On the Incidence Geometry of Grassmann Spaces

Geometriae Dedicata75: 19–31, 1999.© 1999Kluwer 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,81100 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 Grassmannianswhich are useful tools to characterize the incidence geometry of Grassmann varieties and of theirspecial 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 variet-ies is naturally studied is a semilinear space.

A semilinear spaceis a point-line geometry(P,L) satisfying the followingaxioms:any two distinct points lie on at most one line, any line contains at leasttwo points and any point lies on at least one line.A line is called thick if thereare at least three points on it. Two distinct pointsp andq are said to becollinearif there exists a line containingp andq. The symbolp ∼ q means that the twopointsp andq are collinear andp ∨ q denotes the line ofL joining p andq. Moregenerally, two subsetsX andY arecollinear (X ∼ Y ), if each point of one of themis collinear with all points of the other one. For every subsetX of P,X⊥ denotesthe set of points ofP collinear with all points ofX. In the sequel,(P,L) will bea non singularsemilinear space, i.e.P does not contain a point collinear with allother points. Anisomorphismof semilinear spaces is a bijectionf between pointssuch thatf andf −1 transform lines into lines.

The incidence graphof a semilinear space(P,L) is the graphG(P,L) whosevertices are the points, two vertices being adjacent if they are collinear.(P,L) isconnectedif the graphG(P,L) is connected, i.e. for every pairp, q of points ofPthere exists a finite chain of pointsp1 = p, p2, . . . , pt = q such thatpi ∼ pi+1,for i = 1, . . . , t −1. Thedistanced(p, q) between the pointsp andq of P is theirdistance inG(P,L), henced(p, q) = h if h + 1 is the minimum length of finite

? Work supported by National Research Project ‘Strutture Geometriche, Combinatoria eloro applicazioni’ of the Italian ‘Ministero dell’Universita e della Ricerca Scientifica’ and by‘G.N.S.A.G.A.’ of ‘C.N.R.’

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20 EVA FERRARA DENTICE AND NICOLA MELONE

chainsp1 = p, p2, . . . , pt = q such thatpi ∼ pi+1. A subsetX of P is acliqueif it is a clique ofG(P,L), i.e. any two points ofX are collinear. Asubspaceof(P,L) is a cliqueW containing the line joining any two points ofW . Clearly, anonempty intersection of subspaces is a subspace, thus it is possible to define theclosure[X] of a subsetX of P as the intersection of all subspaces containingX.We say that therankof a subspaceW of (P,L) is k if k+1 is the maximum lengthof all saturated chains of subspacesW0 ⊂ W1 ⊂ · · · ⊂ Wt , such thatW0 is a pointandWt = W . It follows that points and lines of(P,L) are subspaces of rank 0and 1, respectively. Finally,(P,L) hasrankd if d − 1 is the maximum rank of itsproper subspaces.

A polar spaceis a semilinear space(P,L) such that for every pointp and everylineL p⊥ contains either all points ofL or exactly one point ofL. A polar space ofrank 2 is ageneralized quadrangle. Clearly, if (P,L) is a generalized quadrangle,for every lineL and every pointp not onL we have|p⊥ ∩ L| = 1.

Let nowP be a projective space. TheGrassmann space of indexh of P is the in-cidence geometry Gr(h,P) whose points are theh-subspaces ofP and whose linesare thepencilsof h-subspaces (a pencil being the set ofh-subspaces contained in a(h+1)-subspace and passing through a(h−1)-subspace). IfP is a projective spacePG(m,K) of finite dimensionm over a skew-fieldK, Gr(h,P) is also denoted byGr(m, h,K). In the caseK is a field, Gr(m, h,K) has a canonical embedding into aprojective space PG(N,K),N = (m+1

h+1

)−1, via thePlücker morphism, which mapsanyh-subspaceS on the point of PG(N,K) whose coordinates are the Grassmanncoordinates ofS. The imageG(m, h,K) = p(Gr(m, h,K)) is an intersection ofquadrics of PG(N,K), i.e. it is a ruled algebraic variety defined by a system ofquadratic equations, called theGrassmann variety of indices(m, h). Two distinctpointsp andq of a Grassmann space Gr(h,P) = (P,L) arecollinear if, and onlyif, the subspacep ∩ q of PG(m,K) has dimensionh − 1. It follows easily thatGr(h,P) = (P,L) is a semilinear space and the Plücker morphism induces anisomorphismp: 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. Inparticular, Gr(m,m − 2,K) is the Grassmann space of lines of the dual space ofPG(m,K).

In 1976 Cooperstein [4] characterized the incidence geometry of the Grassmannvariety 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 intersec-tion properties of the two disjoint families of maximal subspaces of Gr(m, h,K). In1984 Melone and Olanda characterized Gr(1,P) using only one family of maximalsubspaces. The following result is contained in [6].

