such a subspace e x i s t s ) . p and S n S 0) i s e i t h e r a l i n e i n R
(E G) and n o t through p (since (G,
Then, the set
{S
L
E S: S 3
an element o f S through q
9 o r t h e empty set. By prop. I X , the elements i n S through p form a p r o j e c t i v e
R); thus, t h e r e i s always some element i n S through p which i s skew w i t h S * consequently, (S, R) i s a proper p a r t i a l space (see V I ) .
plane i n (S,
(1'
Proof.
Assume t h a t , through p, t h e r e are two d i s t i n c t elements o f
S' and S", both meeting S a t a p o i n t : q
S' n S = q ' , S" n S
q q ' # 9". (Indeed, q ' = q" i m p l i e s t h a t through t h e l i n e (p,q') there are two d i s t i n c t elements of
q
= 9".
say
S,
Then,
= (p,q")
S, a c o n t r a d i c t i o n ) . Since the three
p o i n t s p, q ' , and q" a r e pairwise c o l l i n e a r , there e x i s t s a subspace
a
in
n
through them. The l i n e r i n (S, R) consists o f those elements i n S P ,a through p, meeting S a t a p o i n t on (q',q"); moreover, any p o i n t on (q',q'') q i s j o i n e d t o p by a l i n e belonging t o some element i n r PP' Now, assume S"' i s an element i n S through p, n e i t h e r skew w i t h S q' nor belonging t o r ; then, S"' meets S a t a p o i n t q"' n o t on (q',q"). P ,a q Since the p o i n t s q', q"', and p a r e pairwise c a l l i n e a r , they are contained i n a plane a'
of
n,
and q"' E (q',q"), (p,Ci').
which i s d i s t i n c t from CY (otherwise, a n S a c o n t r a d i c t i o n ) ; these two planes i n
IT
= a' n S q 9 meet a t the l i n e
L e t T and 1' be the maximal subspace i n T through a and a ' ,
respectively. Then,
T
= T' (by prop.
contains the l i n e (p,q')
I , t a k i n g i n t o account t h a t T n T '
= a n a'). The three p o i n t s q ' , q", and q"' belong
140
A . Bichara and G. Tallini
t o T nS thus, the p r o j e c t i v e plane through them i n T i s completely contained q' i n S , which i s impossible (see A21). q F i n a l l y , i t must be proved t h a t i f an element S i n S through p and i n P c i d e n t w i t h S e x i s t s , then there e x i s t s a l i n e r , any element o f which q P ,a i s an element o f S through p and i n c i d e n t w i t h S Set S n S 2 {q'}; q P qthrough the l i n e (p,q') t h e r e i s a maximal subspace T o f T, meeting S a t a q l i n e through q ' , a l l whose p o i n t s are c o l l i n e a r w i t h p ; and t h i s l i n e and
.
p span a plane belonging t o Prbposition X I V .
Let S
P
n.
The statement f o l l o w s ,
be the s e t o f a l l elements i n S through a p o i n t p
i n G. I f S belongs t o S and doesn't pass through p, then the subset
(S'
'L
S
i n (S, R ) means S' i s i n c i d e n t w i t h S i n (G, F ) ) o f S
empty o r a l i n e i n R . Furthermore, S P
Proof.
P i s a maximal subspace i n
i s either (S, R ) .
The f i r s t p a r t o f the statement f o l l o w s from prop. X I I I . Again by
i s a subspace i n (S, R ) , containing some element which i s P non-collinear w i t h S. Therefore, no subspace containing both S and S e x i s t s P i n (S, R ) and S i s a maximal subspace.
prop. X I I I , S
P
Thus, the c o l l e c t i o n P = {S : p E GI o f maximal subspace i n (S, R ) , P any o f them being a p r o j e c t i v e plane (see prop. I X ) , arises. Proposition XV.
The c o l l e c t i o n
P = ISp : p E GI i s proper.
Furthermore,
any two d i s t i n c t elements i n P share a t most one p o i n t o f S.
Proof.
L e t p and q be any two d i s t i n c t p o i n t s i n G. The maximal subspace
and S i n (S, R ) share a l l the elements i n S through both p and q. I f p P q and q are non-collinear i n (G, F), then there i s no element o f S through
S
them. I f p and q are j o i n e d by a l i n e f
E
F, then ( i n ( G , F ) )
there exists
e x a c t l y one maximal subspace belonging t o S which passes through f and so through p and q. I n both cases,
S n S G 1, and the statement follows. I P 91
141
The Grassmann manifold representing the planes in a projective space
7. THE COLLECTION E OF MAXIFlAL SUBSPACES I N (S, l?) F i r s t we prove: Proposition X V I .
L e t S and S ' be two d i s t i n c t elements i n S meeting a t a
p o i n t p i n G. I f S1 and S2 are d i s t i n c t elements i n S both meeting S and S ' a t d i s t i n c t p o i n t s i n G, then S1 and S E S belonging t o the l i n e
3
s2
are c o l l i n e a r i n (S, R ) and any element
(S1, S2) i n R e i t h e r meets both S and S ' a t
d i s t i n c t p o i n t s of G o r belongs t o the l i n e ( S ,
Proof.
S') i n R.
From A I 1 1 i t follows t h a t S1 and S2 meet: I q l = S1 n S2, and thus
2
they a r e c o l l i n e a r i n (S, R ) (see prop. V I ) . Since n e i t h e r S1 nor S p, then q # p. I f q E S, any element
s
contains
2 # S on the l i n e (S1,S2) i n l?, passing
through q, meets S a t t h a t p o i n t . I f q 4 S, t h e s e t o f a l l elements i n S through q and i n c i d e n t w i t h S i s e i t h e r a l i n e i n R o r t h e empty s e t (see prop. XIV); since both S, and S2 pass through q and are i n c i d e n t w i t h S, any element on the l i n e (S1, S 2 ) i n R i s i n c i d e n t w i t h S. By the same argument, any element on (S1, S2) i s proved t o be i n c i d e n t w i t h S ' ; thus, i t i s o n l y
S2) passing through p, then i t belongs t o the l i n e (S, S ' ) . Under these assumptions, S3 contains both p and { S t c E S: S" 3 p q, and, thus, meets S1 a t q. Hence, S3 E L ' , where L ' and S" n S1 # 01; a l s o S and S ' belong t o L ' (by hypothesis). By prop. X I V , S3, S and S' belong t o a l i n e i n R ; so, S3 E (S,S') and the statement i s t o be proved t h a t i f S3 i s an element on (Sly
proved. L e t S and S ' be two d i s t i n c t elements i n S, which are c o l l i n e a r i n (S, R ) ; then S and
S' share a p o i n t p i n G. Denote by o(S,S') t h e s e t con-
s i s t i n g o f a l l elements i n S t h a t e i t h e r belong t o the l i n e (S,S')
i n R,
o r meet both S and S ' a t p o i n t s i n G d i s t i n c t from p. Proposition X V I I .
I f S and S ' a r e d i s t i n c t elements i n S which are c o l l i n e a r
i n (S, R ) , then the s e t
o(S, S ' ) i s a subspace i n (S, R ) and i t properly
contains the l i n e (S, S') i n R. Proof
.
Let T be a member o f T through the p o i n t p, where { p l = S n S';
5y
142
A. Bichara and G. Tallini
A21, T meets S and S' a t the lines f and f ' in F , respectively. I f q
E
f,
q ' E f ' , q , q ' # p , then q # q ' and q and q ' belong to a l i n e f " i n T. Through
f " there i s an element S" E S meeting S and S' a t q and q ' , respectively. Thus, S" belongs t o a(S,S') \ (S,S'); therefore, the l i n e (S,S') i s properly contained i n G(S,S'). To prove t h a t a(S,S')
points S l y S2
E
o(S,S')
i s a subspace in (S, R ) ( i . e . any two d i s t i n c t
are collinear in (S, R ) and the l i n e joining them
i s completely contained in o(S,Sl)), three cases will be considered. ( i ) If S1 and S2 both belong t o the l i n e (S,S'), then there i s nothing
t o prove. ( i i ) If
f1 and
S2 meet S and
S' a t d i s t i n c t points, then the statement
follows from prop. XVI. ( i i i ) If S1 meets both S and S' a t d i s t i n c t points and S2 belongs t o the line (S,S'), then
-
by an argument simular t o t h a t in prop. XVI
- it
is
easy to prove t h a t S and S are collinear and any element on ( S l y S2) belongs 1 2 to o(S,S'). Proposition XVIII.
Let S and S' be two d i s t i n c t elements i n S which are
collinear i n (S, R ) , i . e . they meet a t a point p i n G. Then there e x i s t exactly two maximal subspaces o f (S, R ) t h r o u g h S and
s';
the f i r s t one i s S ( i . e . P
i t consists o f a l l elements i n S through p ) and belongs t o P; the second one i s a(S,S'). Proof.
These two subspaces share exactly the elements on the l i n e ( S , S l ) .
Let H be a subspace of (S, R ) containing both S and S' and so the
l i n e (S, S l ) in R. Then: e i t h e r H consists of elements in S a l l of them through p and H i s a subspace of S , o r there exists some S" E S contained i n H and P not passing through p. Since H i s a subspace and S", S, and S' belong t o H,
S" i s collinear w i t h b o t h S and S ' , i . e . meets S and S' a t d i s t i n c t points. By prop. XIV, the elements in S which are collinear with S" a r e exactly the
P
elements on the line ( S , S l ) i n R; hence, H n S = (S, S'). Therefore, any P element i n H (S, S'), being collinear w i t h both S and S', meets S and S' a t points i n G d i s t i n c t from p (and from each other). Thus H C o(S, S') and the statement follows. Proposition XIX.
Let S and S' be two d i s t i n c t elements i n S, which are col-
143
The Grassmann manifold representing the planes in a projective space l i n e a r i n (S, R ) . IfS1 and S 2 are d i s t i n c t elements i n u(S,S'), then they meet and u(S1, S 2 ) = u ( S , S'). Proof. Since S, and S2 belong t o the subspace and thus have a common p o i n t
(I E
u ( S , S'), they a r e c o l l i n e a r
G. By prop. X V I I I , the subspace
u(S, S ' ) ,
containing S1 and S2, i s contained i n e i t h e r the maximal subspace S , o r the q maximal subspace u(S1, S2). Since the elements i n u ( S , S l ) d o n ' t a l l pass through a same p o i n t ,
u(S, S ' )
5 u(S1,
S 2 ) . E q u a l i t y f o l l o w s from
a(S, S ' )
being a maximal subspace i n (S, R ) .
