On the Incidence Geometry of Grassmann Spaces

Transcription

1 Geometriae Dedicata 75: 19 31, Kluwer Academic Publishers. Printed in the Netherlands. 19 On the Incidence Geometry of Grassmann Spaces EVA FERRARA DENTICE and NICOLA MELONE Dipartimento di Matematica, Seconda Università degli Studi di Napoli, Piazza Duorno, Caserta, Italy (Received: 4 August 1997; revised version: 9 July 1998) Abstract. In this paper we identify some properties on the point-line structure of Grassmannians which are useful tools to characterize the incidence geometry of Grassmann varieties and of their special quotients. Mathematics Subject Classifications (1991): 51M35, 51A50, 51E12. Key words: polar spaces, Grassmann spaces, incidence geometry. 1. Introduction The incidence structure in which the incidence geometry of ruled algebraic varieties is naturally studied is a semilinear space. A semilinear space is a point-line geometry (P, L) satisfying the following axioms: any two distinct points lie on at most one line, any line contains at least two points and any point lies on at least one line. A line is called thick if there are at least three points on it. Two distinct points p and q are said to be collinear if there exists a line containing p and q. The symbol p q means that the two points p and q are collinear and p q denotes the line of L joining p and q. More generally, two subsets X and Y are collinear (X Y), if each point of one of them is collinear with all points of the other one. For every subset X of P,X denotes the set of points of P collinear with all points of X. In the sequel, (P, L) will be a non singular semilinear space, i.e. P does not contain a point collinear with all other points. An isomorphism of semilinear spaces is a bijection f between points such that f and f 1 transform lines into lines. The incidence graph of a semilinear space (P, L) is the graph G(P, L) whose vertices are the points, two vertices being adjacent if they are collinear. (P, L) is connected if the graph G(P, L) is connected, i.e. for every pair p, q of points of P there exists a finite chain of points p 1 = p, p 2,...,p t = q such that p i p i+1, for i = 1,...,t 1. The distance d(p,q) between the points p and q of P is their distance in G(P, L), hence d(p,q) = h if h + 1 is the minimum length of finite Work supported by National Research Project Strutture Geometriche, Combinatoria e loro applicazioni of the Italian Ministero dell Università e della Ricerca Scientifica and by G.N.S.A.G.A. of C.N.R. corrected JEFF INTERPRINT geom1647 (geomkap:mathfam) v tex; 29/08/1995; 13:41; p.1

2 20 EVA FERRARA DENTICE AND NICOLA MELONE chains p 1 = p, p 2,...,p t = q such that p i p i+1. A subset X of P is a clique if it is a clique of G(P, L), i.e. any two points of X are collinear. A subspace of (P, L) is a clique W containing the line joining any two points of W. Clearly, a nonempty intersection of subspaces is a subspace, thus it is possible to define the closure [X] ofasubsetx of P as the intersection of all subspaces containing X. We say that the rank of a subspace W of (P, L) is k if k + 1 is the maximum length of all saturated chains of subspaces W 0 W 1 W t, such that W 0 is a point and W t = W. It follows that points and lines of (P, L) are subspaces of rank 0 and 1, respectively. Finally, (P, L) has rank d if d 1 is the maximum rank of its proper subspaces. A polar space is a semilinear space (P, L) such that for every point p and every line Lp contains either all points of L or exactly one point of L. A polar space of rank2isageneralized quadrangle. Clearly, if (P, L) is a generalized quadrangle, for every line L and every point p not on L we have p L =1. Let now P be a projective space. The Grassmann space of index h of P is the incidence geometry Gr(h, P) whose points are the h-subspaces of P and whose lines are the pencils of h-subspaces (a pencil being the set of h-subspaces contained in a (h+1)-subspace and passing through a (h 1)-subspace). If P is a projective space PG(m, K) of finite dimension m over a skew-field K, Gr(h, P) is also denoted by Gr(m,h,K). In the case K is a field, Gr(m,h,K)has a canonical embedding into a projective space PG(N, K), N = ( m+1 h+1) 1, via the Plücker morphism, which maps any h-subspace S on the point of PG(N, K) whose coordinates are the