J. Geom. 93 (2009), 62–70 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0047-2468/010062-9, published online 15 April 2009 DOI 10.1007/s00022-009-2059-y
Journal of Geometry
Embeddings of Affine Grassmann Spaces Eva Ferrara Dentice and Pia Maria Lo Re Abstract. In this paper we prove that if a Grassmann space Δ = GrA (m, h, K) of the h–subspaces of an affine space A = AG(m, K) has an embedding e into a projective space P G(n, K ) over a skew–field K , and e satisfies two suitable conditions (α) and (β), then K and K are isomorphic fields and Δe is, up to projections, an affine Grassmannian. Mathematics Subject Classification (2000). 51A45; 51M35. Keywords. Projective and affine Grassmann spaces, embeddings.
1. Introduction Two distinct points p and q of a partial linear space Γ = (P, L) are collinear if there exists a line L ∈ L containing them. The symbol p ∼ q means that p and q are collinear. For convenience, we also say that p is collinear to itself. More generally, two subsets X and Y are collinear (X ∼ Y ) if each point of one of them is collinear with every point of the other. Γ is connected if for every pair p, q of points of P there exists a finite chain of points p1 = p, p2 , . . . , pt = q such that pi ∼ pi+1 , for i = 1, . . . , t − 1. A subspace of Γ is a subset W of P such that for every two collinear points of W the line joining them is contained in W . Clearly, any intersection of subspaces is a subspace and P is a subspace of Γ (the improper one), thus it is possible to define the closure [X] of a subset X of P as the intersection of all subspaces of Γ containing X. Moreover, a singular subspace of Γ is a subspace consisting of pairwise collinear points. In the paper [3], lax projective embeddings of connected partial linear spaces into projective spaces and related questions are considered. According to [3], a lax projective embedding e : Γ −→ P(K) of a connected partial linear space Γ = (PΓ , LΓ ) into a projective space P(K) = (PP , LP ) over a skew–field K is an injective mapping e : PΓ −→ PP satisfying the following properties:
Vol. 93 (2009)
Embeddings of Affine Grassmann Spaces
63
(L1 ) [PΓe ] = PP ; (L2 ) every line L ∈ LΓ satisfies [Le ] ∈ LP ; (L3 ) no two distinct lines of Γ are mapped by e into the same line of P(K). If moreover Le is a line of P(K) for every line L of Γ, then e is said to be full. Bichara and Mazzocca characterized in [1] and [2] the Grassmann space of index h of an affine space A of dimension m ≥ 3, where 1 ≤ h ≤ m − 2. This is the partial linear space Gr(h, A) whose points are the h–subspaces of A and whose lines are the proper and improper pencils of h–subspaces of A, a proper pencil being a set of all h–subspaces passing through a fixed (h − 1)–subspace and contained in a fixed incident (h + 1)–subspace, and an improper pencil being a set of all pairwise parallel h–subspaces contained in a fixed (h + 1)–subspace. According to [2], a proper pencil of h–subspaces of A will be called a line of the first kind of Gr(h, A), and an improper pencil will be called a line of the second kind. It follows that the lines of Gr(h, A) are partitioned into two disjoint subsets L1 and L2 , consisting of lines of the first kind and of the second kind, respectively. Two points of Gr(h, A) are collinear if, and only if, they are either two h–subspaces intersecting at a (h − 1)–subspace, or two parallel h–subspaces of A, and the line through them is a line of the first or of the second kind, respectively. It is easy to see that if we denote by P the projective extension of A and by H∞ the hyperplane at infinity of A (removed from P in order to obtain A), then Gr(h, A) is the incidence structure induced by the well known Grassmann space Gr(h, P) of the h–subspaces of P on the set of all h–subspaces of P not in H∞ . Let F (A, B) be the line of Gr(h, P) coinciding with the pencil of h–subspaces of P passing through a (h − 1)–subspace A and contained in a (h + 1)–subspace B. If A ⊆ H∞ and B ⊆ H∞ , then F (A, B) \ {B ∩ H∞ } is a line of the second kind of Gr(h, A) and, conversely, every line of the