8.
FURTHER PROPERTIES OF
P AND C
I n the PLS (S, R ) two c o l l e c t i o n o f maximal subspaces have been defined (see prop. X I V and X V I I I ) :
P = IS : p P
E
GI, a proper c o l l e c t i o n ;
X = {u(S, S') : S, S' E S , S # S', S n S' # 01. Proposition XX.
For any r i n R , t h e r e e x i s t a unique S
u i n E such t h a t r C S and r 5 u.
P
i n P and a unique
P
Proof.
Since any two d i s t i n c t elements on r a r e i n c i d e n t , t h e statement
f o l l o w s from prop. X V I I I and X I X . Proposition X X I . or a l i n e i n R.
If S E P
P and u
E
X, then S n u i s e i t h e r t h e empty s e t
Proof, -
The Set S n u consists o f a l l elements i n S through p which are P c o l l i n e a r w i t h any element i n u. Take S E S n u and l e t S' be any other P element i n a; then (see prop. X I X ) u = u(S, S') and S E S I f {p) = S n S ' ,
P'
then the statement f o l l o w s from prop. X V I I I . I f p 4 S ' , then the s e t o f a l l elements i n S
- by prop.
XIV
-
i n c i d e n t w i t h S ' i s a l i n e r i n R , containing P S. L e t S" be a p o i n t on r, d i s t i n c t from S. Since S" meets S and S ' a t d i s t i n c t
144
A . Bichara and G. Tallini
p o i n t s , S " E u ( S , S'); t h e r e f o r e ,
( b y prop. XIX) u(S, S ' ) = u ( S , S " ) and t h e
p r e v i o u s argument proves t h e statement. Any two d i s t i n c t elements i n
Proposition X X I I .
c s h a r e a t most one p o i n t o f
S. I f u , u ' and u" a r e t h r e e p a i r w i s e d i s t i n c t elements i n l u ' n u''1 = 1, t h e n Iu n u"I
=
1. Consequently, i f {uj : j
i s a f i n i t e sequence of elements i n
1 t h e n Iu n u o ntl Proof.
>
c
such t h a t Iui n uitlI
and Iu n u ' l = 1,
E
{O,l,...,ntl}}
2 1 (i=0,
...,n ) ,
1.
From prop. X V I I I i t f o l l o w s t h a t any two d i s t i n c t elements i n Z share
a t most one p o i n t i n S. l l r i t e I S ' ) = u n u ' and IS") = u ' n u". I f S ' = S", t h e n t h e s t a t e m e n t i s o b v i o u s l y t r u e . I f S' # S " , t h e n S' and S " , b e l o n g i n g t o t h e subspace u ' , a r e c o l l i n e a r i n (S, R ) and l e t r be t h e l i n e j o i n i n g i n P t h r o u g h r, which P meets u ' a t t h e l i n e r ( s e e prop. XXI) and u a t a l i n e s' t h r o u g h S ' and u" them. By p r o p . XX, t h e r e e x i s t s a maximal subspace S
a t a l i n e s" t h r o u g h S". The l i n e s s' and s" i n R a r e d i s t i n c t ( o t h e r w i s e , b y prop. XX, u = u " , w h i c h i s i m p o s s i b l e ) and b e l o n g t o S , a p r o j e c t i v e p l a n e P ( b y prop. I X ) . I t f o l l o w s t h a t t h e r e e x i s t s e x a c t l y one element S i n S bel o n g i n g t o s' n so'; c l e a r l y , S belongs t o u ( a s S
u" (as S
E
s" and s" 5
0");
an element i n u Proof.
i
Let u
1
s' and s' C u ) and t o
thus, IS} C u n u". S i n c e u # u'', b y t h e f i r s t
p a r t o f t h e statement, IS1 = u n u"; t h e r e f o r e , Proposition X X I I I .
€
ICJ n u"1
= 1.
and u2 be two d i s t i n c t elements i n C .
( i = 1, 2 ) , t h e n I S 1 n S21
>
I f Si
1 i m p l i e s (ul n u,l I
>
is
1.
I f S1 = S2, t h e s t a t e m e n t i s o b v i o u s l y t r u e . Assume S1 # S2; t h e n
S1 n S2 c o n s i s t s o f a p o i n t p i n G. L e t fl be a l i n e o f
F in
S1 t h r o u g h p
t h e r e i s e x a c t l y one T E T which 1 meets S2 a t a l i n e f 2 o f F. T i s a 3-dimensional p r o j e c t i v e space ( s e e p r o p . (and such a l i n e does e x i s t ) ; t h r o u g h f
I X ) ; thus, t h e r e e x i s t s a l i n e f i n T t h r o u g h p d i s t i n c t f r o m fl and f2. Through f t h e r e i s an element S i n S , w h i c h i s c l e a r l y d i s t i n c t f r o m S1 and S2. Therefore, t h e maximal subspace
floreover, ul n u(S1, S)
3
u(S
S ) and i' I S 1 l , u(S1,S) n u(S,S2)
From p r o p . X X I I t h e s t a t e m e n t f o l l o w s .
u(S, S 2 ) i n Z e x i s t .
2
I S } , and u(S,S2) n u2 ?IS2}.
145
The Grassmann manifold representing the planes in a projective space Proposition X X I V .
and u ' i n X share e x a c t l y one
Two d i s t i n c t elements u
p o i n t i n S.
Proof,
By prop. X X I I , i t i s enough t o prove t h a t there e x i s t s a c o l l e c t i o n
,...
,...
: j E t0 , n t l I l o f elements i n Z such t h a t lul nu. I 1 (i=O ,n) j 1t 1 and uo = u, untl = u'. Take S E u and S ' E u'. Assume S # S' (otherwise t h e
{u
q2 a p o i n t i n S ' , w i t h
statement i n t r i v i a l ) and l e t q1 be a p o i n t i n S,
F ) i s connected, t h e r e e x i s t a f i n i t e subset o f G, {p1,...,pnt21, and a f i n i t e subset o f F, Ifl ,. . ,fntl I, such t h a t q1 = p1 , phtll (h = 1, ...,n t l ) . Through each l i n e fh i n F q2 = pnt2 and fh 1 {p,, q1 # q2.
Since (G,
.
there i s a maximal subspace Sh i n S (see A211). The c o l l e c t i o n {S1,...,S
ntl e i t h e r are equal o r have
consists o f elements i n S such t h a t Sk and S ktl' the maximal subspace uk = u(Sk,Sktl) t i o n {ak : k = l,...,nI,
>
lat n
1
(t
>
ktl i t may be assumed Sk # Sktl.
H.1.o.g.
e x a c t l y one common p o i n t p
at n u
1,
3 tS I ttl
ttl
l,...,n-1).
(k
Writing u
...,n)
..., n-1);
u and u
=
> 1. Since
Then,
i n I: e x i s t . I n the c o l l e c -
( t = 1, 0
1
ntl
hence,
= u', l e t ' s prove
and S n S 2 E u 1 1 1and q2 E S', {qll, by prop. X I I I , Iuo n ulI > 1. Since q2 E Sntl 'ntl 'n and S' E u', lan n u') > 1 and t h e statement i s proved.
t h a t luo n ull
9.
1 and lan n untlI
S E uo, S
THE PROOFS OF THEOREH 1 AN0 2 By the previous sections, t h e space (G,
F ) i s associated v i t h t h e PPLS
(S, R) (see prop. X I I I ) s a t i s f y i n g :
Given t h r e e pairwise c o l l i n e a r points, t h e r e e x i s t s a subspace conA;) t a i n i n g them (see prop. X and XI).
No l i n e i s a maximal subspace; moreover, t h e r e e x i s t two c o l l e c t i o n s , say c and P, o f maximal subspaces i n (S, R) such t h a t an:' maximal subspace belongs e i t h e r t o C o r t o P (see prop. X I V , X V I I I and XX) and A')
(I) (11) (1111
a,
U'
E
c,
u # u'
u E Z, n E P
v
r E R
=s>
* d!
u
*
l o n o ' \ = 1 (see prop. X X I V ) ;
either E
C,
u n n = 0 or
j! n
E
P :
r
5
u n n E R (see prop. X X I ) ;
u, r
5n
(see prop. X X ) .
Therefore (see [ 31 ), ts being the s e t o f a l l elements i n C through S E S
Its( > 2 ; furthermore, I: =
I t s : S E S I i s a proper c o l l e c t i o n and the p a i r
,
A . Bichara and C. Tallini
146
i s a p r o j e c t i v e space. The mapping p: L
(C,L)
+
S, defined by
p ( l l s ) = S, i s
one-to-one and onto and maps p e n c i l s o f l i n e s i n L onto elements o f R and i t s -1 inverse mapping p maps l i n e s i n R onto p e n c i l s o f l i n e s i n L. F i n a l l y , P maps r u l e d planes onto elements o f P and p-’ maps elements i n P onto r u l e d planes. be an element i n et be the c o l l e c t i o n o f a l l planes i n L). L e t -P andLLTI the s e t o f a l l l i n e s i n then p(LTI) i s an element S i n P. Consider
: P’+
F
the mapping
(C,
TI
IT;
P
G, defined by
I t i s easy t o check t h a t :
Proposition XXV.
F i s one-to-one and onto.
Next, L e t L be a l i n e i n L. The image under F o f t h e s e t o f a l l
Proposition XXVI. planes i n
7 through L
i s the element S = p ( L ) i n S. The image under F - l o f an
element S i n S i s the s e t o f a l l planes i n
7 through
the l i n e
L
p-l(S) i n
L. Therefore, S i s the c o l l e c t i o n o f the images under F o f t h e sets o f a l l
planes i n (C,
Proof. L c
TI
If Q
L ) through any l i n e i n L.
7 and
IT E
$? E
p(LTI) = S
L T I ep ( L )
B
then P’ p(LTI) Q S
- by E
S
P
- F(n)
(9.1) Q
p
E
= p.
Moreover,
S, and the statement follows.
Two d i s t i n c t elements p1 and p2 i n G are c o l l i n e a r i n
Proposition X X V I I .
(G, F ) i f f they are t h e images under F o f two planes i n (C,
L)
which meet
a t a l i n e o f L. Proof.
Set ri
F
-1
(p.), i .1
1,2.