Grassmann coordinates of S. TheimageG(m,h,K) = p(gr(m,h,k)) is an intersection of quadrics of PG(N, K), i.e. it is a ruled algebraic variety defined by a system of quadratic equations, called the Grassmann variety of indices (m, h). Two distinct points p and q of a Grassmann space Gr(h, P) = (P, L) are collinear if, and only if, the subspace p q of PG(m, K) has dimension h 1. It follows easily that Gr(h, P) = (P, L) is a semilinear space and the Plücker morphism induces an isomorphism p:gr(m,h,k) G(m,h,K)of semilinear spaces. We can also observe that the Grassmann spaces Gr(m,h,K) and Gr(m, m h 1,K)are isomorphic semilinear spaces, an isomorphism being the duality. In particular, Gr(m, m 2,K) is the Grassmann space of lines of the dual space of PG(m, K). In 1976 Cooperstein [4] characterized the incidence geometry of the Grassmann variety G(m,h,q) of PG(m, q). Subsequently, Tallini and Bichara ([8], [1], [2]) characterized Grassmann spaces of indices (m, h) of PG(m, K) involving intersection properties of the two disjoint families of maximal subspaces of Gr(m,h,K).In 1984 Melone and Olanda characterized Gr(1, P) using only one family of maximal subspaces. The following result is contained in [6]. THEOREM 1. (Melone Olanda [6]). Let (P, L) be a semilinear space whose lines are not maximal subspaces and let be a covering of maximal subspaces of (P, L) satisfying the following property tex; 29/08/1995; 13:41; p.2

3 ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 21 * For every S and for every point p S any subspace of through p intersects S in a unique point and these points trace out a line formed by all points of S collinear with p. Then (P, L) is isomorphic to the Grassmann space of index 1 of a projective space. In 1983 A. M. Cohen proved the following result. THEOREM 2 (Cohen [3]). Let (P, L) be a connected semilinear space with thick lines whose subspaces have finite rank, and suppose that the following holds. (1) For any L L and p P with p L > 1 the line L is entirely contained in p (i.e. (P, L) is a so-called gamma space). (2) For any two x,y P with d(x,y) = 2, the subset x y is a subspace isomorphic to a nondegenerate generalized quadrangle. (3) For p P, L L such that p L = Ø but p L = Ø, the subset p L is a line. Then (P, L) is one of the following structures. (i) A nondegenerate polar space of rank 3. (ii) A Grassmann space Gr(m,h,K). (iii) A quotient Gr(2d 1,d 1,K)/ σ, where σ is an involutory automorphism of Gr(2d 1,d 1,K) induced by a polarity on the underlying projective space PG(2d 1,K)of Witt index at most d 5. We note that Gr(2d 1,d 1,K)/ σ is the incidence structure whose points are the orbits of the points of Gr(2d 1,d 1,K) with respect to the action of the group σ ={Id,σ} and whose lines are the subsets {x σ x L},whereL is a line of Gr(2d 1,d 1,K)(cfr. [3]). Geometric characterizations of Grassmann spaces by relatively simple properties of the point-line incidence structure start with the work of Cooperstein [4] and have been carried on by Lo Re and Olanda [5], Cohen [3] and Shult [7]. These results fully display the spirit of synthetic geometry in the sense of Steiner, that is one obtains an exact and elaborate structure with many subspaces from a few simple axioms mentioning only points and lines. In this paper we identify some properties of the point-line structure of Grassmann spaces which are useful tools to characterize the incidence geometry of the Grassmann space of the lines of a projective space (see Theorem 3). Moreover, as a consequence of Theorem 2, we obtain a characterization of the Grassmann space of h-subspaces of a projective space and of its special quotient (Theorem 4). THEOREM 3. Let (P, L) be a semilinear space satisfying the following properties. (I) For every line L the subset L is not a clique and it does not contain three pairwise noncollinear points tex; 29/08/1995; 13:41; p.3