second kind is F (A, B) \ {B ∩ H∞ }, with A ⊆ H∞ and B ⊆ H∞ . In the sequel, for every pair (A, B) consisting of a (h − 1)–subspace A contained in a (h + 1)–subspace B of P, F ∗ (A, B) will denote either F (A, B) ∈ L1 or F (A, B) \ {B ∩ H∞ } ∈ L2 , according to A, B ⊆ H∞ or A ⊆ H∞ and B ⊆ H∞ . If A is an affine space AG(m, K) of finite dimension m over a skew–field K, then the Grassmann space Gr(h, A) is also denoted by GrA (m, h, K). If K is a field, then the Grassmann space Gr(m, h, K) of the h–subspaces of a projective space P G(m, K) has a full projectiveembedding ℘ , the Pl¨ ucker morphism, into a pro − 1, and the image Gr(m, h, K)℘ = Gm,h,K jective space P G(M, K), M = m+1 h+1 is the so–called projective Grassmannian of the h–subspaces of P G(m, K). It follows that, if A = AG(m, K) is the affine space obtained by P G(m, K) by removing a fixed hyperplane H∞ , then the restriction of ℘ to the Grassmann space GrA (m, h, K) clearly provides a lax projective embedding of GrA (m, h, K) into P G(M, K). Indeed, the image GrA (m, h, K)℘ is a proper subset Am,h,K of Gm,h,K spanning P G(M, K) and such that, for every line of Gm,h,K , either is contained in Am,h,K or there exists a point x ∈ \ Am,h,K such that \ {x} is contained in
64
E. Ferrara Dentice and P. M. Lo Re
J. Geom.
Am,h,K . In the sequel, in the case K is a field, we will refer to Am,h,K as the affine Grassmannian of the h–subspaces of AG(m, K). In this paper, the following theorem is proved. Theorem 1.1. Let GrA (m, h, K) be the Grassmann space of the h–subspaces of an affine space A = AG(m, K) of finite dimension m ≥ 3 over a skew–field K different from GF (2), 1 ≤ h ≤ m−2, let L1 and L2 be the sets of lines of the first and of the second kind of GrA (m, h, K), respectively, and let e : GrA (m, h, K) −→ P G(n, K ) be a lax projective embedding, K a skew–field, satisfying the following conditions: (α) For every line L of GrA (m, h, K), Le either is a line of P G(n, K ) or is a punctured line 1 of P G(n, K ), according to L ∈ L1 or L ∈ L2 . (β) For every pair (L, M ) of lines of GrA (m, h, K), if L = F ∗ (A, B), M = F ∗ (C, D) and [Le ] ∩ [M e ] = ∅, then A ∨ C ⊆ B ∩ D. Then K and K are isomorphic fields and GrA (m, h, K)e is, up to projections, an affine Grassmannian over K .
2. The proof of Theorem 1.1 Let Δ = GrA (m, h, K) be the Grassmann space of the h–subspaces of an affine space A = AG(m, K) over a skew–field K, let PΔ be the point–set of Δ, let L1 and L2 be the sets of lines of the first and of the second kind of Δ, respectively, and let e : Δ −→ P G(n, K ) be a lax projective embedding, K a skew–field, satisfying conditions (α) and (β). It is easy to see that GrA (m, h, K) contains three pairwise disjoint families of maximal singular subspaces, say Σ1 , Σ2 and T . Every element of Σ1 is the family of all h–subspaces passing through a fixed (h − 1)–subspace (proper star), every element of Σ2 is the family of all pairwise parallel h–subspaces (improper star), and, finally, every element of T is the family of all h–subspaces contained in a fixed (h + 1)–subspace of AG(m, K) (dual star, or, in the case h = 1, ruled plane). From the intersection properties of the families of maximal singular subspaces of GrA (m, h, K) described in [2], we have that every line of the first kind (respectively, of the second kind) is contained in exactly one maximal singular subspace of Σ1 (resp. Σ2 ) and exactly one maximal singular subspace of T . Furthermore, every proper star is isomorphic to a projective space P G(m − h, K), every improper star is isomorphic to an affine space AG(m − h, K) and every dual star is isomorphic to a punctured projective space of dimension h + 1 over the opposite skew–field K op . Since every singular subspace of GrA (m, h, K) is contained in a maximal singular subspace, it follows that every singular subspace of GrA (m, h, K) is either a projective space, or a punctured projective space, or an affine space, all coordinatized by either the skew–field K or the opposite skew–field K op . 1A
punctured projective space is a projective space with one point deleted.