By prop. X X V I , F(ni) E S , i = 1,2;
t h e subspace S o f (G,
I f rl n n
2
i s a l i n e L i n L, s e t S = ~ ( l ) .
hence, the p o i n t s p1 and p2, belonging t o
F), a r e c o l l i n e a r i n (G, F).
Conversely, i f p1 and p2 a r e j o i n e d by the l i n e f i n
F , l e t S be the
The Grassmann manifold representing the planes in a projective space
maximal subspace i n S through f.
Obviously, pl,
147
p, E S; hence n 1 and n 2
meet a t the l i n e p - l ( S ) i n L and the statement follows. Proposition X X V I I I .
If
7
7 which
i s the s e t o f a l l t h e planes i n
a 3-dimensional subspace i n (C, L ) , then the image under F o f
7
belong t o
belongs t o T .
Furthermore, the inverse image under F o f an element i n T i s t h e s e t o f a l l the planes i n a 3 - f l a t o f
Proof.
First it w i l
be proved t h a t i f T
F),
E
T, then F-’(T) c o n s i s t s o f a c o l -
contained i n the same 3 - f l a t i n (1, L ) . Since T i s a
l e c t i o n o f planes a1 subspace of (G,
(c, L ) .
t s p o i n t s a r e pairwise c o l l i n e a r ; hence, F-’(T)
o f planes p a i r w i s e meeting a t l i n e s i n L (see prop. X X V I I ) .
consists
Thus, F-’(T) con-
s i s t s o f e i t h e r planes i n a 3 - f l a t o f (I, L ) , o r planes through a l i n e . L e t ’ s prove t h e l a t t e r case i s impossible. L e t
pl,
-
p,
and p be t h r e e independent 3 by A2 T c a n ’ t be a l i n e ) . The
-
p o i n t s i n T (such p o i n t s do e x i s t because -1 -1 planes F (p,) = n1 and F (p,) = n, meet a t a l i n e L i n L. F-l(p3) meets
IT^
The plane r3=
and IT, a t l i n e s d i s t i n c t from L and from each other. (Indeed,
i f n l s IT,, and n3 passed through L , then t h e t h r e e p o i n t s pl, would belong t o the element
p ( L ) = S i n S, which would share w i t h T t h e p o i n t s
on the l i n e s j o i n i n g a l l t h e p a i r s o f p o i n t s pi, this i s
impossible (see A,I).)
the 3 - f l a t j o i n i n g nl and n,;
p2, and p3
pj,
i # j, i,j
1 , 2 , 3 , and
Thus, F - l ( T ) consists o f a l l t h e planes i n t h i s space w i l l be denoted by ( n l ,
IT,). Let n
i = 1,2,3. Then n n n = L i is i is = 1,2,3, are t h r e e l i n e s i n L, two of them a t l e a s t being d i s t i n c t (as nl, IT, be a plane i n (nl, n,),
d i s t i n c t from n
and n3 d o n ’ t form a p e n c i l ) . Assume Ll # 1, (a s i m i l a r argument holds i n t h e other cases).
The t h r e e p o i n t s p = F(n), p1 and p2 a r e p a i r w i s e c o l l i n e a r
(as IT, nl and n, p a i r w i s e meet a t l i n e s i n L). subspace i n (G,
F) containing p, pl, and p.,
Thus, by A1, t h e r e e x i s t s a
L e t H be a maximal subspace i n
(G, F ) through these points. H cannot belong t o S, because n = F-’(p), -1 -1 and n, = F (p,) d o n ’ t pass through t h e same l i n e (see prop. n1 = F (p,)
XXVI).
Hence, H i s an element i n T through p1 and p, and so H = T (T being
the o n l y maximal subspace i n T through the l i n e (pl,
p,))
and p E T.
Thus, the image under F o f any plane i n IT^, IT,) i s an element i n T, -1 and conversely, i.e. F (T) i s t h e s e t o f a l l planes i n (nl, n,).
148
A . Bichara and G. Tallini
planes i n
By a s i m i l a r argument, the image under F o f t h e set o f a l l t h e
a 3-flat o f
(c, L) i s proved t o be an element T i n T, and t h e p r o o f i s complete.
Proposition X X I X .
Three planes
IT^, IT2 , and IT3 i n
(C,
L ) form a p e n c i l ( i . e .
they belong t o the same 3 - f l a t and have a common l i n e ) i f f t h e i r images under p2, and p3’ are t h r e e p o i n t s i n G which belong t o the same l i n e i n F .
F, pl,
IT^, and IT3 form a p e n c i l , then they pass through a l i n e .t i n L and belong t o t h e 3 - f l a t (v1, IT^) spanned by v and IT Set S = P ( L ) and 1 2‘ Proof.
If v1 ,
l e t T be t h e image under F o f the s e t o f a l l the planes i n X X V I and X X V I I I , {p1,p2,p31
5
S, {p1,p2,p3)
e i t h e r the empty s e t o r a l i n e i n
5
T.
IT^).
(n1,
Since (by A21)
By prop.
T nS is
F , t h e p o i n t s ply p2, and p3 belong t o a
line. Conservely, i f
ply
p2 and p3 belong t o t h e l i n e f i n
T E T be the two maximal subspaces i n (G,
IT^,
n2, and
IT^
pass through the l i n e p-’(S)
a l l the planes i n a 3 - f l a t i n
(c, L).
F, l e t S
E
S and
F ) through f. The t h r e e planes and belong t o t h e s e t F-’(T) o f
Therefore,
IT,,
n2 and
and the statement i s proved.
IT
3
form a p e n c i l
From the r e s u l t s i n t h i s section theorem 1 f o l l o w s . F i n a l l y , i f (G, F ) i s a non-connected PPLS s a t i s f y i n g A1 and A2 and G P i s the connected component o f p (p E G) ( i . e . the s e t o f a l l t h e p o i n t s t h a t
F), then
can be reached from p by a polygonal path c o n s i s t i n g o f l i n e s i n
F containing a p o i n t i n G i s completely contained i n G and any P P subspace i n (G, F ) which i s n o t skew w i t h G i s completely contained i n G P’ P Hence, denoting by F t h e s e t o f a l l t h e l i n e s i n F which are contained i n G P P’ the p a i r ( G F ) i s a connected PPLS s a t i s f y i n g axioms A1 and A,. Theorem 2
any l i n e i n
follows.
P’
P
ACKNOWLEDGEHENT.
L
This research was p a r t i a l l y supported by GNSAGA o f CNR.
The Grassmann manifold representing the planes in a projective space
149
REFERENCES
[ 11 [
B. Segre, Lectures on modern geometry, CremoneseEd. Roma (1961)
21 G. T a l l i n i , Spazi p a r z i a l i d i r e t t e , spazi p o l a r i , Geometrie subimmerse, Quaderni Sem. Geom. Combinatorie, 1 s t . Mat. Univ. Roma, n. 14 (gennaio 1979).
[3]
G. T a l l i n i , On a c h a r a c t e r i z a t i o n o f t h e Grassmann m a n i f o l d r e p r e s e n t i n g t h e l i n e s i n a p r o j e c t i v e space, i n : P.J. Cameron, J.!4.P. H i r s h f e l d , D.R. Hughes (eds.) F i n i t e Geometries and designs. London l l a t h . SOC. Lect. Notes S e r i e s n. 49. Cambridge U n i v e r s i t y Press (1981) 354-358.
ON A CHARACTERIZATION OF THE GRASSHANN MANIFOLD REPRESENTIRG THE PLANES I N A PROJECTIVE SPACE
Alessandro Bichara and Giuseppe Tal l i n i I s t i t u t o Matematico "G. Castelnuovo", U n i v e r s i t i i d i Roma, I t a l y 1.
INTRODUCTION The Grassmann manifold representing the planes i n a p r o j e c t i v e space w i l l
be characterized as a p a r t i a l l i n e space (G,
F ) whose maximal subspaces s a t i s f y
some s u i t a b l e conditions (see s e c t i o n 2 ) . The p r o o f uses the f o l l o w i n g : from
F ) i t i s possible t o construct - i n a natural way - another p a r t i a l l i n e space, say (S, R), which turns o u t t o be isomorphic t o the Grassmann manifold (G,
representing the l i n e s i n a p r o j e c t i v e space [ 3 1 . The r e s u l t s t h a t w i l l be proved here can be generalized t o t h e Grassmann manifold representing the d - f l a t s i n a p r o j e c t i v e space. The p r o o f o f such a general theorem, which goes by i n d u c t i o n ( s t a r t i n g from d = 2 , since the step from d
1 t o d = 2 i s q u i t e special), i s the subject o f another paper by the
authors.
2.
PRELIMINARIES L e t G be a non empty set, whose elements w i l l be c a l l e d p o i n t s , and
F a
(proper) non empty c o l l e c t i o n o f (proper) subsets o f G, which w i l l be c a l l e d
l i n e s . The p a i r (G,
F ) i s s a i d t o be a p a r t i a l l i n e space (PLS) i f the f o l l o w -
ing hold ( [ Z I ) : ( i ) Any two d i s t i n c t p o i n t s i n G belong t o a t most one l i n e i n F. ( i i ) Any l i n e i n F contains a t l e a s t two p o i n t s o f G. ( i i i ) F i s a covering o f G. Two d i s t i n c t p o i n t s p and p ' i n G w i l l be s a i d t o be c o l l i n e a r i f they belong 129
A . Bichara and G. Tallini
130
t o a l i n e , denoted by (p,p'),
i n F; by p
- p'
i t i s meant t h a t p and p '
(p# p ' ) are c o l l i n e a r ; otherwise, p and p ' are non-collinear: p f p ' . (G, F)
w i l l be c a l l e d a proper p a r t i a l l i n e space (PPLS) i f t h e r e
e x i s t two non-collinear p o i n t s i n G. A p o i n t subset H o f a PPLS (G, F) i s c a l l e d a subspace i f any two o f i t s p o i n t s a r e c o l l i n e a r and the l i n e j o i n i n g them i s completely contained i n H; a subspace H o f (G, F) i s a maximal subspace i f i t i s n o t p r o p e r l y contained i n any subspace o f (G, F ) . A PLS (G, F) w i l l be c a l l e d i r r e d u c i b l e i f t h e f o l l o w i n g holds: Any l i n e i n F c o n t a i n s a t l e a s t t h r e e p o i n t s o f G.