THEOREM 1. (Melone–Olanda [6]).Let(P,L) be a semilinear space whose linesare not maximal subspaces and let6 be a covering of maximal subspaces of(P,L)

satisfying the following property

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ON THE INCIDENCE GEOMETRY OF GRASSMANN SPACES 21

* For everyS ∈ 6 and for every pointp 6∈ S any subspace of6 throughpintersectsS in a unique point and these points trace out a line formed by allpoints ofS collinear withp.

Then(P,L) is isomorphic to the Grassmann space of index1 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 thicklines whose subspaces have finite rank, and suppose that the following holds.

(1) For anyL ∈ L andp ∈ P with |p⊥ ∩ L| > 1 the lineL is entirely containedin p⊥ (i.e. (P,L) is a so-called gamma space).

(2) For any twox, y ∈ P with d(x, y) = 2, the subsetx⊥ ∩ y⊥ is a subspaceisomorphic to a nondegenerate generalized quadrangle.

(3) For p ∈ P ,L ∈ L such thatp⊥∩L = Ø butp⊥∩L⊥ 6= Ø, the subsetp⊥∩L⊥is a line.

Then(P,L) is one of the following structures.

(i) A nondegenerate polar space of rank3.(ii) A Grassmann spaceGr(m, h,K).(iii) A quotientGr(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 projectivespacePG(2d − 1,K) of Witt index at mostd − 5.

We note that Gr(2d − 1, d − 1,K)/〈σ 〉 is the incidence structure whose pointsare the orbits of the points of Gr(2d − 1, d − 1,K) with respect to the action ofthe group〈σ 〉 = {Id, σ } and whose lines are the subsets{x〈σ 〉|x ∈ L}, whereL is aline of Gr(2d − 1, d − 1,K) (cfr. [3]).

Geometric characterizations of Grassmann spaces by relatively simple proper-ties of the point-line incidence structure start with the work of Cooperstein [4] andhave been carried on by Lo Re and Olanda [5], Cohen [3] and Shult [7]. Theseresults fully display the spirit of synthetic geometry in the sense of Steiner, thatis one obtains an exact and elaborate structure with many subspaces from a fewsimple axioms mentioning only points and lines.

In this paper we identify some properties of the point-line structure of Grass-mann spaces which are useful tools to characterize the incidence geometry of theGrassmann space of the lines of a projective space (see Theorem 3). Moreover, asa consequence of Theorem 2, we obtain a characterization of the Grassmann spaceof 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 proper-ties.

(I) For every lineL the subsetL⊥ is not a clique and it does not contain threepairwise noncollinear points.

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(II) For every pairL,M of noncollinear and nonintersecting lines, either1 6|L⊥ ∩M⊥| 6 2 andL⊥ ∩M⊥ is not contained inL∪M, orL⊥ ∩M⊥ is a lineintersectingL ∪M.

Then(P,L) is isomorphic to the Grassmann spaceGr(1,P) of a projective spaceP.

THEOREM 4.Let (P,L) be a connected semilinear space whose subspaces havefinite rank and suppose that the following holds.

(I) For every lineL the subsetL⊥ is not a clique and it does not contain threepairwise noncollinear points.

(II ′) For every pair L,M of noncollinear and nonintersecting lines, either|L⊥ ∩ M⊥| 6 2 andL⊥ ∩ M⊥ is not contained inL ∪ M, or L⊥ ∩ M⊥ isa line.

Then (P,L) is isomorphic either to a Grassmann spaceGr(m, h,K), or to aquotientGr(2d − 1, d − 1,K)/〈σ 〉, whereσ is an involutory automorphism ofGr(2d − 1, d − 1,K) induced by a polarity on the underlying projective spacePG(2d − 1,K) of Witt index at mostd − 5.

2. Some Properties of Semilinear Spaces Satisfying (I) and (II′)

Since property (II′) easily follows from property (II), we will assume in the sequelthat(P,L) is a semilinear space satisfying properties (I) and (II′).

PROPOSITION 2.1.No line is a maximal subspace, and two maximal subspacesintersect in at most one line. Consequently, for every lineL and for every pointp not onL and collinear withL there is exactly one maximal subspaceS(p,L)containingp andL.