4 22 EVA FERRARA DENTICE AND NICOLA MELONE (II) For every pair L, M of noncollinear and nonintersecting lines, either 1 L M 2 and L M is not contained in L M,orL M is a line intersecting L M. Then (P, L) is isomorphic to the Grassmann space Gr(1, P) of a projective space P. THEOREM 4. Let (P, L) be a connected semilinear space whose subspaces have finite rank and suppose that the following holds. (I) For every line L the subset L is not a clique and it does not contain three pairwise noncollinear points. (II ) For every pair L, M of noncollinear and nonintersecting lines, either L M 2 and L M is not contained in L M, orl M is a line. Then (P, L) is isomorphic either to a Grassmann space Gr(m,h,K),ortoa quotient Gr(2d 1,d 1,K)/ σ, whereσ is an involutory automorphism of Gr(2d 1,d 1,K) induced by a polarity on the underlying projective space PG(2d 1,K)of Witt index at most d Some Properties of Semilinear Spaces Satisfying (I) and (II ) Since property (II ) easily follows from property (II), we will assume in the sequel that (P, L) is a semilinear space satisfying properties (I) and (II ). PROPOSITION 2.1. No line is a maximal subspace, and two maximal subspaces intersect in at most one line. Consequently, for every line L and for every point p not on L and collinear with L there is exactly one maximal subspace S(p,L) containing p and L. Proof. By condition (I), for every line L the subset L contains two noncollinear points, hence one of them, say p, does not lie on L. In this case, L is properly contained in the subspace W =[L, p]. LetnowS and S be two distinct maximal subspaces such that S S contains a line L and a point x not on L, hence the closure X =[L, x] is a subspace of S S.IfM is a line of S not in S S,thenM is not collinear with S,sinceS is a maximal subspace, and there exists a point z of S noncollinear with M.LetN be a line of S passing through z and intersecting X not in M, then the lines M and N are nonintersecting and noncollinear (since z is a point of N noncollinear with M), but X M N, contradicting property (II ). From Proposition 2.1 we have the following result. PROPOSITION 2.2. If L and M are two noncollinear and nonintersecting lines and there is a point on one of them collinear with all the points of the other one, tex; 29/08/1995; 13:41; p.4

5 ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 23 then L M is a line. It follows that if L M contains at most two points, then L M does not contain any point of L M. Proof. By property (II ), if the line L contains a point collinear with M, then L M 2. Let us suppose that L M ={a, b}, with a L and b L M. From b L it follows b a and N = b a and M are two nonintersecting lines. By condition (II ), N and M are collinear, as a and b are two points of N collinear with M. If N is not contained into the maximal subspace S(a,M), then the maximal subspace S passing through N and M intersects S at least in the plane [M,a], contradicting Proposition 2.1. It follows that N is contained into S(a,M). Since b L, N is contained in S(b,L), hence N L and N L M,a contradiction. Moreover, the following results hold. PROPOSITION 2.3. For every maximal subspace S and every point p not on S such that p S = Ø, the subset p S is a line. Proof. Since p S, the maximal subspace S contains a point q noncollinear with p. Letp S be nonempty and z be a point of S collinear with p. By Proposition 2.1, S contains a point x not on the line q z. The lines L = p z and M = q x are nonintersecting and noncollinear, since the point q of M and the point p of L are noncollinear. Moreover, by Proposition 2.2, L M is a line R passing through z. If S does not contain R, then the maximal subspace containing R and M intersects S at least into the plane [M,z], a contradiction. Hence p is collinear with the line R of S and p S R. Ifp S contains a point y not in R, then the lines p y and R are nonintersecting and collinear, since p and y are collinear with R. In this case, the maximal subspace containing R and p y intersects S into [R,y], a contradiction. It follows that p S = R. PROPOSITION 2.4. (P, L) is a gamma space, i.e. for every line L and every point p with p L > 1, the line L is entirely contained in p. Proof. Suppose that p does not lie on L and let x and y be two distinct points of L collinear with p and S be a maximal subspace passing through L. Ifp is a point of S, thenp is collinear with L, otherwise, from p x and p y and by Proposition 2.3 p is collinear with the line L of S. PROPOSITION 2.5. Each line is contained in exactly two maximal subspaces. Proof. By condition (I), for every line L the subset L contains at least two noncollinear points x and y, hence L is contained at least in the two distinct maximal subspaces S = S(x,L) and S = S(y,L). Suppose that S is a maximal subspace passing through L and different from S and S and let z be a point of S L. By property (I), either z is collinear with x or it is collinear with y, hence either z S or z S properly contains the line L, a contradiction, by Proposition tex; 29/08/1995; 13:41; p.5