(ii')
F i n a l l y , a PPLS (G, F) i s s a i d t o be connected i f For any two d i s t i n c t p o i n t s p, p ' i n G t h e r e e x i s t s a polygonal
(iv)
path j o i n i n g them. L e t P(r,K)
(r
> 4)
be an r-dimensional p r o j e c t i v e space over a f i e l d K.
L e t G be t h e c o l l e c t i o n o f a l l t h e planes i n P(r,K) and F c o n s i s t s o f p e n c i l s o f planes i n P(r,K)
(a p e n c i l o f planes being the s e t o f a l l t h e planes
through a l i n e contained i n a 3 - f l a t P(3,K) i n P(r,K)),
Then (G,
F) i s a
proper i r r e d u c i b l e PLS which i s isomorphic t o Grassmann manifold G
rY2,K (As i t i s w e l l known, Gr,2,K i s an i r r e rtl ducible algebraic (3(r-2))-dimensional manifold i n an(( ) 1)-dimensional representing the planes i n P(r,K).
-
p r o j e c t i v e space; moreover, t h i s manifold i s an i n t e r s e c t i o n o f quadrics.) Any n o n - t r i v i a l ( i . e . p r o p e r l y containing some element o f F) subspace in
(G, F ) i s e i t h e r
( a ) the c o l l e c t i o n o f a l l the planes through a l i n e L belonging t o an h - f l a t (through L), h 3 4 (such a c o l l e c t i o n w i l l a l s o be c a l l e d an L - s t a r ) or (b) the c o l l e c t i o n o f a l l planes belonging t o a 3 - f l a t ; o r ( c ) the c o l l e c t i o n o f a l l the planes through a p o i n t i n a 3 - f l a t ;
such
a s e t o f planes w i l l be c a l l e d a s t a r o f planes. Therefore, a maximal subspace i n (G, F) i s e i t h e r an il-star, and the c o l l e c t i o n o f such maximal subspaces w i l l be denoted by S, o r the s e t o f a l l planes i n a 3 - f l a t i n P(r,K),
and the c o l l e c t i o n o f these maximal sub-
spaces w i l l be denoted by T. The PPLS (G, F) i s connected and s a t i s f i e s : Ale
If t h r e e p o i n t s i n G are pairwise c o l l i n e a r , then t h e r e e x i s t s a
The Grassmann manifold representing the planes in a projective space
131
subspace i n (G, F) through them.
No l i n e i n F i s a maximal subspace. Furthermore, t h e r e e x i s t two A2. c o l l e c t i o n s , say S and T, o f maximal subspaces i n (G, F ) and any maximal subspace belongs e i t h e r t o S o r t o T. Moreover: (1)
SES,TET*eitherSnT=fl,
(11)
VfEF*3!SES93!
(111)
I f S,
SO, S"
element i n S, d i s t i n c t from
E
TET:
or
SnTEF.
f c S , f C -T .
S p a i r w i s e meet a t d i s t i n c t p o i n t s , then any
S" and meeting both S and S' a t d i s t i n c t p o i n t s ,
meet S" too.
T
E
There e x i s t three subspaces f, n,
A3. T and
T
covers f and i s covered by
Axioms A1,
Remark 1.
7
i n (G, F) such t h a t f
E
F,
T.
A2, A3 and t h e connectedness hypothesis, besides being
s a t i s f i e d by Grassmann manifold representing t h e planes i n P(r,K),
are a l s o
s a t i s f i e d by the generalized Grassmann manifold representing the planes i n an e i t h e r i r r e d u c i b l e (and p o s s i b l y Pascalian) o r r e d u c i b l e p r o j e c t i v e space P even o f i n f i n i t e dimension. Remark 2.
The s e t t h e o r e t i c union o f Grassmann manifolds representing t h e
planes i n skew p r o j e c t i v e spaces i s a non-connected PLS s a t i s f y i n g axioms A1, A2, and A3. I n order t o characterize the (generalized) Grassmann manifold representi n g the planes i n one o r more p r o j e c t i v e spaces (see Remark 2 ) , PPLS's (G, F) s a t i s f y i n g A1, A2, and A
w i 11 be proved:
3
w i l l be studied. Namely, the f o l l o w i n g r e s u l t s
I f (G, F) i s a connected PPLS s a t i s f y i n g axioms A1, A2, and A3,
Theorem 1.
then t h e r e e x i s t a p r o j e c t i v e space (1, L ) and a mapping F from the c o l l e c t i o n %
P o f a l l planes i n (1, L ) i n t o G such t h a t : (i)
F i s one-to-one and onto;
( i i ) F consists o f e x a c t l y t h e images under F o f the p e n c i l s o f planes in
?; ( i i i ) S consists o f e x a c t l y the images under F o f L - s t a r s i n (1, L),
L E L; and ( i v ) T consists o f e x a c t l y t h e images under F o f the c o l l e c t i o n s o f planes
132 in
A . Bichara and G. Tallini
?,
each o f them being formed by a l l t h e planes i n a 3 - f l a t i n (G, F ) . Thus, (C,
L ) i s i r r e d u c i b l e i f f (G, F ) i s i r r e d u c i b l e . F i n a l l y ,
if
(I, L ) i s f i n i t e l y generated and Pascalian, then (G, F ) i s isomorphic t o Grassmann m a n i f o l d r e p r e s e n t i n g t h e planes i n (C, Theorem 2 .
L).
L e t (G, F ) be a PPLS s a t i s f y i n g axioms A
A2, and A3. Then each 1' connected component o f (G, F ) i s a PPLS s a t i s f y i n g axioms A1 and A2. I f f o r each connected component o f
(G, F ) A3 holds, then (G,
F) i s the set
t h e o r e t i c union o f g e n e r a l i z e d Grassmann m a n i f o l d s r e p r e s e n t i n g t h e planes i n p a i r w i s e skew p r o j e c t i v e spaces.
3. SOME PROPERTIES OF A CONNECTED PPLS Let
(G, F )
be a connected PPLS s a t i s f y i n g axioms A1, A 2 , and A
sect. 2 . Then: P r o p o s i t i o n I. The c o l l e c t i o n s S and T a r e skew. L e t M and
3
in
PI' ( w i t h M # M I )
be two maximal subspaces having two d i s t i n c t common p o i n t s p and p ' ; then
M and M ' belong t o d i f f e r e n t c o l l e c t i o n s o f maximal subspaces. Thus, two d i s t i n c t maximal subspaces belonging t o t h e same c o l l e c t i o n have a t most one common p o i n t . Proof.
I f a maximal subspace M" were contained i n S n T, then
M" n M"
MI';
a c o n t r a d i c t i o n t o A21, as MI' p r o p e r l y c o n t a i n s a l i n e . Since p and p ' belong t o M y they a r e c o l l i n e a r ; l e t f be t h e l i n e j o i n i n g them; o b v i o u s l y , f
5M
and f c
M' , t h e two maximal subspaces belong
t o d i f f e r e n t c o l l e c t i o n s ' a n d t h e statement i s proved. P r o p o s i t i o n 11.
L e t S and S ' be two d i s t i n c t maximal subspaces i n S, having
a common p o i n t p. I f T i s a maximal subspace i n T , meeting-S and S ' a t t h e l i n e s f and f ' , r e s p e c t i v e l y , then f and f ' meet a t p, so T passes through p. Proof.
f and f ' are d i s t i n c t l i n e s (otherwise, two d i s t i n c t maximal subspaces
133
The Grassrnann manifold representing the planes in a projective space S and S' i n S would meet a t t h e l i n e Since any l i n e i n
f = f ' , which i s impossible by prop. I ) .
F has a t l e a s t two points, t h e r e e x i s t p o i n t s p '
E f and
p" E f ' such t h a t p, p ' and p" are p a i f w i s e d i s t i n c t . These p o i n t s are p a i r wise c o l l i n e a r (indeed, p, p '
E
s,
p, p"
€
s',
and p ' , p"
T);
€
thus, by A1,
there e x i s t s a subspace H through them and N i s contained i n a maximal subspace M. I f 11 belonged t o
M
s,
then M = S (by prop. I, M n
2 Hence M
s i m i l a r l y , M = S ' ( s i n c e t1 n S '
= S);
t o S, then S = S' a c o n t r a d i c t i o n .
{p,p"l).
s ' 3 {p,p')
implies
Therefore, i f I1 belonged
$ S and, by A*,
H E T . Since !1
belongs t o T and contains the d i s t i n c t p o i n t s p ' and p" i n T E T , by prop. I,M = T. Moreover, p E M and 14 = T imply t o S; thus,
{p)
5T n S
f
and
p E T. The p o i n t p i n T belongs
p E f; s i m i l a r l y , p E f ' ; therefore, t h e
d i s t i n c t l i n e s f and f ' meet a t t h e p o i n t p and t h e statement i s proved. Proposition 111. Any T i n T i s a p r o j e c t i v e space. Proof. I t i s enough t o prove t h a t i n the l i n e space T Veblen-Nedderburn axiom holds : L e t fl and f
be two l i n e s i n T meeting a t the p o i n t p3; i f f3 and f4 2 are d i s t i n c t l i n e s i n T, each o f them meeting both fl and f2 a t p o i n t s d i s t i n c t from p3, then f3 and f4 meet a t a p o i n t . = f2 n f4; then
Through
..,4)
t h e r e i s e x a c t l y one maximal subspace S E S (see i A211). Such maximal subspaces a r e p a i r w i s e d i s t i n c t ( i f i # j and Si = sj, fi ( i = 1,.
then the maximal subspace Si would share w i t h T E T the p o i n t s i n fi which i s impossible by A21). Now, fi c Si,
i = 1,2
and {p
l 3
= fl
U
f
j'
n f2 imply
I p3 } -C 1S n S2; since S1 # S2, by prop. I, {p31 = S1 n S2. By the same argument, {pl} = S2 n Sg and {pz} = S1 n S3; moreover, S meets S1 and
4 S2 a t t h e p o i n t s q1 and q2, respectively. The t h r e e maximal subspaces S1, S2, and S pairwise meet a t d i s t i n c t p o i n t s (see (3.1)) and S4 meets S1 and S
2
3 a t d i s t i n c t points; therefore, by A2111, S4 meets S3 a t a p o i n t
134
A . Blchara and C. Tallini
q: {q} = S3 n S4. Since T meets l y , and S3 n
S4
S3 and S4 a t the l i n e s f3 and f 4' respective-
= {q}, by prop. 11, f 3 and f 4 meet a t the p o i n t q and the
statement i s proved. The p r o j e c t i v e space which a r e members o f T , c o n t a i n p r o j e c t i v e planes forming a c o l l e c t i o n 11 o f subsets o f G. Clearly, any element i n 11 i s a subspace i n (G,
F ) which i s contained i n a maximal subspace belonging t o
Proposition I V .