Vol. 93 (2009)
Embeddings of Affine Grassmann Spaces
65
The incidence structure of the images via e of all singular subspaces of Δ is established by the following result. Proposition 2.1. For every singular subspace S of Δ, S e is either a projective subspace of P G(n, K ), or a punctured projective subspace of P G(n, K ), or a projective subspace of P G(n, K ) with one hyperplane deleted. Proof. Because of the injectivity of e, for every singular subspace S of Δ the point– line geometry (S, L(S)), whose points are those of S and whose lines are the lines of Δ contained in S, is isomorphic to (S e , L(S)e := {Le | L ∈ L(S)}). It follows that (S e , L(S)e ) is either a projective space, or a punctured projective space, or an affine space, all coordinatized by either K or K op and contained in P G(n, K ). If S e is a projective space, then S is a projective space and every line of L(S) is a line of the first kind of Δ. From condition (α), it follows that [Le ] = Le ⊆ S e , for every L ∈ L(S). For every two distinct points xe and y e of S e , let R be the line of L(S) passing through x and y. From property (L2 ), the line of P G(n, K ) joining xe and y e coincides with [Re ]. Hence [Re ] = Re ⊆ S e , and S e is a projective subspace of P G(n, K ). This implies that K and K are isomorphic skew–fields. If S e is a punctured projective space, then S is a punctured projective space, thus we can suppose that S = P G(d, K op )\{∞} and the lines of the second kind of L(S) are precisely the lines of P G(d, K op ) passing through ∞. From condition (α), every line of the second kind of L(S) is mapped by e into a line of S e coinciding with a punctured line of P G(n, K ). Vice versa, for every line Le of L(S)e coinciding with a punctured line of P G(n, K ), L is a line of the second kind of S. It follows that e := S e ∪{λ} all the punctured lines of S e have the same hole λ ∈ P G(n, K ) and S op e , either is a projective space P G(d, K ). For every two distinct points p, q ∈ S e e e there exist x, y ∈ S such that p = x and q = y , or p = x and q = λ. In the first case, from condition (L2 ), the line of P G(n, K ) passing through p and q coincides with [Le ], L being the line of L(S) passing through x and y. It follows that either [Le ] = Le or [Le ] = Le ∪ {λ}, according to L is a line of the first kind or of the e . In the second case, the line of P G(n, K ) passing second kind, and [Le ] ⊆ S e passing through through p and q coincides with the line of the projective space S e e x and λ, thus it contains at least a further point z , z ∈ S \ {x}. Denoted by M the line of L(S) passing through x and z, M is a line of the second kind and [M e ] = M e ∪ {λ} is the line of P G(n, K ) passing through p and q. It follows that e , hence S e is a projective subspace of P G(n, K ). This implies that K op [M e ] ⊆ S and K are isomorphic skew–fields. Finally, let us suppose that S e is an affine space contained in P G(n, K ), i.e. S is an affine space AG(d, K). Clearly, every line of L(S) is a line of the second kind, hence, from condition (α), every line of S e is a punctured line of P G(n, K ). Let Λ be the subset of all points of P G(n, K ) coinciding with the holes of the lines Le of S e , for every L ∈ L(S). Since |K| > 2, Λ coincides with the set of all directions of the lines of the affine space S e , hence the set S e = S e ∪ Λ is a projective space