6 24 EVA FERRARA DENTICE AND NICOLA MELONE PROPOSITION 2.6. Each line contains at least three points. Proof. Let L be a line containing only two points a and b and let S and S the two maximal subspaces passing through L. A line M of S passing through the point a and a line N of S passing through b are nonintersecting and noncollinear, otherwise every point of M different from a would be collinear with a and N in S, contradicting Proposition 2.3. If M N ={a, b}, thenm N is contained in M N, contradicting (II ). Hence M N contains a point c external to M and N. From Proposition 2.3 we have that c does not lie on S (respectively, on S ), otherwise it would be collinear with a and N in S (resp., with b and M in S). It follows that the maximal subspace S(c,L) is different from S and S, contradicting Proposition 2.5. PROPOSITION 2.7. For any two points x,y with d(x,y) = 2 the subset x y forms a generalized quadrangle. Proof. Firstly, we observe that x y contains at least two incident lines. Let M be a line through x and N a line through y intersecting at a point z and let S 1 and S 2 be the two maximal subspaces containing M. Sincey z, by Proposition 2.3, y S 1 and y S 2 are two lines R and R passing through z and different from M,soR R x y. Now, we can consider a point p and a line L of x y such that p L.LetS = S(x,L) and S = S(y,L) be the two maximal subspaces passing through L and suppose that p is collinear with L. By Proposition 2.5 it is either p S or p S, contradicting Proposition 2.3. It follows that p is not collinear with L and so p S S.Sincep is collinear with x S and y S,by Proposition 2.3 p S is a line M through x and p S is a line M through y. The lines M and M are not collinear, otherwise x y and d(x,y) = 1. Moreover, they intersect L at the same point w, otherwise M and M are two nonintersecting and noncollinear lines such that {p} L M (M ), contradicting condition (II ). Hence w p L and p L ={w}, sincep is not collinear with L, and (P, L) is a gamma space (by Proposition 2.4). From the previous Propositions it follows that a semilinear space satisfying (I) and (II ) fullfills the hypothesis of Theorem of Shult [7]. Since no line of (P, L) is a maximal subspace (Prop. 2.1), from Propositions 2.4, 2.6 and 2.7 and from the Corollary of Lemma 3.12 of [4], the following result holds. PROPOSITION 2.8. Every maximal subspace of (P, L) is a projective spaces. 3. Some Example of Semilinear Spaces Satisfying (I) and (II ) Gr(h, P) = (P, L) is a semilinear connected space, since two points p, q of a Grassmann space Gr(h, P) = (P, L) are collinear if and only if the dimension of the subspace p q of P is h 1. Moreover, if L is the line of L whose points are the h-subspaces of P containing a (h 1)-subspace T and contained into a tex; 29/08/1995; 13:41; p.6

7 ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 25 (h + 1)-subspace S, then a point p of P is collinear with L if and only if p is a h-subspace of P either containing T or contained in S. In the sequel, we will denote by L(T, S) the line of Gr(h, P) whose points are the h-subspaces of P containing T and contained in S. PROPOSITION 3.1. The Grassmann space Gr(h,P) = (P,L) satisfies property (I). Proof. Let L(T, S) be a line of L and p and q be two h-subspaces of P such that T p S and T q S. Clearly, p and q are two points of P collinear with L,butp and q are not collinear since p q = T q has dimension h 2. So L is not a clique. Moreover, three points p 1,p 2,p 3 of P collinear with L are three h- subspaces of P, each one of them either contains T or is contained in S. It follows that at least two subspaces p 1,p 2,p 3 contain T (respectively, are contained in S), hence L does not contain three pairwise noncollinear points. PROPOSITION 3.2. The Grassmann space Gr(h,P) = (P,L) satisfies property (II ). Moreover, Gr(h, P) satisfies property (II) if and only if either h = 1 or P has finite dimension m and h = m 2. Proof. If L = L(T, S) and M = L(T,S ) are two noncollinear and nonintersecting lines of L, thens = S