T.
L e t T and T ' be two elements i n T through a p o i n t p i n G.
, and S3 a r e three p a i r w i s e d i s t i n c t elements i n S through p 2 f { E F. I f t h e l i n e s fi, and such t h a t T n Si = fi E F , and T ' n S i i = 1,2,3, belong t o t h e same plane a i n T, then a l s o the l i n e s f; belong
Assume S1, S
t o a unique plane i n T I .
Proof.
I t i s enough the prove
L e t a' be the plane i n T' through f ' and f;. 1 t h a t f ' belongs t o a ' .
3
L e t f and f ' be two l i n e s n o t through p, the former i n a , t h e l a t t e r i n a ' . Since a and a' a r e p r o j e c t i v e planes, f meets fl,
meets
fi and f;.
Set {qi}
= f n fi,
i = 1,2,3,
I t i s easy t o check t h a t the f i v e p o i n t s qi,
and
'
f2 and f3, and f '
{q!} = f ' n f!, j = 1,2.
J
J
are pairwise d i s t i n c t ;
qj moreover, i f S and S' are the maximal subspaces i n
9 through
f and f ' ,
, S 2 , S 3 are p a i r w i s e d i s t i n c t 1 and {qi) = S n S i , Iq!) S' n Sj ' Since S1, S2, and S3 pairwise meet a t J d i s t i n c t p o i n t s and S' ( # S) meets S, and S2 a t d i s t i n c t p o i n t s , S and S' r e s p e c t i v e l y , then the f i v e subspaces S, S', S
have a common p o i n t q, which i s obviously d i s t i n c t from q1 and q ' Thus, 1' t h e three maximal subspaces S, S1 and S' pairwise meet a t d i s t i n c t p o i n t s
3 ( # S') meets S and S1 a t t h e d i s t i n c t p o i n t s q3 and p ( r e s p e c t i v e l y ) . Therefore, S3 and S' have a common p o i n t q ' . Since Sg n T ' = f i , S' n T ' = f '
and S
and I q ' ) = S3 n S', by prop. 11, f ' and f ' meet a t 4. Hence, f ' and f i a r e 3 coplanar. The plane through them contains f ' and t h e p o i n t p on f i ; thus, i t i s a'
and f i belongs t o a ' .
The Grassmann manifold representing the planes in a pmjective space
F)
4. THE PARTIAL LINE SPACE (S, R ) ASSOCIATED WITH (G, Take p E G, a E
n,
with p
135
o f S, c o n s i s t i n g P ,a o f those max'imal subspaces i n S meeting a a t l i n e s i n F through p, i . e .
r = {SES: P ,a i s uniquely defined. Proposition V. I f a , a'
(4.1
E
n
E
a; then the subset r
and S n a E F 1
S 3 p
and p
E
,
a , p ' E a ' , then
1
Proof. Since a i s a p r o j e c t i v e plane,
through t h e p o i n t p i n a there are a t
l e a s t two d i s t i n c t l i n e s fl and f 2 o f
F.
The maximal subspace S1 and S2 i n
r and now i t w i l l be shown P ,a t h a t S1 # S2. I f S1 = S2, then t h i s member o f S would share w i t h a maximal
S through fl and f 2 ( r e s p e c t i v e l y ) belong t o
subspace T i n T a s e t I containing the d i s t i n c t l i n e s
fl and f2, which
and S1 # S2; (4.1) f o l l o w s . i s impossible by A21. Therefore, S1, S2 E r P ,a L e t S1 and S2 be two d i s t i n c t elements i n r n r S1 and S2 p,a p',a'. meet a a t l i n e s i n F through p and a' a t l i n e s i n F through p'; thus p and p ' belong t o
S, n S2. By prop. I, p = p ' .
Mhen a
a ' , (4.2) i s obviously true. Assume a
#
a ' . L e t T and T ' be
the maximal subspaces i n T through a and a ' respectively; then (otherwise, S1 would meet T a t two l i n e s i n
T # T'
F through p, one belonging t o
a, t h e other t o a ' , which c o n t r a d i c t s A21) and T n T ' =
{PI.
Furthermore,
by prop. I V , any element Sg i n r , d i s t i n c t from S1 and S2, meets T' a t P ,a being coplanar w i t h S1 n T ' and S2 n T ' belongs t o a l i n e i n F, which
-
r
P,a"
hence, rp,a
5
-
rpl,al. By the same argument,
(4.2) follows.
r p',a'
C r
-
p,a
and
{r : a E ll , p ~ a o} f subsets o f S i s defined; P ,a since i t i s n o t a proper c o l l e c t i o n (see prop. V ) , l e t R be the proper Thus, the c o l l e c t i o n
c o l l e c t i o n associated w i t h it.
136
A . Bichara and G. Tallini
The pair (S, R ) is a PLS. Moreover, two d i s t i n c t elements
Proposition VI.
in S a r e collinear i n (S, R ) i f f they have a common point i n G. Proof. Let -
S and S ' be two d i s t i n c t elements in S. If S n S '
{p}, then l e t
f be a line in S through p; through f there i s a maximal subspace T E T meeting
s'
a t one point a t l e a s t ; thus, i t meets
s'
a t a line f '
F. In the
E
projective space T , the d i s t i n c t l i n e s f and f ' a r e joined by a plane therefore, S and S ' belong to r
.
CI E
II ;
P 9Q If S n S ' = 0, no element in R through S and S' exists.
Let f be a l i n e in S E S. Through f there i s a maximal subspace T E T . If p E f , l e t f ' be a l i n e in T through p , d i s t i n c t from f . Through f ' there i s a maximal subspace S'
E
S , which i s obviously d i s t i n c t from S and meets
S a t p ; thus, through S there i s an element of R (joining S and S'). Hence, R i s a cover of S. Moreover, f o r any r E R ,
I rl > 2
two elements i n R have a t most one common p o i n t
(s, R )
(see (4.1) and any (by (4.2 ); i t follows t h a t
i s a PLS.
Let p be a point in G . The collection S
of a l l elements n S through P p i s a subspace of (S, R ) . (Indeed, any two d i s t i n c t elements in S are P
collinear i n (S, R )
and the l i n e through them i s completely contained in S ) . P Let p be a point i n G and T an element i n T through p.
Proposition VII.
s in
(S, R ) i s isomorphic t o the s t a r F consisting P P ,T of the lines in F through p and belonging to the projective space T. Thus,
Then the subspace
s i s a projective space and i s of f i n i t e dimension h i f f T i s of f i n i t e P dimension
h
t
1.
Proof. -
Any element in S meets T a t a l i n e i n F (see A21). P P,T Let cp be the mapping defined by
q:SES+SnTEF Clearly,
cp
P,T'
i s one-to-one and onto (see A211). Moreover,
cp
-1
maps pencils
onto lines, i n R , belonging t o S (Indeed, any pencil i n F conP ,T P' P,T s i s t s of a l l lines in T through p , b e l o n g i n g t o a plane n; such a pencil in F
i s the image under
cp
of the l i n e r
).
P Now, i t will be proved t h a t cp maps lines in S onto pencils i n F P P,T' ,Q
137
The Grassmann manifold representing the planes in a projective space be a l i n e i n S and S and S ' two d i s t i n c t p o i n t s on it; they p,a' P meet T a t ( d i s t i n c t ) l i n e s f and f ' , which are j o i n e d by a plane a through Let r
= r (see prop. V); furthermore, v(rp,a) = v ( r p ) Psa' p,a ,,I i s the p e n c i l c o n s i s t i n g of the l i n e s i n F through p and belonging t o a . P,T It f o l l o w s t h a t cp i s an isomorphism between S and the s t a r o f l i n e s through P p i n t h e p r o j e c t i v e space T.
p i n T; then, r
From prop. Proposition
VII,
VIII.
i t f o l l o w s immediately:
L e t T and T ' be any two d i s t i n c t elements i n T through
a p o i n t p i n G. Then T i s o f f i n i t e dimension h t 1 i f f TI i s o f f i n i t e dimension
h t 1.
Next, we prove: Proposition
IX. Any T i n
T i s a 3-dimensional p r o j e c t i v e space. Furthermore,
the c o l l e c t i o n S o f a l l elements i n S through a p o i n t p i n G i s a p r o j e c t P i v e plane, which i s a subspace o f (S, R ) .
Proof. I f T Assume T #
i s the space
r, and
7
l e t q1
i n A3,
then T i s a 3-dimensional p r o j e c t i v e space.
-
be a p o i n t i n T
and q
2 (G, F ) i s connected, t h e r e e x i s t both a f i n i t e subset
p o i n t s i n G and a f i n i t e subset P1 = 41 3 Pn = q2 and l i n e fi
fi
3
{fl,
{pi,
..., f n - l 1
i = 1, 2,
pitll,
a p o i n t i n T. Ip
l,..., p n l
o f lines i n
. . ., n - 1.
have t h e common p o i n t p1 = 9,;
since
T
of
F such t h a t Through any
there i s e x a c t l y one T . i n T (see A211). The subspaces 1
Since
and T1
has f i n i t e dimension equal t o three,
V I I I , T1 i s a 3-dimensional p r o j e c t i v e space. Since TinTitl 2 [pitll ( i = 1, ...,n - 2 ) and Tn-l n T2 {pnl, a l l Tiis ( i = 1, n-1) and T
by prop.
...,
are 3-dimensional p r o j e c t i v e spaces. F i n a l l y , t o prove t h a t S i s a p r o j e c t i v e plane, i t i s enough t o r e c a l l P t h a t t h e r e e x i s t s an element T i n T through p (through p t h e r e i s a t l e a s t one l i n e f i n F which i s contained i n a maximal subspace i n T); T i s a 3-dimensional p r o j e c t i v e space and from prop. V I I the statement follows.
138 5.
A . Bichara and G. TaNini
THE SUBSPACES OF (S, R ) F i r s t we prove: Given t h r e e p a i r w i s e c o l l i n e a r p o i n t s i n (S,R), Sly
P r o p o s i t i o n X.