Proof.By condition (I), for every lineL the subsetL⊥ contains two noncollinearpoints, hence one of them, sayp, does not lie onL. In this case,L is properlycontained in the subspaceW = [L,p]. Let nowS andS ′ be two distinct maximalsubspaces such thatS ∩ S ′ contains a lineL and a pointx not onL, hence theclosureX = [L, x] is a subspace ofS ∩S ′. If M is a line ofS not inS ∩S ′, thenMis not collinear withS ′, sinceS ′ is a maximal subspace, and there exists a pointz

of S ′ noncollinear withM. LetN be a line ofS ′ passing throughz and intersectingX not inM, then the linesM andN are nonintersecting and noncollinear (sincezis a point ofN noncollinear withM), butX ⊆ M⊥ ∩ N⊥, contradicting property(II ′).

From Proposition 2.1 we have the following result.

PROPOSITION 2.2.If L andM are two noncollinear and nonintersecting linesand there is a point on one of them collinear with all the points of the other one,

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thenL⊥ ∩M⊥ is a line. It follows that ifL⊥ ∩M⊥ contains at most two points,thenL ∪M does not contain any point ofL⊥ ∩M⊥.

Proof. By property (II′), if the lineL contains a point collinear withM, then|L⊥∩M⊥| > 2. Let us suppose thatL⊥∩M⊥ = {a, b}, with a ∈ L andb 6∈ L∪M.From b ∼ L it follows b ∼ a andN = b ∨ a andM are two nonintersectinglines. By condition (II′), N andM are collinear, asa andb are two points ofNcollinear withM. If N is not contained into the maximal subspaceS(a,M), thenthe maximal subspaceS ′ passing throughN andM intersectsS at least in the plane[M,a], contradicting Proposition 2.1. It follows thatN is contained intoS(a,M).Sinceb ∼ L, N is contained inS(b,L), henceN ∼ L andN ⊆ L⊥ ∩ M⊥, acontradiction.

Moreover, the following results hold.

PROPOSITION 2.3.For every maximal subspaceS and every pointp not onSsuch thatp⊥ ∩ S 6= Ø, the subsetp⊥ ∩ S is a line.

Proof. Sincep 6∈ S, the maximal subspaceS contains a pointq noncollinearwith p. Let p⊥ ∩ S be nonempty andz be a point ofS collinear with p. ByProposition 2.1,S contains a pointx not on the lineq ∨ z. The linesL = p∨ z andM = q ∨ x are nonintersecting and noncollinear, since the pointq of M and thepointp of L are noncollinear. Moreover, by Proposition 2.2,L⊥ ∩M⊥ is a lineRpassing throughz. If S does not containR, then the maximal subspace containingR andM intersectsS at least into the plane[M, z], a contradiction. Hencep iscollinear with the lineR of S andp⊥ ∩ S ⊇ R. If p⊥ ∩ S contains a pointy notin R, then the linesp ∨ y andR are nonintersecting and collinear, sincep andyare collinear withR. In this case, the maximal subspace containingR andp ∨ yintersectsS into [R, y], a contradiction. It follows thatp⊥ ∩ S = R.

PROPOSITION 2.4.(P,L) is a gamma space, i.e. for every lineL and every pointp with |p⊥ ∩ L| > 1, the lineL is entirely contained inp⊥.

Proof. Suppose thatp does not lie onL and letx andy be two distinct pointsof L collinear withp andS be a maximal subspace passing throughL. If p is apoint of S, thenp is collinear withL, otherwise, fromp ∼ x andp ∼ y and byProposition 2.3p is collinear with the lineL of S.

PROPOSITION 2.5.Each line is contained in exactly two maximal subspaces.Proof.By condition (I), for every lineL the subsetL⊥ contains at least two non-

collinear pointsx andy, henceL is contained at least in the two distinct maximalsubspacesS = S(x,L) andS ′ = S(y,L). Suppose thatS ′′ is a maximal subspacepassing throughL and different fromS andS ′ and letz be a point ofS ′′ − L. Byproperty (I), eitherz is collinear withx or it is collinear withy, hence eitherz⊥ ∩Sor z⊥ ∩ S ′ properly contains the lineL, a contradiction, by Proposition 2.3.

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PROPOSITION 2.6.Each line contains at least three points.Proof. Let L be a line containing only two pointsa andb and letS andS ′ the

two maximal subspaces passing throughL. A line M of S passing through thepoint a and a lineN of S ′ passing throughb are nonintersecting and noncollinear,otherwise every point ofM different froma would be collinear witha andN in S ′,contradicting Proposition 2.3. IfM⊥ ∩ N⊥ = {a, b}, thenM⊥ ∩ N⊥ is containedin M ∪ N , contradicting (II′). HenceM⊥ ∩ N⊥ contains a pointc external toMandN . From Proposition 2.3 we have thatc does not lie onS (respectively, onS ′),otherwise it would be collinear witha andN in S ′ (resp., withb andM in S). Itfollows that the maximal subspaceS(c, L) is different fromS andS ′, contradictingProposition 2.5.