and S S contains at most one of T and T. We preliminarly observe that (i) L M is at most a line. L M contains a line if and only if S S has dimension either h or h 1 and contains exactly one of T and T, say T S S. In this case, we have L(T, S ) L M and the equality easily follows. (ii) If L M contains a point p of L M, thenl M is a line. If p is a point of L collinear with M, then either p S and L M = L(T, S ) or T p and L M = L(T,S). From (ii) it follows that (iii) If L and M are two noncollinear and nonintersecting lines of Gr(h, P) such that L M 2 then (L M ) (L M) = Ø. Moreover, we have the following result (iv) The Grassmann space Gr(h, P) = (P, L) is a gamma space. Let L be the line L(T, S) and p be a point of P collinear with two distinct points q 1 and q 2 of L,i.e.dim(p q 1 ) = h 1 = dim(p q 2 ).Ifp q 1 = p q 2,then p contains T, otherwise p = (p q 1 ) (p q 2 ) is contained in S. (v) If L M 3, thenl M is a line tex; 29/08/1995; 13:41; p.7

8 26 EVA FERRARA DENTICE AND NICOLA MELONE By property (I), L M contains two collinear points a and b. From (iv) and (i) it follows that L M = a b. From (iii) and (v) we have that Gr(h, P) = (P, L) satisfies condition (II ). Now we can prove that Gr(h, P) = (P, L) satisfies (II) if and only if either h = 1orP has finite dimension m and h = m 2. If either h = 1 or Gr(h, P) = Gr(m, m 2,K), K skew-field, one easily prove that for every pair of noncollinear and nonintersecting lines L and M it is L M 1. Conversely, suppose that Gr(h, P) = Gr(m, m 2, K) and h>1 and observe that we can consider the case dim P h+3. Let S and S be two (h+1)-subspaces of P intersecting into a (h 1)-subspace W and let T and T be two (h 1)- subspaces of P contained in S and S respectively, such that T W = T W.The two lines L = L(T, S) and M = L(T,S ) are noncollinear and nonintersecting and L M = Ø, contradicting (II). From Propositions 3.1 and 3.2 we have the following results. PROPOSITION 3.3. Theorem 3. The Grassmann space Gr(1, P) satisfies the hypotesis of PROPOSITION 3.4. The Grassmann space Gr(h,P) satisfies the hypotesis of Theorem 4 if and only if P has finite dimension m. Let (P, L) be a semilinear space and G be a group of automorphisms of (P, L). Denote by x G the orbit of a point x P,i.e.x G ={x σ P σ G}. Suppose that L x G for every point x P and for every line L L.Thequotient of (P, L) in G is the incidence structure (P, L)/G whose points are the orbits in P of G and whose lines are of the form {x G x L} for L L. Clearly, if p and q are two collinear points of (P, L) and L is the line p q, then the points p G and q G of (P, L)/G are collinear, since the line {x G x L} contains them. Now we can assume that (P, L) is a semilinear connected space and let (P,L ) be the quotient (P, L)/ σ, whereσ is an involutory automorphism of (P, L). In the sequel, for every point x of P and for every line L of L we will denote by x the point x σ of P and by L the line {x σ x L} of L. The following property holds. PROPOSITION 3.5. If d(p,p σ ) 3 for every point p P, then the quotient (P,L ) is a semilinear connected space. Moreover, if the subspaces of (P, L) have finite rank, then the subspaces of (P,L ) have finite rank too. Proof. Note that for every point p P we have p = p σ, otherwise d(p,p σ ) = 0. Moreover, if L σ denotes the set {x σ,x L} for every L L,thenL L σ = Ø, otherwise d(x,x σ ) 2, for every x L. It easily follows that the point p of P belongs to the line L of L if and only if either p L or p σ L. If two distinct points x and y of P lie on two distinct lines of L, then either two distinct points of P lie on two distinct lines, or d(x,x σ ) = 2, or d(y,y σ ) = 2. If x and y are two tex; 29/08/1995; 13:41; p.8