S2, and
S3, t hro ugh t h e same p o i n t p i n G (when c o n sidered as subspaces o f ( F , G ) ) ,
t h e r e e x i s t s a p r o j e c t i v e p l a n e i n (S, R ) through them. Proof. -
By prop. I X , S
and S3.
P
i s a p r o j e c t i v e p l a ne i n (S, R), t hrough Sly
S2,
Given t h r e e independent p a i r w i s e c o l l i n e a r p o i n t s i n (S,
Proposition X I .
n o t t hro ugh t h e same p o i n t i n G (when c o n s i dered as subspaces o f
R),
(G, F ) ) ,
t h e r e e x i s t s , i n (S, R ) , a p r o j e c t i v e p l a n e t hrough them.
P roof .
Under t h e assumptions, t h e t h r e e p o i n t s (which o b v i o u s l y e x i s t )
a r e p a i r w i s e d i s t i n c t . (Indeed,
p1 = p2 would i m p l y p1 E S1
and p1 E S2,
= p ) . Thus, t h e t h r e e l i n e s (pi,p.) i # j, i,j = 1,2,3, J 2 3 i n F are p a i r w i s e d i s t i n c t (otherwise, S , and S 3 would c o n t a i n t h e l i n e 1 ' s2 (pl,p2) = (pl,p3), which c o n t r a d i c t s A 1 1 ) . Consequently, t h e r e e x i s t s a 2 subspace i n (G, F ) c o n t a i n i n g ply p2, and p3 (see A1); t h i s subspace i s a
and t h en
p1 = p
member a o f subspace
T
n.
( S i n c e i t meets S1 a t a l i n e , i t i s cont ained i n a maximal
E T and i s t h e p r o j e c t i v e p l a n e spanned i n
Consider t h e f o l l o w i n g subset o f
:
ab t u r n s o u t t o be a p r o j e c t i v e p l a n e . I n f a c t , i f S i
elements i n lines
~ l h ,t h e n
fl and f
T b y pl, p2, and p3).
and S;
a r e two d i s t i n c t
t h e y meet t h e p r o j e c t i v e p l a n e a a t two ( d i s t i n c t )
o f F and f, and f 2 meet a t a p o i n t p i n a ; thus, S i and S; 2 belong t o t h e l i n e r i n R. L e t r and r , p # p ' , be two Pya Pya P' ,a
The Grassmann manifold representing the planes in a projective space ( d i s t i n c t ) l i n e s i n a s . Since t h e l i n e (p,p')
in
F on
CY
139
e x i s t s , the maximal
subspace S E S through i t i s the o n l y element belonging t o
r P,"
The statement f o l l o w s .
nr PI,"'
As a c o r o l l a r y t o prop. X and X I : Proposition X I I .
Any subspace o f (S, R ) i s a p r o j e c t i v e space.
6. THE COLLECTION
P
OF MAXIMAL SUBSPACES I N (S, R)
L e t p be a p o i n t i n G and S
Proposition X I I I .
(1
F ) i s a PPLS, such a subspace e x i s t s ) . p and S n S 0) i s e i t h e r a l i n e i n R
(E G) and n o t through p (since (G,
Then, the set
{S
L
E S: S 3
an element o f S through q
9 o r t h e empty set. By prop. I X , the elements i n S through p form a p r o j e c t i v e
R); thus, t h e r e i s always some element i n S through p which i s skew w i t h S * consequently, (S, R) i s a proper p a r t i a l space (see V I ) .
plane i n (S,
(1'
Proof.
Assume t h a t , through p, t h e r e are two d i s t i n c t elements o f
S' and S", both meeting S a t a p o i n t : q
S' n S = q ' , S" n S
q q ' # 9". (Indeed, q ' = q" i m p l i e s t h a t through t h e l i n e (p,q') there are two d i s t i n c t elements of
q
= 9".
say
S,
Then,
= (p,q")
S, a c o n t r a d i c t i o n ) . Since the three
p o i n t s p, q ' , and q" a r e pairwise c o l l i n e a r , there e x i s t s a subspace
a
in
n
through them. The l i n e r i n (S, R) consists o f those elements i n S P ,a through p, meeting S a t a p o i n t on (q',q"); moreover, any p o i n t on (q',q'') q i s j o i n e d t o p by a l i n e belonging t o some element i n r PP' Now, assume S"' i s an element i n S through p, n e i t h e r skew w i t h S q' nor belonging t o r ; then, S"' meets S a t a p o i n t q"' n o t on (q',q"). P ,a q Since the p o i n t s q', q"', and p a r e pairwise c a l l i n e a r , they are contained i n a plane a'
of
n,
and q"' E (q',q"), (p,Ci').
which i s d i s t i n c t from CY (otherwise, a n S a c o n t r a d i c t i o n ) ; these two planes i n
IT
= a' n S q 9 meet a t the l i n e
L e t T and 1' be the maximal subspace i n T through a and a ' ,
respectively. Then,
T
= T' (by prop.
contains the l i n e (p,q')
I , t a k i n g i n t o account t h a t T n T '
= a n a'). The three p o i n t s q ' , q", and q"' belong
140
A . Bichara and G. Tallini
t o T nS thus, the p r o j e c t i v e plane through them i n T i s completely contained q' i n S , which i s impossible (see A21). q F i n a l l y , i t must be proved t h a t i f an element S i n S through p and i n P c i d e n t w i t h S e x i s t s , then there e x i s t s a l i n e r , any element o f which q P ,a i s an element o f S through p and i n c i d e n t w i t h S Set S n S 2 {q'}; q P qthrough the l i n e (p,q') t h e r e i s a maximal subspace T o f T, meeting S a t a q l i n e through q ' , a l l whose p o i n t s are c o l l i n e a r w i t h p ; and t h i s l i n e and
.
p span a plane belonging t o Prbposition X I V .
Let S
P
n.
The statement f o l l o w s ,
be the s e t o f a l l elements i n S through a p o i n t p
i n G. I f S belongs t o S and doesn't pass through p, then the subset
(S'
'L
S
i n (S, R ) means S' i s i n c i d e n t w i t h S i n (G, F ) ) o f S
empty o r a l i n e i n R . Furthermore, S P
Proof.
P i s a maximal subspace i n
i s either (S, R ) .
The f i r s t p a r t o f the statement f o l l o w s from prop. X I I I . Again by
i s a subspace i n (S, R ) , containing some element which i s P non-collinear w i t h S. Therefore, no subspace containing both S and S e x i s t s P i n (S, R ) and S i s a maximal subspace.
prop. X I I I , S
P
Thus, the c o l l e c t i o n P = {S : p E GI o f maximal subspace i n (S, R ) , P any o f them being a p r o j e c t i v e plane (see prop. I X ) , arises. Proposition XV.
The c o l l e c t i o n
P = ISp : p E GI i s proper.
Furthermore,
any two d i s t i n c t elements i n P share a t most one p o i n t o f S.
Proof.
L e t p and q be any two d i s t i n c t p o i n t s i n G. The maximal subspace
and S i n (S, R ) share a l l the elements i n S through both p and q. I f p P q and q are non-collinear i n (G, F), then there i s no element o f S through
S
them. I f p and q are j o i n e d by a l i n e f
E
F, then ( i n ( G , F ) )
there exists
e x a c t l y one maximal subspace belonging t o S which passes through f and so through p and q. I n both cases,
S n S G 1, and the statement follows. I P 91
141
The Grassmann manifold representing the planes in a projective space
7. THE COLLECTION E OF MAXIFlAL SUBSPACES I N (S, l?) F i r s t we prove: Proposition X V I .
L e t S and S ' be two d i s t i n c t elements i n S meeting a t a
p o i n t p i n G. I f S1 and S2 are d i s t i n c t elements i n S both meeting S and S ' a t d i s t i n c t p o i n t s i n G, then S1 and S E S belonging t o the l i n e
3
s2
are c o l l i n e a r i n (S, R ) and any element
(S1, S2) i n R e i t h e r meets both S and S ' a t
d i s t i n c t p o i n t s of G o r belongs t o the l i n e ( S ,
Proof.
S') i n R.
From A I 1 1 i t follows t h a t S1 and S2 meet: I q l = S1 n S2, and thus
2
they a r e c o l l i n e a r i n (S, R ) (see prop. V I ) . Since n e i t h e r S1 nor S p, then q # p. I f q E S, any element
s
contains
2 # S on the l i n e (S1,S2) i n l?, passing
through q, meets S a t t h a t p o i n t . I f q 4 S, t h e s e t o f a l l elements i n S through q and i n c i d e n t w i t h S i s e i t h e r a l i n e i n R o r t h e empty s e t (see prop. XIV); since both S, and S2 pass through q and are i n c i d e n t w i t h S, any element on the l i n e (S1, S 2 ) i n R i s i n c i d e n t w i t h S. By the same argument, any element on (S1, S2) i s proved t o be i n c i d e n t w i t h S ' ; thus, i t i s o n l y
S2) passing through p, then i t belongs t o the l i n e (S, S ' ) . Under these assumptions, S3 contains both p and { S t c E S: S" 3 p q, and, thus, meets S1 a t q. Hence, S3 E L ' , where L ' and S" n S1 # 01; a l s o S and S ' belong t o L ' (by hypothesis). By prop. X I V , S3, S and S' belong t o a l i n e i n R ; so, S3 E (S,S') and the statement i s t o be proved t h a t i f S3 i s an element on (Sly
proved. L e t S and S ' be two d i s t i n c t elements i n S, which are c o l l i n e a r i n (S, R ) ; then S and
S' share a p o i n t p i n G. Denote by o(S,S') t h e s e t con-
s i s t i n g o f a l l elements i n S t h a t e i t h e r belong t o the l i n e (S,S')
i n R,
o r meet both S and S ' a t p o i n t s i n G d i s t i n c t from p. Proposition X V I I .
I f S and S ' a r e d i s t i n c t elements i n S which are c o l l i n e a r
i n (S, R ) , then the s e t
o(S, S ' ) i s a subspace i n (S, R ) and i t properly
contains the l i n e (S, S') i n R. Proof
.