PROPOSITION 2.7.For any two pointsx, y with d(x, y) = 2 the subsetx⊥ ∩ y⊥forms a generalized quadrangle.

Proof.Firstly, we observe thatx⊥ ∩ y⊥ contains at least two incident lines. LetM be a line throughx andN a line throughy intersecting at a pointz and letS1 andS2 be the two maximal subspaces containingM. Sincey ∼ z, by Proposition 2.3,y⊥ ∩ S1 andy⊥ ∩ S2 are two linesR andR′ passing throughz and different fromM, soR ∪R′ ⊆ x⊥ ∩ y⊥. Now, we can consider a pointp and a lineL of x⊥ ∩ y⊥such thatp 6∈ L. LetS = S(x,L) andS ′ = S(y,L) be the two maximal subspacespassing throughL and suppose thatp is collinear withL. By Proposition 2.5 itis eitherp ∈ S or p ∈ S ′, contradicting Proposition 2.3. It follows thatp is notcollinear withL and sop 6∈ S ∪ S ′. Sincep is collinear withx ∈ S andy ∈ S ′, byProposition 2.3p⊥ ∩ S is a lineM throughx andp⊥ ∩ S ′ is a lineM ′ throughy.The linesM andM ′ are not collinear, otherwisex ∼ y andd(x, y) = 1. Moreover,they intersectL at the same pointw, otherwiseM andM ′ are two nonintersectingand noncollinear lines such that{p} ∪ L ⊆ M⊥ ∩ (M ′)⊥, contradicting condition(II ′). Hencew ∈ p⊥ ∩ L andp⊥ ∩ L = {w}, sincep is not collinear withL, 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 pointsp, q of aGrassmann space Gr(h,P) = (P,L) are collinear if and only if the dimension ofthe subspacep ∩ q of P is h − 1. Moreover, ifL is the line ofL whose pointsare theh-subspaces ofP containing a (h − 1)-subspaceT and contained into a

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(h + 1)-subspaceS, then a pointp of P is collinear withL if and only if p is ah-subspace ofP either containingT or contained inS. In the sequel, we will denoteby L(T , S) the line of Gr(h,P) whose points are theh-subspaces ofP containingT and contained inS.

PROPOSITION 3.1.The Grassmann spaceGr(h,P) = (P,L) satisfies property(I).

Proof.LetL(T , S) be a line ofL andp andq be twoh-subspaces ofP such thatT ⊂ p 6⊆ S andT 6⊂ q ⊂ S. Clearly,p andq are two points ofP collinear withL, butp andq are not collinear sincep ∩ q = T ∩ q has dimensionh− 2. SoL⊥is not a clique. Moreover, three pointsp1, p2, p3 of P collinear withL are threeh-subspaces ofP, each one of them either containsT or is contained inS. It followsthat at least two subspacesp1, p2, p3 containT (respectively, are contained inS),henceL⊥ does not contain three pairwise noncollinear points.

PROPOSITION 3.2.The Grassmann spaceGr(h,P) = (P,L) satisfies property(II ′). Moreover,Gr(h,P) satisfies property(II) if and only if eitherh = 1 or P hasfinite dimensionm andh = m− 2.

Proof. If L = L(T , S) andM = L(T ′, S ′) are two noncollinear and nonin-tersecting lines ofL, thenS 6= S ′ and S ∩ S ′ contains at most one ofT andT ′.

We preliminarly observe that

(i) L⊥ ∩M⊥ is at most a line.

L⊥ ∩ M⊥ contains a line if and only ifS ∩ S ′ has dimension eitherh or h − 1and contains exactly one ofT and T ′, sayT ⊆ S ∩ S ′. In this case, we haveL(T , S ′) ⊆ L⊥ ∩M⊥ and the equality easily follows.

(ii) If L⊥ ∩M⊥ contains a pointp ofL ∪M, thenL⊥ ∩M⊥ is a line.

If p is a point ofL collinear withM, then eitherp ⊂ S ′ andL⊥ ∩M⊥ = L(T , S ′)or T ′ ⊂ p andL⊥ ∩M⊥ = L(T ′, S).

From (ii) it follows that

(iii) If L andM are two noncollinear and nonintersecting lines ofGr(h,P) suchthat |L⊥ ∩M⊥| 6 2 then (L⊥ ∩M⊥) ∩ (L ∪M) = Ø.Moreover, we have the following result

(iv) The Grassmann spaceGr(h,P) = (P,L) is a gamma space.

Let L be the lineL(T , S) andp be a point ofP collinear with two distinct pointsq1 andq2 of L, i.e. dim(p ∩ q1) = h− 1= dim(p ∩ q2). If p ∩ q1 = p ∩ q2, thenp containsT , otherwisep = (p ∩ q1) ∨ (p ∩ q2) is contained inS.