9 ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 27 distinct points of a line L of L, thenx and y are two distinct points of the line L of L, otherwise x = y and x = y would imply x = y σ and d(y,y σ ) = 1, a contradiction. It follows that any line of L contains at least two points. Finally, if x is a point of P and L is a line of L passing through the point x, then the line L contains x and any point of P lies on at least one line. Finally, we can prove the following result. PROPOSITION 3.6. If (P, L) satisfies property (I) and (II ) and d(p,p σ ) 5 for every p P,then(P,L ) satisfies property (I) and (II ). Proof. We first observe that: ( ) For every point x and for every line L of (P,L ) we have x L if and only if either x L or x σ L. It follows that two lines L and M are collinear if and only if either L M or L σ M. As a matter of fact, x L iff x z for every z L and, hence, iff either x z or x σ z,foreveryz L.Ifz 1 and z 2 are two distinct points of L such that x z 1 and x σ z 2,thend(x,x σ ) 3, a contradiction. Now we can prove that (P,L ) satisfies property (I). Let L be a line of L and suppose that L is a clique. If x and y are two noncollinear points of L,thenx and y are two points of L, and they are collinear, since L is a clique, hence x y σ. It follows that d(y,y σ ) = 3, a contradiction. Let us suppose that L contain three pairwise noncollinear points p 1,p 2,p 3,i.e.p i, p j and p i pj σ, for every i = j, i, j = 1, 2, 3. By ( ), it is either p i L or pi σ L for i = 1, 2, 3, a contradiction, since L does not contains three pairwise noncollinear points. Finally, we can prove that (P,L ) satisfies property (II ). Let L and M be two noncollinear nonintersecting lines of (P,L ).By( ), the lines L and M and the lines L σ and M are noncollinear and nonintersecting, too. If (L ) (M ) contains a line N and a point x not on L, then either L M (or (L σ ) M ) contains a line and an external point, or d(x,x σ ) = 4(ord(z,z σ ) = 4, for every z N), hence (L ) (M ) is at most a line. Moreover, if (L ) (M ) 2 and (L ) (M ) contains a point of L M, then from ( ) and Proposition 2.2 it follows that (L ) (M ) is a line, a contradiction. Finally, we prove that if (L ) (M ) 3, then (L ) (M ) is a line. Denoted by p 1,p 2,p 3 three distinct points of (L ) (M ), using property ( ) and eventually substituting either L by L σ or M by M σ, we can suppose that p 1 L M. If there exists an index j = 2, 3 such that p j L and pj σ M, then we have d(p j,pj σ ) = 4, hence for every j = 2, 3 p j L if, and only if, p j M. If p 2,p 3 L (respectively, p2 σ, pσ 3 L), thenp 1,p 2,p 3 L M (resp., p 1,p2 σ, pσ 3 L M )andl M is a line. Consequently, (L ) (M ) is a line. On the other hand, if p 2, p3 σ L (respectively, pσ 2, p 3 L), then p 1,p 2, p3 σ L M (p 1, p2 σ, p 3 L M ) and (L ) (M ) is a line, too tex; 29/08/1995; 13:41; p.9

10 28 EVA FERRARA DENTICE AND NICOLA MELONE The following results show that Grassmann spaces with involutory automorphisms whose quotients satisfy the previous proposition exist. LEMMA 3.7. For every pair p, q of distinct points of a Grassmann space Gr(h, P) = (P, L), d(p,q) = t if, and only if, dim(p q) = h t. Proof. We proceed by induction on t 1. The case t = 1 is trivial. Suppose t>1and the assumption true for any w<t.ifd(p,q) = t, then there exists a minimum chain of points of Pp= p 1, p 2,...,p t, p t+1 = q such that p i p i+1. Clearly, it is d(p,p t ) = t 1 so, by induction hypothesis, dim(p p t ) = h t +1. From p t p t+1 = q it follows dim(q p t ) = h 1, hence dim((p p t ) (q p t )) h t. Moreover, dim(p q) h t,since(p p t ) (q p t ) p q.if dim(p q) = h w >h t, then, by the induction hypothesis, d(p,q) = w<t, a contradiction. It follows that dim(p q) = h t. Conversely, if dim(p q) = h t and d(p,q) = t, by the necessary condition just proved, it is dim(p q) = h t = h t, hence d(p,q) = t. Let P = PG(2d 1,K)be a (2d 1)-dimension projective space over a skewfield K. A polarity σ of P having Witt index at most d 5 transforms every (d 1)-subspace S of P into a (d 1)-subspace of the dual projective space P, where is the family of hyperplanes of P containing a (d 1)-subspace S.It follows that σ induces an involutory automorphism of the Grassmann space G = Gr(2d 1,d 1,K). Since σ has Witt index at most d 5, for every point p of G the subspace p p σ has codimension at least 5 in p, hence dim(p p σ ) d 6 and, by Lemma 3.5, d(p,p σ ) 5. Finally, G is a semilinear connected space whose subspaces have finite rank satisfying Propositions (3.5) and (3.6), hence the following result holds. PROPOSITION 3.8. If σ is an involutory automorphism of a Grassmann space Gr(2d 1,d 1,K) induced by a polarity of PG(2d 1,K) of Witt index at most d 5, then the quotient Gr(2d 1,d 1,K)/ σ satisfies the hypothesis of Theorem A Characterization of Gr(1, P) In this section, (P, L) denotes a semilinear space satisfying properties (I) and (II). PROPOSITION 4.1. If S and S are two maximal subspaces of (P, L) intersecting in a line L, then either S or S is a projective plane. Proof. Suppose that S contains a line M nonintersecting L and S contains a line N nonintersecting L. Hence, M and N are nonintersecting lines and, by Proposition 2.3, they are not collinear, but M N contains L, contradicting (II). Hence, either any line of S or any line of S meets L and the assumption follows from Proposition tex; 29/08/1995; 13:41; p.10