Let T be a member o f T through the p o i n t p, where { p l = S n S';
5y
142
A. Bichara and G. Tallini
A21, T meets S and S' a t the lines f and f ' in F , respectively. I f q
E
f,
q ' E f ' , q , q ' # p , then q # q ' and q and q ' belong to a l i n e f " i n T. Through
f " there i s an element S" E S meeting S and S' a t q and q ' , respectively. Thus, S" belongs t o a(S,S') \ (S,S'); therefore, the l i n e (S,S') i s properly contained i n G(S,S'). To prove t h a t a(S,S')
points S l y S2
E
o(S,S')
i s a subspace in (S, R ) ( i . e . any two d i s t i n c t
are collinear in (S, R ) and the l i n e joining them
i s completely contained in o(S,Sl)), three cases will be considered. ( i ) If S1 and S2 both belong t o the l i n e (S,S'), then there i s nothing
t o prove. ( i i ) If
f1 and
S2 meet S and
S' a t d i s t i n c t points, then the statement
follows from prop. XVI. ( i i i ) If S1 meets both S and S' a t d i s t i n c t points and S2 belongs t o the line (S,S'), then
-
by an argument simular t o t h a t in prop. XVI
- it
is
easy to prove t h a t S and S are collinear and any element on ( S l y S2) belongs 1 2 to o(S,S'). Proposition XVIII.
Let S and S' be two d i s t i n c t elements i n S which are
collinear i n (S, R ) , i . e . they meet a t a point p i n G. Then there e x i s t exactly two maximal subspaces o f (S, R ) t h r o u g h S and
s';
the f i r s t one i s S ( i . e . P
i t consists o f a l l elements i n S through p ) and belongs t o P; the second one i s a(S,S'). Proof.
These two subspaces share exactly the elements on the l i n e ( S , S l ) .
Let H be a subspace of (S, R ) containing both S and S' and so the
l i n e (S, S l ) in R. Then: e i t h e r H consists of elements in S a l l of them through p and H i s a subspace of S , o r there exists some S" E S contained i n H and P not passing through p. Since H i s a subspace and S", S, and S' belong t o H,
S" i s collinear w i t h b o t h S and S ' , i . e . meets S and S' a t d i s t i n c t points. By prop. XIV, the elements in S which are collinear with S" a r e exactly the
P
elements on the line ( S , S l ) i n R; hence, H n S = (S, S'). Therefore, any P element i n H (S, S'), being collinear w i t h both S and S', meets S and S' a t points i n G d i s t i n c t from p (and from each other). Thus H C o(S, S') and the statement follows. Proposition XIX.
Let S and S' be two d i s t i n c t elements i n S, which are col-
143
The Grassmann manifold representing the planes in a projective space l i n e a r i n (S, R ) . IfS1 and S 2 are d i s t i n c t elements i n u(S,S'), then they meet and u(S1, S 2 ) = u ( S , S'). Proof. Since S, and S2 belong t o the subspace and thus have a common p o i n t
(I E
u ( S , S'), they a r e c o l l i n e a r
G. By prop. X V I I I , the subspace
u(S, S ' ) ,
containing S1 and S2, i s contained i n e i t h e r the maximal subspace S , o r the q maximal subspace u(S1, S2). Since the elements i n u ( S , S l ) d o n ' t a l l pass through a same p o i n t ,
u(S, S ' )
5 u(S1,
S 2 ) . E q u a l i t y f o l l o w s from
a(S, S ' )
being a maximal subspace i n (S, R ) .
8.
FURTHER PROPERTIES OF
P AND C
I n the PLS (S, R ) two c o l l e c t i o n o f maximal subspaces have been defined (see prop. X I V and X V I I I ) :
P = IS : p P
E
GI, a proper c o l l e c t i o n ;
X = {u(S, S') : S, S' E S , S # S', S n S' # 01. Proposition XX.
For any r i n R , t h e r e e x i s t a unique S
u i n E such t h a t r C S and r 5 u.
P
i n P and a unique
P
Proof.
Since any two d i s t i n c t elements on r a r e i n c i d e n t , t h e statement
f o l l o w s from prop. X V I I I and X I X . Proposition X X I . or a l i n e i n R.
If S E P
P and u
E
X, then S n u i s e i t h e r t h e empty s e t
Proof, -
The Set S n u consists o f a l l elements i n S through p which are P c o l l i n e a r w i t h any element i n u. Take S E S n u and l e t S' be any other P element i n a; then (see prop. X I X ) u = u(S, S') and S E S I f {p) = S n S ' ,
P'
then the statement f o l l o w s from prop. X V I I I . I f p 4 S ' , then the s e t o f a l l elements i n S
- by prop.
XIV
-
i n c i d e n t w i t h S ' i s a l i n e r i n R , containing P S. L e t S" be a p o i n t on r, d i s t i n c t from S. Since S" meets S and S ' a t d i s t i n c t
144
A . Bichara and G. Tallini
p o i n t s , S " E u ( S , S'); t h e r e f o r e ,
( b y prop. XIX) u(S, S ' ) = u ( S , S " ) and t h e
p r e v i o u s argument proves t h e statement. Any two d i s t i n c t elements i n
Proposition X X I I .
c s h a r e a t most one p o i n t o f
S. I f u , u ' and u" a r e t h r e e p a i r w i s e d i s t i n c t elements i n l u ' n u''1 = 1, t h e n Iu n u"I
=
1. Consequently, i f {uj : j
i s a f i n i t e sequence of elements i n
1 t h e n Iu n u o ntl Proof.
>
c
such t h a t Iui n uitlI
and Iu n u ' l = 1,
E
{O,l,...,ntl}}
2 1 (i=0,
...,n ) ,
1.
From prop. X V I I I i t f o l l o w s t h a t any two d i s t i n c t elements i n Z share
a t most one p o i n t i n S. l l r i t e I S ' ) = u n u ' and IS") = u ' n u". I f S ' = S", t h e n t h e s t a t e m e n t i s o b v i o u s l y t r u e . I f S' # S " , t h e n S' and S " , b e l o n g i n g t o t h e subspace u ' , a r e c o l l i n e a r i n (S, R ) and l e t r be t h e l i n e j o i n i n g i n P t h r o u g h r, which P meets u ' a t t h e l i n e r ( s e e prop. XXI) and u a t a l i n e s' t h r o u g h S ' and u" them. By p r o p . XX, t h e r e e x i s t s a maximal subspace S
a t a l i n e s" t h r o u g h S". The l i n e s s' and s" i n R a r e d i s t i n c t ( o t h e r w i s e , b y prop. XX, u = u " , w h i c h i s i m p o s s i b l e ) and b e l o n g t o S , a p r o j e c t i v e p l a n e P ( b y prop. I X ) . I t f o l l o w s t h a t t h e r e e x i s t s e x a c t l y one element S i n S bel o n g i n g t o s' n so'; c l e a r l y , S belongs t o u ( a s S
u" (as S
E
s" and s" 5
0");
an element i n u Proof.
i
Let u
1
s' and s' C u ) and t o
thus, IS} C u n u". S i n c e u # u'', b y t h e f i r s t
p a r t o f t h e statement, IS1 = u n u"; t h e r e f o r e , Proposition X X I I I .
€
ICJ n u"1
= 1.
and u2 be two d i s t i n c t elements i n C .
( i = 1, 2 ) , t h e n I S 1 n S21
>
I f Si
1 i m p l i e s (ul n u,l I
>
is
1.
I f S1 = S2, t h e s t a t e m e n t i s o b v i o u s l y t r u e . Assume S1 # S2; t h e n
S1 n S2 c o n s i s t s o f a p o i n t p i n G. L e t fl be a l i n e o f
F in
S1 t h r o u g h p
t h e r e i s e x a c t l y one T E T which 1 meets S2 a t a l i n e f 2 o f F. T i s a 3-dimensional p r o j e c t i v e space ( s e e p r o p . (and such a l i n e does e x i s t ) ; t h r o u g h f
I X ) ; thus, t h e r e e x i s t s a l i n e f i n T t h r o u g h p d i s t i n c t f r o m fl and f2. Through f t h e r e i s an element S i n S , w h i c h i s c l e a r l y d i s t i n c t f r o m S1 and S2. Therefore, t h e maximal subspace
floreover, ul n u(S1, S)
3
u(S
S ) and i' I S 1 l , u(S1,S) n u(S,S2)
From p r o p . X X I I t h e s t a t e m e n t f o l l o w s .
u(S, S 2 ) i n Z e x i s t .
2
I S } , and u(S,S2) n u2 ?IS2}.
145
The Grassmann manifold representing the planes in a projective space Proposition X X I V .
and u ' i n X share e x a c t l y one
Two d i s t i n c t elements u
p o i n t i n S.
Proof,
By prop. X X I I , i t i s enough t o prove t h a t there e x i s t s a c o l l e c t i o n
,...
,...
: j E t0 , n t l I l o f elements i n Z such t h a t lul nu. I 1 (i=O ,n) j 1t 1 and uo = u, untl = u'. Take S E u and S ' E u'. Assume S # S' (otherwise t h e
{u
q2 a p o i n t i n S ' , w i t h
statement i n t r i v i a l ) and l e t q1 be a p o i n t i n S,
F ) i s connected, t h e r e e x i s t a f i n i t e subset o f G, {p1,...,pnt21, and a f i n i t e subset o f F, Ifl ,. . ,fntl I, such t h a t q1 = p1 , phtll (h = 1, ...,n t l ) . Through each l i n e fh i n F q2 = pnt2 and fh 1 {p,, q1 # q2.
Since (G,
.
there i s a maximal subspace Sh i n S (see A211). The c o l l e c t i o n {S1,...,S
ntl e i t h e r are equal o r have
consists o f elements i n S such t h a t Sk and S ktl' the maximal subspace uk = u(Sk,Sktl) t i o n {ak : k = l,...,nI,
>
lat n
1
(t
>
ktl i t may be assumed Sk # Sktl.
H.1.o.g.
e x a c t l y one common p o i n t p
at n u
1,
3 tS I ttl
ttl
l,...,n-1).
(k
Writing u
...,n)
..., n-1);
u and u
=
> 1. Since
Then,
i n I: e x i s t . I n the c o l l e c -
( t = 1, 0
1
ntl
hence,
= u', l e t ' s prove
and S n S 2 E u 1 1 1and q2 E S', {qll, by prop. X I I I , Iuo n ulI > 1. Since q2 E Sntl 'ntl 'n and S' E u', lan n u') > 1 and t h e statement i s proved.
t h a t luo n ull
9.