(v) If |L⊥ ∩M⊥| > 3, thenL⊥ ∩M⊥ is a line.

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By property (I),L⊥ ∩M⊥ contains two collinear pointsa andb. From (iv) and (i)it follows thatL⊥ ∩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 = 1 orP has finite dimensionm andh = 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 linesL andM it is|L⊥ ∩M⊥| > 1.

Conversely, suppose that Gr(h,P) 6= Gr(m,m − 2,K) andh > 1 and observethat we can consider the case dimP > h+3. LetS andS ′ be two (h+1)-subspacesof P intersecting into a (h − 1)-subspaceW and letT andT ′ be two (h − 1)-subspaces ofP contained inS andS ′ respectively, such thatT ∩W 6= T ′ ∩W . Thetwo linesL = L(T , S) andM = L(T ′, S ′) are noncollinear and nonintersectingandL⊥ ∩M⊥ = Ø, contradicting (II).

From Propositions 3.1 and 3.2 we have the following results.

PROPOSITION 3.3. The Grassmann spaceGr(1,P) satisfies the hypotesis ofTheorem3.

PROPOSITION 3.4. The Grassmann spaceGr(h,P) satisfies the hypotesis ofTheorem4 if and only ifP has finite dimensionm.

Let (P,L) be a semilinear space andG be a group of automorphisms of(P,L).Denote byxG theorbit of a pointx ∈ P , i.e.xG = {xσ ∈ P |σ ∈ G}. Suppose thatL 6⊂ xG for every pointx ∈ P and for every lineL ∈ L. Thequotientof (P,L) inG is the incidence structure(P,L)/G whose points are the orbits inP of G andwhose lines are of the form{xG|x ∈ L} for L ∈ L. Clearly, if p andq are twocollinear points of(P,L) andL is the linep ∨ q, then the pointspG andqG of(P,L)/G are collinear, since the line{xG|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). Inthe sequel, for every pointx of P and for every lineL of L we will denote byx′the pointx〈σ 〉 of P ′ and byL′ the line{x〈σ 〉|x ∈ L} of L′. The following propertyholds.

PROPOSITION 3.5.If d(p, pσ ) > 3 for every pointp ∈ P , then the quotient(P ′, L′) is a semilinear connected space. Moreover, if the subspaces of(P,L) havefinite rank, then the subspaces of(P ′, L′) have finite rank too.

Proof.Note that for every pointp ∈ P we havep 6= pσ , otherwised(p, pσ ) =0. Moreover, ifLσ denotes the set{xσ , x ∈ L} for everyL ∈ L, thenL ∩Lσ = Ø,otherwised(x, xσ ) 6 2, for everyx ∈ L. It easily follows that the pointp′ of P ′belongs to the lineL′ of L′ if and only if eitherp ∈ L or pσ ∈ L. If two distinctpointsx′ andy′ of P ′ lie on two distinct lines ofL′, then either two distinct pointsof P lie on two distinct lines, ord(x, xσ ) = 2, ord(y, yσ ) = 2. If x andy are two

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distinct points of a lineL of L, thenx′ andy′ are two distinct points of the lineL′ of L′, otherwisex′ = y′ andx 6= y would imply x = yσ andd(y, yσ ) = 1, acontradiction. It follows that any line ofL′ contains at least two points. Finally, ifx′ is a point ofP ′ andL is a line ofL passing through the pointx, then the lineL′containsx′ and any point ofP ′ lies on at least one line.

Finally, we can prove the following result.

PROPOSITION 3.6.If (P,L) satisfies property(I) and(II ′) andd(p, pσ ) > 5 foreveryp ∈ P , then(P ′, L′) satisfies property(I) and(II ′).

Proof.We first observe that:

(∗) For every pointx′ and for every lineL′ of (P ′, L′) we havex′ ∼ L′ if and onlyif either x ∼ L or xσ ∼ L. It follows that two linesL′ andM ′ are collinear ifand only if eitherL ∼ M or Lσ ∼ M.

As a matter of fact,x′ ∼ L′ iff x′ ∼ z′ for everyz ∈ L and, hence, iff eitherx ∼ z or xσ ∼ z, for everyz ∈ L. If z1 andz2 are two distinct points ofL such thatx ∼ z1 andxσ ∼ z2, thend(x, xσ ) 6 3, a contradiction.