66
E. Ferrara Dentice and P. M. Lo Re
J. Geom.
(namely, S e is the projective extension of S e , and Λ is the hyperplane at infinity of S e ). For every two distinct points p, q ∈ S e , one of the following cases occour. (i) There exist x, y ∈ S such that p = xe and q = y e . In this case, from condition (L2 ), the line of P G(n, K ) passing through p and q coincides with the line [Re ], R being the line of L(S) passing through x and y. Since R is a line of the second kind, Re is a punctured line of L(S)e , whose hole is a point of Λ. It follows that [Re ] ⊆ S e . (ii) There exists x ∈ S such that p = xe and q ∈ Λ. In this case, the line of P G(n, K ) passing through p and q coincides with the line of the projective space S e passing through xe and q, thus it contains at least a further point z e , z ∈ S \ {x}. Denoted by M the line of L(S) passing through x and z, [M e ] = M e ∪ {q} is the line of P G(n, K ) passing through p and q. It follows that [M e ] ⊆ S e . (iii) p, q ∈ Λ. Denoted by x a fixed point of S, from the case (ii), the two lines of P G(n, K ) passing through xe and p and xe and q, respectively, are two lines [Le ] = Le ∪ {p} and [M e ] = M e ∪ {q}, being L and M two lines of L(S) passing through x. It is easy to see that the projective plane π of P G(n, K ) spanned by p, q and xe intersects S e at an affine plane σ. Since |K| > 2, the line at infinity of σ coincides with the line of P G(n, K ) passing through p and q. It follows that ⊆ Λ, hence ⊆ Se. From cases (i), (ii), (iii), every line passing through any two distinct points p, q ∈ S e is contained in S e , hence S e is a projective subspace of P G(n, K ). Furthermore, a fundamental consequence of the proof of the previous result is the following. Proposition 2.2. K and K are isomorphic skew–fields and K op and K are isomorphic skew–fields. Hence K ∼ = K . = K op ∼ From the previous Proposition, in the sequel we can identify K and K . (The isomorphism K op ∼ = K implies now that K admits an anti-automorphism.) Let P = P G(m, K) be the projective extension of A, H∞ be the hyperplane at infinity of A and Γ = (PΓ , LΓ ) be the Grassmann space of the h–subspaces of P. Recall that Δ is the point–line geometry induced by Γ on the set of all h– subspaces of P not in H∞ . The goal of the paper is to construct a full embedding e : Γ −→ P G(n, K) coinciding with e on Δ. More precisely, we need to define in a suitable way the images of all points of Γ not in GrA (m, h, K). Proposition 2.3. Let X be a h–subspace of H∞ and B be a (h + 1)–subspace of P passing through X and not contained in H∞ . The family of all h–subspaces of B different from X is a dual star T of Δ and T e is a punctured projective subspace
Vol. 93 (2009)
Embeddings of Affine Grassmann Spaces
67
of P G(n, K), say Π \ {p}. The point p ∈ P G(n, K) is the common hole of all punctured projective subspaces which arise as images via e of all dual stars of Δ passing through X. Proof. Let B be a (h + 1)–subspace of P passing through X, not contained in H∞ and different from B, T be the dual star of Δ defined by B and Π \ {q} be the punctured projective space (T )e . Let A and A be two distinct (h − 1)– subspaces of X, L = F ∗ (A, B) and M = F ∗ (A , B ). From condition (α), Le and M e are punctured lines of Π and Π , respectively, with holes p and q, resp. The line R = F ∗ (A , B) is contained in T , thus Re ⊆ Π and p is the hole of Re . On the other hand, R is contained in the improper star SA of Δ consisting of all h–subspaces of A having the direction of A . More precisely, R and M are two disjoint lines of the affine plane σ of SA consisting of all those pairwise parallel h–subspaces of A having the direction of A and contained in (B ∨ B ) \ H∞ . From Proposition 2.1, it follows that σ e is an affine plane. Hence Re and M e have the same hole. Thus, both p and q are holes of the punctured line Re , and p = q. From Proposition 2.3, the hole of all those punctured projective subspaces of P G(n, K) defined by the dual stars of Δ having the same h–subspace at infinity X depends only on X, so it can be denoted by pX . Thus, the embedding e can be naturally extended to all points of the Grassmann space Γ which are not points of Δ. More precisely, the following mapping e : PΓ −→ P G(n, K) is well defined. e x if x ∈ PΔ e (2.1) x = pX if x = X is an h−subspace of H∞ . Proposition 2.4. For every two distinct h–subspaces X, Y of H∞ such that dim(X ∩ Y ) = h − 1 the points pX and pY of P G(n, K) are different. Proof. Let B be a (h+1)–subspace of P containing X and B be a (h+1)–subspace containing Y such that B, B ⊆ H∞ . From Proposition 2.3, there is no loss of generality in assuming that B and B intersect at a h–subspace W not contained in H∞ . Furthermore, let T and T be the dual stars of Δ defined by B and B , respectively. Then T e = Π \ {pX } and (T )e = Π \ {pY } are punctured projective subspaces of P G(n, K). Clearly, W e = pX , pY . If pX = pY , then pX ∨W e ⊆ Π∩Π , hence T e ∩ (T )e would contain a point Z e different from W e , a contradiction. Proposition 2.5. The mapping e : PΓ −→ P G(n, K) is injective.