1 and lan n untlI
S E uo, S
THE PROOFS OF THEOREH 1 AN0 2 By the previous sections, t h e space (G,
F ) i s associated v i t h t h e PPLS
(S, R) (see prop. X I I I ) s a t i s f y i n g :
Given t h r e e pairwise c o l l i n e a r points, t h e r e e x i s t s a subspace conA;) t a i n i n g them (see prop. X and XI).
No l i n e i s a maximal subspace; moreover, t h e r e e x i s t two c o l l e c t i o n s , say c and P, o f maximal subspaces i n (S, R) such t h a t an:' maximal subspace belongs e i t h e r t o C o r t o P (see prop. X I V , X V I I I and XX) and A')
(I) (11) (1111
a,
U'
E
c,
u # u'
u E Z, n E P
v
r E R
=s>
* d!
u
*
l o n o ' \ = 1 (see prop. X X I V ) ;
either E
C,
u n n = 0 or
j! n
E
P :
r
5
u n n E R (see prop. X X I ) ;
u, r
5n
(see prop. X X ) .
Therefore (see [ 31 ), ts being the s e t o f a l l elements i n C through S E S
Its( > 2 ; furthermore, I: =
I t s : S E S I i s a proper c o l l e c t i o n and the p a i r
,
A . Bichara and C. Tallini
146
i s a p r o j e c t i v e space. The mapping p: L
(C,L)
+
S, defined by
p ( l l s ) = S, i s
one-to-one and onto and maps p e n c i l s o f l i n e s i n L onto elements o f R and i t s -1 inverse mapping p maps l i n e s i n R onto p e n c i l s o f l i n e s i n L. F i n a l l y , P maps r u l e d planes onto elements o f P and p-’ maps elements i n P onto r u l e d planes. be an element i n et be the c o l l e c t i o n o f a l l planes i n L). L e t -P andLLTI the s e t o f a l l l i n e s i n then p(LTI) i s an element S i n P. Consider
: P’+
F
the mapping
(C,
TI
IT;
P
G, defined by
I t i s easy t o check t h a t :
Proposition XXV.
F i s one-to-one and onto.
Next, L e t L be a l i n e i n L. The image under F o f t h e s e t o f a l l
Proposition XXVI. planes i n
7 through L
i s the element S = p ( L ) i n S. The image under F - l o f an
element S i n S i s the s e t o f a l l planes i n
7 through
the l i n e
L
p-l(S) i n
L. Therefore, S i s the c o l l e c t i o n o f the images under F o f t h e sets o f a l l
planes i n (C,
Proof. L c
TI
If Q
L ) through any l i n e i n L.
7 and
IT E
$? E
p(LTI) = S
L T I ep ( L )
B
then P’ p(LTI) Q S
- by E
S
P
- F(n)
(9.1) Q
p
E
= p.
Moreover,
S, and the statement follows.
Two d i s t i n c t elements p1 and p2 i n G are c o l l i n e a r i n
Proposition X X V I I .
(G, F ) i f f they are t h e images under F o f two planes i n (C,
L)
which meet
a t a l i n e o f L. Proof.
Set ri
F
-1
(p.), i .1
1,2.
By prop. X X V I , F(ni) E S , i = 1,2;
t h e subspace S o f (G,
I f rl n n
2
i s a l i n e L i n L, s e t S = ~ ( l ) .
hence, the p o i n t s p1 and p2, belonging t o
F), a r e c o l l i n e a r i n (G, F).
Conversely, i f p1 and p2 a r e j o i n e d by the l i n e f i n
F , l e t S be the
The Grassmann manifold representing the planes in a projective space
maximal subspace i n S through f.
Obviously, pl,
147
p, E S; hence n 1 and n 2
meet a t the l i n e p - l ( S ) i n L and the statement follows. Proposition X X V I I I .
If
7
7 which
i s the s e t o f a l l t h e planes i n
a 3-dimensional subspace i n (C, L ) , then the image under F o f
7
belong t o
belongs t o T .
Furthermore, the inverse image under F o f an element i n T i s t h e s e t o f a l l the planes i n a 3 - f l a t o f
Proof.
First it w i l
be proved t h a t i f T
F),
E
T, then F-’(T) c o n s i s t s o f a c o l -
contained i n the same 3 - f l a t i n (1, L ) . Since T i s a
l e c t i o n o f planes a1 subspace of (G,
(c, L ) .
t s p o i n t s a r e pairwise c o l l i n e a r ; hence, F-’(T)
o f planes p a i r w i s e meeting a t l i n e s i n L (see prop. X X V I I ) .
consists
Thus, F-’(T) con-
s i s t s o f e i t h e r planes i n a 3 - f l a t o f (I, L ) , o r planes through a l i n e . L e t ’ s prove t h e l a t t e r case i s impossible. L e t
pl,
-
p,
and p be t h r e e independent 3 by A2 T c a n ’ t be a l i n e ) . The
-
p o i n t s i n T (such p o i n t s do e x i s t because -1 -1 planes F (p,) = n1 and F (p,) = n, meet a t a l i n e L i n L. F-l(p3) meets
IT^
The plane r3=
and IT, a t l i n e s d i s t i n c t from L and from each other. (Indeed,
i f n l s IT,, and n3 passed through L , then t h e t h r e e p o i n t s pl, would belong t o the element
p ( L ) = S i n S, which would share w i t h T t h e p o i n t s
on the l i n e s j o i n i n g a l l t h e p a i r s o f p o i n t s pi, this i s
impossible (see A,I).)
the 3 - f l a t j o i n i n g nl and n,;
p2, and p3
pj,
i # j, i,j
1 , 2 , 3 , and
Thus, F - l ( T ) consists o f a l l t h e planes i n t h i s space w i l l be denoted by ( n l ,
IT,). Let n
i = 1,2,3. Then n n n = L i is i is = 1,2,3, are t h r e e l i n e s i n L, two of them a t l e a s t being d i s t i n c t (as nl, IT, be a plane i n (nl, n,),
d i s t i n c t from n
and n3 d o n ’ t form a p e n c i l ) . Assume Ll # 1, (a s i m i l a r argument holds i n t h e other cases).
The t h r e e p o i n t s p = F(n), p1 and p2 a r e p a i r w i s e c o l l i n e a r
(as IT, nl and n, p a i r w i s e meet a t l i n e s i n L). subspace i n (G,
F) containing p, pl, and p.,
Thus, by A1, t h e r e e x i s t s a
L e t H be a maximal subspace i n
(G, F ) through these points. H cannot belong t o S, because n = F-’(p), -1 -1 and n, = F (p,) d o n ’ t pass through t h e same l i n e (see prop. n1 = F (p,)
XXVI).
Hence, H i s an element i n T through p1 and p, and so H = T (T being
the o n l y maximal subspace i n T through the l i n e (pl,
p,))
and p E T.
Thus, the image under F o f any plane i n IT^, IT,) i s an element i n T, -1 and conversely, i.e. F (T) i s t h e s e t o f a l l planes i n (nl, n,).
148
A . Bichara and G. Tallini
planes i n
By a s i m i l a r argument, the image under F o f t h e set o f a l l t h e
a 3-flat o f
(c, L) i s proved t o be an element T i n T, and t h e p r o o f i s complete.
Proposition X X I X .
Three planes
IT^, IT2 , and IT3 i n
(C,
L ) form a p e n c i l ( i . e .
they belong t o the same 3 - f l a t and have a common l i n e ) i f f t h e i r images under p2, and p3’ are t h r e e p o i n t s i n G which belong t o the same l i n e i n F .
F, pl,
IT^, and IT3 form a p e n c i l , then they pass through a l i n e .t i n L and belong t o t h e 3 - f l a t (v1, IT^) spanned by v and IT Set S = P ( L ) and 1 2‘ Proof.
If v1 ,
l e t T be t h e image under F o f the s e t o f a l l the planes i n X X V I and X X V I I I , {p1,p2,p31
5
S, {p1,p2,p3)
e i t h e r the empty s e t o r a l i n e i n
5
T.
IT^).
(n1,
Since (by A21)
By prop.
T nS is
F , t h e p o i n t s ply p2, and p3 belong t o a
line. Conservely, i f
ply
p2 and p3 belong t o t h e l i n e f i n
T E T be the two maximal subspaces i n (G,
IT^,
n2, and
IT^
pass through the l i n e p-’(S)
a l l the planes i n a 3 - f l a t i n
(c, L).
F, l e t S
E
S and
F ) through f. The t h r e e planes and belong t o t h e s e t F-’(T) o f
Therefore,
IT,,
n2 and
and the statement i s proved.
IT
3
form a p e n c i l
From the r e s u l t s i n t h i s section theorem 1 f o l l o w s . F i n a l l y , i f (G, F ) i s a non-connected PPLS s a t i s f y i n g A1 and A2 and G P i s the connected component o f p (p E G) ( i . e . the s e t o f a l l t h e p o i n t s t h a t
F), then
can be reached from p by a polygonal path c o n s i s t i n g o f l i n e s i n
F containing a p o i n t i n G i s completely contained i n G and any P P subspace i n (G, F ) which i s n o t skew w i t h G i s completely contained i n G P’ P Hence, denoting by F t h e s e t o f a l l t h e l i n e s i n F which are contained i n G P P’ the p a i r ( G F ) i s a connected PPLS s a t i s f y i n g axioms A1 and A,. Theorem 2
any l i n e i n
follows.
P’
P
ACKNOWLEDGEHENT.
L
This research was p a r t i a l l y supported by GNSAGA o f CNR.
The Grassmann manifold representing the planes in a projective space
149
REFERENCES
[ 11 [
B. Segre, Lectures on modern geometry, CremoneseEd. Roma (1961)
21 G. T a l l i n i , Spazi p a r z i a l i d i r e t t e , spazi p o l a r i , Geometrie subimmerse, Quaderni Sem. Geom. Combinatorie, 1 s t . Mat. Univ. Roma, n. 14 (gennaio 1979).
[3]
G. T a l l i n i , On a c h a r a c t e r i z a t i o n o f t h e Grassmann m a n i f o l d r e p r e s e n t i n g t h e l i n e s i n a p r o j e c t i v e space, i n : P.J. Cameron, J.!4.P. H i r s h f e l d , D.R. Hughes (eds.) F i n i t e Geometries and designs. London l l a t h . SOC. Lect. Notes S e r i e s n. 49. Cambridge U n i v e r s i t y Press (1981) 354-358.