Now we can prove that(P ′, L′) satisfies property (I). LetL′ be a line ofL′ andsuppose thatL′⊥ is a clique. Ifx andy are two noncollinear points ofL⊥, thenx′andy′ are two points ofL′⊥, and they are collinear, sinceL′⊥ is a clique, hencex ∼ yσ . It follows that d(y, yσ ) = 3, a contradiction. Let us suppose thatL′⊥contain three pairwise noncollinear pointsp′1, p

′2, p

′3, i.e.pi, 6∼ pj andpi 6∼ pσj ,

for everyi 6= j , i, j = 1,2,3. By (∗), it is eitherpi ∼ L orpσi ∼ L for i = 1,2,3,a contradiction, sinceL⊥ does not contains three pairwise noncollinear points.

Finally, we can prove that(P ′, L′) satisfies property (II′). Let L′ andM ′ betwo noncollinear nonintersecting lines of(P ′, L′). By (∗), the linesL andM andthe linesLσ andM are noncollinear and nonintersecting, too. If(L′)⊥ ∩ (M ′)⊥contains a lineN ′ and a pointx′ not onL, then eitherL⊥ ∩M⊥ (or (Lσ )⊥ ∩M⊥)contains a line and an external point, ord(x, xσ ) = 4 (or d(z, zσ ) = 4, for everyz ∈ N), hence(L′)⊥ ∩ (M ′)⊥ is at most a line. Moreover, if|(L′)⊥ ∩ (M ′)⊥| 6 2and(L′)⊥ ∩ (M ′)⊥ contains a point ofL′ ∪M ′, then from (∗) and Proposition 2.2it follows that(L′)⊥ ∩ (M ′)⊥ is a line, a contradiction.

Finally, we prove that if|(L′)⊥ ∩ (M ′)⊥| > 3, then(L′)⊥ ∩ (M ′)⊥ is a line.Denoted byp′1, p

′2, p

′3 three distinct points of(L′)⊥ ∩ (M ′)⊥, using property (∗)

and eventually substituting eitherL byLσ orM byMσ , we can suppose thatp1 ∈L⊥ ∩M⊥. If there exists an indexj = 2,3 such thatpj ∼ L andpσj ∼ M, thenwe haved(pj , pσj ) = 4, hence for everyj = 2,3pj ∼ L if, and only if,pj ∼ M.If p2, p3 ∼ L (respectively,pσ2 , pσ3 ∼ L), thenp1, p2, p3 ∈ L⊥ ∩ M⊥ (resp.,p1, p

σ2 , pσ3 ∈ L⊥ ∩M⊥) andL⊥ ∩M⊥ is a line. Consequently,(L′)⊥ ∩ (M ′)⊥ is

a line. On the other hand, ifp2, pσ3 ∼ L (respectively,pσ2 , p3 ∼ L), thenp1, p2,pσ3 ∈ L⊥ ∩M⊥(p1, pσ2 , p3 ∈ L⊥ ∩M⊥) and(L′)⊥ ∩ (M ′)⊥ is a line, too.

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The following results show that Grassmann spaces with involutory automorph-isms whose quotients satisfy the previous proposition exist.

LEMMA 3.7. For every pair p, q of distinct points of a Grassmann spaceGr(h,P) = (P,L), d(p, q) = t if, and only if,dim(p ∩ q) = h− t .

Proof. We proceed by induction ont > 1. The caset = 1 is trivial. Supposet > 1 and the assumption true for anyw < t . If d(p, q) = t , then there exists aminimum chain of points ofP p = p1, p2, . . . , pt , pt+1 = q such thatpi ∼ pi+1.Clearly, it isd(p, pt ) = t−1 so, by induction hypothesis, dim(p∩pt) = h− t+1.Frompt ∼ pt+1 = q it follows dim(q ∩ pt) = h − 1, hence dim((p ∩ pt) ∩ (q ∩pt)) > h− t . Moreover, dim(p ∩ q) > h− t , since(p∩pt)∩ (q ∩pt) ⊆ p ∩ q. Ifdim(p∩q) = h−w > h−t , then, by the induction hypothesis,d(p, q) = w < t , acontradiction. It follows that dim(p∩q) = h− t . Conversely, if dim(p∩q) = h− tandd(p, q) = t ′, by the necessary condition just proved, it is dim(p∩q) = h−t ′ =h− t , henced(p, q) = t .

LetP = PG(2d − 1,K) be a (2d − 1)-dimension projective space over a skew-fieldK. A polarity σ of P having Witt index at mostd − 5 transforms every (d −1)-subspaceS of P into a (d − 1)-subspace6 of the dual projective spaceP∗,where6 is the family of hyperplanes ofP containing a (d − 1)-subspaceS ′. Itfollows thatσ induces an involutory automorphism of the Grassmann spaceG =Gr(2d−1, d−1,K). Sinceσ has Witt index at mostd−5, for every pointp ofGthe subspacep ∩ pσ has codimension at least 5 inp, hence dim(p ∩ pσ) 6 d − 6and, by Lemma 3.5,d(p, pσ ) > 5. Finally,G is a semilinear connected spacewhose subspaces have finite rank satisfying Propositions (3.5) and (3.6), hence thefollowing result holds.