11 ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 29 Let S 0 be a fixed maximal subspace of (P, L) and suppose that S 0 has rank r 3, in the case (P, L) contains subspaces of rank greater than 2. The following result is easily proved. PROPOSITION 4.2. Every maximal subspace metting S 0 in a line is a projective plane. Proof. If S 0 has rank 2, then every maximal subspace of (P, L) has rank 2 and, by Proposition 2.8, they are projective planes. On the contrary, if S 0 has rank r 3 and S is a maximal subspace of (P, L) intersecting S 0 at a line, by Proposition 4.1, S is a projective plane. Let be the family of maximal subspaces of (P, L) whose elements are S 0 and all maximal subspaces intersecting S 0 at a point, and call star every maximal subspace of. The following Proposition shows that contains maximal subspaces different from S 0. PROPOSITION 4.3. \S 0 is nonempty. Proof. Let L be a line of S 0 and S the maximal subspace passing through L and different from S 0.IfM is a line of S intersecting L at a point x then the maximal subspace S passing through M and different from S intersects S 0 at the point x. Indeed, if (S 0 S ) {x} would contain a point y collinear with the two lines L and M of S, a contradiction, by Proposition 2.3. In order to prove Theorem 3, we need two Lemmas. LEMMA 4.4. The stars of (P, L) satisfy the following properties. (i) Two different stars meet exactly at one point. (ii) If S is a star and S is a maximal subspace of (P, L) intersecting S at a point, then S is a star, too. Proof. (i) Let S and S be two different stars of. If either S or S is S 0,the proof is trivial, so we can assume that both S and S are different from S 0.Letxbe the point S S 0 and y the point S S 0 and suppose that x = y. IfS S is a line R, then a point z of R\{x} is collinear with a line of S 0 (by Proposition 2.4) and there are at least three distinct maximal subspaces passing through R, contradicting Proposition 2.6. Hence, x and y are different. If S S is a line L,thenx is collinear with y and the line L of S, a contradiction. Suppose that S and S are disjoint. By Proposition 2.4, y S is a line M passing through x and x S is a line M passing through y. The lines M and M are clearly disjoint. If M is not collinear with M, then M (M ) = N,andS 1 = S(x,M ), S 2 = S(y,M) and S 0 are three pairwise distinct maximal subspaces through N, a contradiction, by Proposition 2.6. Hence, the lines M and M are collinear and, by Proposition 4.1, the maximal subspace S containing M,M and N is a projective plane, a contradiction, since S contains the two nonintersecting lines M and M tex; 29/08/1995; 13:41; p.11

12 30 EVA FERRARA DENTICE AND NICOLA MELONE (ii) If either S = S 0 or S = S 0 the proof is trivial. If both S and S are different from S 0, we can substitute S and S 0 and the proof is the same of case (i). LEMMA 4.5. is a covering of maximal subspaces of (P, L). Proof. Let p be a point of P not in S 0 and suppose that p S 0 is empty. A fixed line M through p and a fixed line L of S 0 are noncollinear and nonintersecting, so L M contains at least a point q not on L M. Moreover, the point q does not lie on S 0, since it is collinear with p. The maximal subspace S = S(q,L) containing q and L is different from S 0 and it is a projective plane, by Proposition 4.2. Since p q, p S is a line N through q intersecting L at a point, contradicting the assumption p S 0 = Ø. It follows that p S 0 is a line L 0 and the maximal subspace S = S(p,L 0 ) is a projective plane. Let R be a line of S passing through p and intersecting L 0 at a point x and let S be the maximal subspace passing through R and different from S. The subspace S intersects S 0 exactly at the point x, otherwise a point of S S 0 different from x would be collinear with R and L 0 in S, contradicting Proposition 2.4. Hence S is a star passing through p, andthe proof is