Proof. Let X and Y be two distinct points of PΓ . In order to prove that X e = Y e , we examine the following cases. Case 1. X and Y are two distinct points of PΔ .
Then X e = X e and Y e = Y e are distinct, since e is injective.
68
E. Ferrara Dentice and P. M. Lo Re
J. Geom.
Case 2. X ∈ PΓ \ PΔ , Y ∈ PΔ and X and Y are two h–subspaces of P such that dim(X ∩ Y ) = h − 1. Let B be the (h + 1)–subspace X ∨ Y of P. Then the family of all h–subspaces of P contained in B and different from X is a dual star T of Δ, and T e is a punctured projective subspace Π \ {pX } of P G(n, K). It follows that X e = pX and Y e = Y e ∈ T e , hence X e = Y e . Case 3. X and Y are two distinct points of PΓ \ PΔ and X and Y are two h– subspaces of P such that dim(X ∩ Y ) = h − 1.
In this case Proposition 2.4 holds, hence X e = pX = pY = Y e . Case 4. X and Y are two distinct points of PΓ \ PΔ and X and Y are two h– subspaces of P such that dim(X ∩ Y ) ≤ h − 2. Let I be the subspace X ∩ Y of P, B be a (h + 1)–subspace of P passing through X and not contained in H∞ , x be a point of B \ X and D be the (h + 1)–subspace x∨Y . Furthermore, let A be a (h−1)–subspace contained in X and C be a (h−1)– subspace contained in Y such that A ∩ C = I. It follows that dim(A ∨ C) ≥ h and dim(B∩D) ≤ h−1. Let L and M be the lines F ∗ (A, B) and F ∗ (C, D), respectively. From condition (α) it follows that [Le ] = Le ∪ {pX } and [M e ] = M e ∪ {pY }. If pX = pY , then [Le ] ∩ [M e ] = {pX }, contradicting condition (β). It follows that pX = pY , i.e. X e = Y e . Case 5. X ∈ PΓ \ PΔ , Y ∈ PΔ , and X and Y are two h–subspaces of P such that dim(X ∩ Y ) ≤ h − 2. Let I be the subspace X ∩ Y of P, y be a point of Y \ H∞ and B be the (h + 1)– subspace y ∨ X of P. Clearly, B ∩ Y ⊆ H∞ , hence dim(B ∩ Y ) = dim ((B ∩ Y ) ∩ H∞ ) + 1. From (B ∩ Y ) ∩ H∞ = (B ∩ H∞ ) ∩ Y = X ∩ Y = I and y ∨ I ⊆ B ∩ Y , it follows that B ∩ Y = y ∨ I. Moreover, let x be a point of B \ (X ∪ Y ) and D = x ∨ Y . Since Y is a hyperplane of D, it is dim(B ∩ D) = dim(B ∩ Y ) + 1. From x ∨ y ∨ I ⊆ B ∩ D, it follows that B ∩ D = x ∨ y ∨ I. Hence dim(B ∩ D) ≤ h . (2.2) Let A be a (h − 1)–subspace of X, and let us consider the two lines L = F ∗ (A, B) and M = F ∗ (Y ∩ H∞ , D) of L2 . From condition (α), it is [Le ] = Le ∪ {pX } and [M e ] = M e ∪ {pD∩H∞ } (clearly, Y e ∈ M e ). Suppose, to the contrary, that X e = Y e , i. e. pX = Y e . It follows that [Le ] ∩ [M e ] = ∅, thus, from condition (β), A ∨ (Y ∩ H∞ ) ⊆ B ∩ D .
(2.3)
Now, it is dim(A∨(Y ∩H∞ )) = 2h−2 − dim(A∩Y ), hence, from A∩Y ⊆ X∩Y = I, it follows that dim A ∨ (Y ∩ H∞ ) ≥ h . (2.4) From (2), (3) and (4) it follows that A ∨ (Y ∩ H∞ ) = B ∩ D, a contradiction, since A ∨ (Y ∩ H∞ ) ⊆ H∞ and B ∩ D ⊆ H∞ .