PROPOSITION 3.8.If σ is an involutory automorphism of a Grassmann spaceGr(2d − 1, d − 1,K) induced by a polarity ofPG(2d − 1,K) of Witt index atmostd − 5, then the quotientGr(2d − 1, d − 1,K)/〈σ 〉 satisfies the hypothesis ofTheorem4.

4. A Characterization of Gr(1,PP)

In this section,(P,L) denotes a semilinear space satisfying properties (I) and (II).

PROPOSITION 4.1.If S andS ′ are two maximal subspaces of(P,L) intersectingin a lineL, then eitherS or S ′ is a projective plane.

Proof. Suppose thatS contains a lineM nonintersectingL and S ′ containsa lineN nonintersectingL. Hence,M andN are nonintersecting lines and, byProposition 2.3, they are not collinear, butM⊥ ∩N⊥ containsL, contradicting (II).Hence, either any line ofS or any line ofS ′ meetsL and the assumption followsfrom Proposition 2.8.

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Let S0 be a fixed maximal subspace of(P,L) and suppose thatS0 has rankr > 3, in the case(P,L) contains subspaces of rank greater than 2. The followingresult is easily proved.

PROPOSITION 4.2.Every maximal subspace mettingS0 in a line is a projectiveplane.

Proof. If S0 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, ifS0 has rankr > 3andS is a maximal subspace of(P,L) intersectingS0 at a line, by Proposition 4.1,S is a projective plane.

Let6 be the family of maximal subspaces of(P,L) whose elements areS0 andall maximal subspaces intersectingS0 at a point, and callstar every maximal sub-space of6. The following Proposition shows that6 contains maximal subspacesdifferent fromS0.

PROPOSITION 4.3.6\S0 is nonempty.Proof.LetL be a line ofS0 andS the maximal subspace passing throughL and

different fromS0. If M is a line ofS intersectingL at a pointx then the maximalsubspaceS ′ passing throughM and different fromS intersectsS0 at the pointx.Indeed, if(S0 ∩ S ′) − {x} would contain a pointy collinear with the two linesLandM 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 andS ′ is a maximal subspace of(P,L) intersectingS at a point,

thenS ′ is a star, too.

Proof. (i) Let S andS ′ be two different stars of6. If either S or S ′ is S0, theproof is trivial, so we can assume that bothS andS ′ are different fromS0. Let x bethe pointS ∩ S0 andy the pointS ′ ∩ S0 and suppose thatx = y. If S ∩ S ′ is a lineR, then a pointz of R\{x} is collinear with a line ofS0 (by Proposition 2.4) andthere are at least three distinct maximal subspaces passing throughR, contradictingProposition 2.6. Hence,x andy are different. IfS∩S ′ is a lineL, thenx is collinearwith y and the lineL of S ′, a contradiction. Suppose thatS andS ′ are disjoint. ByProposition 2.4,y⊥∩S is a lineM passing throughx andx⊥∩S ′ is a lineM ′ passingthroughy. The linesM andM ′ are clearly disjoint. IfM is not collinear withM ′,thenM⊥∩(M ′)⊥ = N , andS1 = S(x,M ′), S2 = S(y,M) andS0 are three pairwisedistinct maximal subspaces throughN , a contradiction, by Proposition 2.6. Hence,the linesM andM ′ are collinear and, by Proposition 4.1, the maximal subspaceS∗containingM,M ′ andN is a projective plane, a contradiction, sinceS∗ containsthe two nonintersecting linesM andM ′.

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(ii) If either S = S0 or S ′ = S0 the proof is trivial. If bothS andS ′ are differentfrom S0, we can substituteS andS0 and the proof is the same of case (i).

LEMMA 4.5. 6 is a covering of maximal subspaces of(P,L).Proof.Letp be a point ofP not inS0 and suppose thatp⊥∩S0 is empty. A fixed

lineM throughp and a fixed lineL of S0 are noncollinear and nonintersecting, soL⊥∩M⊥ contains at least a pointq not onL∪M. Moreover, the pointq does not lieonS0, since it is collinear withp. The maximal subspaceS = S(q,L) containingq andL is different fromS0 and it is a projective plane, by Proposition 4.2. Sincep ∼ q, p⊥ ∩ S is a lineN throughq intersectingL at a point, contradicting theassumptionp⊥ ∩ S0 = Ø. It follows thatp⊥ ∩ S0 is a lineL0 and the maximalsubspaceS ′ = S(p,L0) is a projective plane. LetR be a line ofS ′ passing throughp and intersectingL0 at a pointx and letS ′′ be the maximal subspace passingthroughR and different fromS ′. The subspaceS ′′ intersectsS0 exactly at the pointx, otherwise a point ofS ′′ ∩ S0 different fromx would be collinear withR andL0

in S ′, contradicting Proposition 2.4. HenceS ′′ is a star passing throughp, and theproof is complete.