complete. Finally, we can characterize the incidence geometry of Grassmann spaces Gr(1, P). Precisely, the following result holds. THEOREM 4.6. Let (P, L) be a semilinear space satisfying the properties (I) and (II). Then(P, L) is isomorphic to the Grassmann space of the lines of a projective space. Proof. By Proposition 2.1, no line of (P, L) is a maximal subspace and, by Lemma 4.5, is a covering of maximal subspaces of (P, L). By Theorem 1 of Melone Olanda, it is sufficient to prove condition ( ). Let S be a star of (P, L) and p be a point not on S. By Lemma 4.5, there is a star S p passing through p and intersecting S at a point x (by case (i) of Lemma 4.4). From Proposition 2.4, it follows that p S is a line L through x. By case (i) of Lemma 4.4, every star passing through p intersects S at a point of L. Viceversa, if z is a point of L and M is the line p z,thenm is contained exactly in two maximal subspaces S 1 e S 2.If both S 1 and S 2 intersect S at lines, then S 1 S = S 2 S = L, since the two lines S 1 S and S 2 S are collinear with p. In this case the line L would be contained into three pairwise distinct maximal subspaces, contradicting Proposition 2.6. Hence either S 1 or S 2 intersects S at the point z, suppose S 1. By case (ii) of Lemma 4.4, S 1 isastar. 5. The Proof of Theorem 4 In this section (P, L) will be a semilinear and connected space satisfying conditions (I) and (II ). The following results hold tex; 29/08/1995; 13:41; p.12

13 ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 31 PROPOSITION 5.1. If (P, L) is a polar space of rank m 3 satisfying (I) and (II ),then(p, L) is isomorphic to the Grassmann space Gr(3, 1,K),whereK is a skew-field. Proof. By Proposition 2.8, every maximal subspace of (P, L) is a projective space. Let S 0 be a fixed maximal subspace of (P, L) having rank r = max{m 1, 2} and let be the family of all maximal subspaces of (P, L) whose elements are S 0 and all maximal subspaces intersecting S 0 at a point. Observe that Proposition 4.3 and Lemma 4.4 hold and Lemma 4.5 holds too, since in the polar space (P, L) the case p S 0 = Ø is always false. Arguing as in the Proof of Theorem 4.6, we prove that (P, L) is isomorphic to the Grassmann space Gr(1, P) of the lines of a projective space P. Clearly, if m > 3then Gr(1, P) contains a point p and a line L such that p L = Ø, hence m = 3 and the proof is complete. PROPOSITION 5.2. If L is a line and p is a point of (P, L) such that p L is empty, then p L is either empty or a line. Proof. Suppose that p L contains a point x and denote by S the maximal subspace S(x,L).Thenp S is a line M through x and M is contained in p L. If p L contains a point y not on M,theny does not lie on S, otherwise p S contains y and M, contradicting Proposition 2.3. Moreover, y is not collinear with x, otherwise y would be collinear with x and L in S, a contradition, by Proposition 2.3. The distance between x and y is clearly 2, but x y contains the point p and the line L such that p L = Ø, contradicting Proposition 2.7. By Propositions 2.4, 2.6, 2.7 and 5.2, (P, L) satisfies hypothesis (1), (2) and (3) of Theorem 2 of Cohen, hence by Proposition 5.1 Theorem 4 is completely proved. References 1. Bichara, A. and Tallini, G.: On a characterization of the Grassmann manifold representing the planes in a projective space, Ann. Discrete Math. 14 (1982), Bichara, A. and Tallini, G.: On a characterization of the Grassmann manifold representing the h-dimensional subspaces in a projective space, Ann. Discrete Math. 18 (1983), Cohen, A. M.: On a theorem of Cooperstein, European J. Combin. 4 (1983), Cooperstein, B. N.: A characterization of some Lie incidence structures, Geom. Dedicata 6 (1977), Lo Re, P. M. and Olanda, D.: Grassmann spaces, J. Geom. 17 (1981), Melone, N. and Olanda, D.: A characteristic property of the Grassmann manifold representing lines of a projective space, European J. Combin. 5 (1984), Shult, E. E.: A remark on Grassmann spaces and Half-spin geometries, European J. Combin. 15 (1994), Tallini, G.: On a characterization of the Grassmann Manifold representing the lines in a projective space, in: P. Cameron, J. Hirschfeld and D. Hughes (eds), Finite Geometries and Designs, London Math. Soc. Lecture Notes Ser. 49, Cambridge University Press, Cambridge, 1981, pp tex; 29/08/1995; 13:41; p.13