Vol. 93 (2009)
Embeddings of Affine Grassmann Spaces
69
Now, taking into account (2.1) and Proposition 2.4, we determine the image of all lines of Γ which are not lines of Δ. Proposition 2.6. Let F (A, B) be a line of Γ such that A ⊆ B ⊆ H∞ , and let X, Y be two distinct points of F (A, B). Then the points of the line pX ∨ pY of P G(n, K) correspond bijectively to all points pZ , Z ∈ F (A, B). Proof. From Proposition 2.3, we can certainly assume that B is a (h+1)–subspace of P containing X and B is a (h + 1)–subspace containing Y such that B ∩ B = W ⊆ H∞ . The family of all pairwise parallel h–subspaces of A having the direction of A and contained in the (h + 2)–dimensional affine space (B ∨ B ) \ H∞ provides an affine plane σ contained in the improper star SA ∈ Σ2 . Let Z be a h–subspace of F (A, B) different from X and Y , and B be the (h + 1)–subspace Z ∨ W of P. The lines L = F ∗ (A, B ), M = F ∗ (A, B ) and N = F ∗ (A, B ) of L2 are contained in σ and contain W . It follows that Le , M e and N e are three pairwise distinct lines of the affine plane σ e passing through the point W e , hence the holes pX , pY and pZ are pairwise different (by Proposition 2.4), and they are collinear on the line at infinity of σ e . Conversely, let p be a point of the line pX ∨ pY of P G(n, K). The line of σ e passing through W e and having the direction of p is the image of a line R of σ. It follows ˜ passing through W and contained in B ∨B that there exists a (h+1)–subspace B ˜ \ {B ˜ ∩ H∞ }. Hence p = (B ˜ ∩ H∞ )e . such that R = F (A, B) From Proposition 2.6, the extension e of the embedding e is well defined on the lines of Γ not in Δ. More precisely, the following holds. ⎧ ∗ if A ⊆ H∞ (i.e. F ∗ (A,B) ∈ L1 ) ⎨ F (A,B)e e ∗ e F (A,B) = F (A,B) ∪ {pX} if A ⊆ X = B ∩ H∞ (i.e. F ∗ (A,B) ∈ L2 ) ⎩ if A = X ∩Y and X ∨ Y = B ⊆ H∞ . pX ∨ p Y
(2.5)
End of the proof. From (2.1), (2.5) and Proposition 2.5, the mapping e : Γ −→ P G(n, K) is a full projective embedding of Γ. From Havlicek [4] and Zanella [5], it follows that K is a field and Γe is either a projective Grassmannian or a proper projection of a projective Grassmannian. It follows that, Δe = Δe is either an affine Grassmannian over the field K, or a proper projection of an affine Grassmannian.
References [1] A. Bichara and F. Mazzocca, On a characterization of Grassmann space representing the lines in an affine space, Simon Stevin, A Quart. J. of Pure and Appl. Math. 56 (1982), 129–141.
70
E. Ferrara Dentice and P. M. Lo Re
J. Geom.
[2] A. Bichara and F. Mazzocca, On a characterization of the Grassmann spaces associated with an affine space, Ann. Discr. Math. 18 (1983), 95–112. [3] E. Ferrara Dentice, G. Marino, A. Pasini, Lax projective embeddings of polar spaces, Milan J. Math. 72 (2004), 335–377. [4] H. Havlicek, Zur Theorie linearer Abbildungen, I, II, J. Geom. 16 (1981), 152–180. [5] C. Zanella, Embeddings of Grassmann Spaces, J. Geom. 52 (1995), 193–201. Eva Ferrara Dentice Seconda Universit` a degli Studi di Napoli – S.U.N. Dipartimento di Matematica via Vivaldi, 43 I-81100 Caserta Italy e-mail:
[email protected] Pia Maria Lo Re Universit` a degli Studi di Napoli – “Federico II” Dipartimento di Matematica “R. Caccioppoli” via Cinthia I-80126 Napoli Italy e-mail:
[email protected] Received: 22 February 2008. Revised: 11 March 2009.