Finally, we can characterize the incidence geometry of Grassmann spacesGr(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 projectivespace.

Proof. By Proposition 2.1, no line of(P,L) is a maximal subspace and, byLemma 4.5,6 is a covering of maximal subspaces of(P,L). By Theorem 1 ofMelone–Olanda, it is sufficient to prove condition (∗). Let S be a star of(P,L)andp be a point not onS. By Lemma 4.5, there is a star Sp passing throughpand intersectingS at a pointx (by case (i) of Lemma 4.4). From Proposition 2.4,it follows thatp⊥ ∩ S is a lineL throughx. By case (i) of Lemma 4.4, every starpassing throughp intersectsS at a point ofL. Viceversa, ifz is a point ofL andMis the linep ∨ z, thenM is contained exactly in two maximal subspacesS1 eS2. IfbothS1 andS2 intersectS at lines, thenS1 ∩ S = S2 ∩ S = L, since the two linesS1∩S andS2∩S are collinear withp. In this case the lineLwould be contained intothree pairwise distinct maximal subspaces, contradicting Proposition 2.6. HenceeitherS1 or S2 intersectsS at the pointz, supposeS1. By case (ii) of Lemma 4.4,S1 is a star.

5. The Proof of Theorem 4

In this section(P,L) will be a semilinear and connected space satisfying condi-tions (I) and (II′). The following results hold.

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PROPOSITION 5.1.If (P,L) is a polar space of rankm > 3 satisfying(I) and(II ′), then(P,L) is isomorphic to the Grassmann spaceGr(3,1,K), whereK is askew-field.

Proof. By Proposition 2.8, every maximal subspace of(P,L) is a projectivespace. LetS0 be a fixed maximal subspace of(P,L) having rankr = max{m−1,2}and let6 be the family of all maximal subspaces of(P,L) whose elements areS0

and all maximal subspaces intersectingS0 at a point. Observe that Proposition 4.3and Lemma 4.4 hold and Lemma 4.5 holds too, since in the polar space(P,L) thecasep⊥ ∩ S0 = Ø is always false. Arguing as in the Proof of Theorem 4.6, weprove that(P,L) is isomorphic to the Grassmann space Gr(1,P) of the lines of aprojective spaceP. Clearly, ifm > 3 then Gr(1,P) contains a pointp and a lineLsuch thatp⊥ ∩ L = Ø, hencem = 3 and the proof is complete.

PROPOSITION 5.2.If L is a line andp is a point of(P,L) such thatp⊥ ∩ L isempty, thenp⊥ ∩ L⊥ is either empty or a line.

Proof. Suppose thatp⊥ ∩ L⊥ contains a pointx and denote byS the maximalsubspaceS(x,L). Thenp⊥∩S is a lineM throughx andM is contained inp⊥∩L⊥.If p⊥ ∩L⊥ contains a pointy not onM, theny does not lie onS, otherwisep⊥ ∩Scontainsy andM, contradicting Proposition 2.3. Moreover,y is not collinear withx, otherwisey would be collinear withx andL in S, a contradition, by Proposi-tion 2.3. The distance betweenx andy is clearly 2, butx⊥ ∩ y⊥ contains the pointp and the lineL such thatp⊥ ∩ 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 theplanes in a projective space,Ann. Discrete Math.14 (1982), 129–150.

2. Bichara, A. and Tallini, G.: On a characterization of the Grassmann manifold representing theh-dimensional subspaces in a projective space,Ann. Discrete Math.18 (1983), 113–132.

3. Cohen, A. M.: On a theorem of Cooperstein,European J. Combin.4 (1983), 107–126.4. Cooperstein, B. N.: A characterization of some Lie incidence structures,Geom. Dedicata6

(1977), 205–258.5. Lo Re, P. M. and Olanda, D.: Grassmann spaces,J. Geom.17 (1981), 50–60.6. Melone, N. and Olanda, D.: A characteristic property of the Grassmann manifold representing

lines of a projective space,European J. Combin.5 (1984), 323–330.7. Shult, E. E.: A remark on Grassmann spaces and Half-spin geometries,European J. Combin.

15 (1994), 47–52.8. Tallini, G.: On a characterization of the Grassmann Manifold representing the lines in a project-

ive 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. 354–358.

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On the Incidence Geometry of Grassmann Spaces

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