Lax Projective Embeddings of Polar Spaces

Milan j. math. 72 (2004), 335–377

DOI 10.1007/s00032-004-0028-3

c© 2004 Birkhauser Verlag Basel/Switzerland Milan Journal of Mathematics

Lax Projective Embeddingsof Polar Spaces

Eva Ferrara Dentice, Giuseppe Marino andAntonio Pasini

Abstract. Let Γ be a non-degenerate polar space of rank n ≥ 3 whereall of its lines have at least three points. We prove that, if Γ admitsa lax embedding e : Γ → Σ in a projective space Σ defined over askewfield K, then Γ is a classical and defined over a sub-skewfield K0 ofK. Accordingly, Γ admits a full embedding e0 in a K0-projective spaceΣ0. We also prove that, under suitable hypotheses on e and e0, thereexists an embedding e : Σ0 → Σ such that ee0 = e and e preservesdimensions.

Mathematics Subject Classification (2000). Primary 51A45; Secondary51A50.

Keywords. Polar spaces, projective spaces, weak embeddings.

1. Introduction

1.1. Full and lax embeddings

A projective embedding of a connected point-line geometry Γ is an injectivemapping e from the point-set PΓ of Γ to the point-set of a desarguesianprojective space Σ such that

(E1) the image e(PΓ) of PΓ spans Σ,(E2) for every line L of Γ, e(L) spans a line of Σ,(E3) no two distinct lines of Γ are mapped by e into the same line of Σ.

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If moreover e(L) is a line of Σ for every line L of Γ, then e is said to befull. If e is non-full, or we don’t know if it is full or we don’t care of that,then we say that e is lax.

Full projective embeddings have been intensively studied, in particularfor point-line geometries arising either from buildings of Lie type or fromdiagram geometries of Petersen and tilde type. The interested reader is re-ferred to Ivanov [13] and Ivanov and Shpectorov [14] for the latter topic,but we are not going to insist on it here. We only warn that, as all lines ofa Petersen or tilde geometry have size 3, a projective embedding of such ageometry is a representation in an elementary abelian 2-group. This nat-urally leads to the investigation of representations in possibly non-abeliangroups generated by involutions (Ivanov and Shpectorov [14]), whence totechniques and problems fairly different from those which we usually meetin the literature on projective embeddings of Lie-type geometries. An at-tempt to fuse both approaches in a unique more general perspective is donein [20], but we will not expose it here.

Turning to geometries of Lie type, full embeddings of polar spacesare well understood since a long time (Tits [35, chapter 8], Buekenhoutand Lefevre [2], Dienst [9]; see also Van Maldeghem [36, section 8.5] fora survey of the case of generalized quadrangles, and Johnson [15] for thegeneral case). Less is known on full embeddings of generalized n-gons withn > 4, in particular generalized hexagons and generalized octagons, even inthe finite case. We refer to Van Maldeghem [36, section 8.6] for informationson the cases of n = 6 and n = 8. Not so much has lately been added towhat one can find in [36]; we only quote a few papers by Thas and VanMaldeghem, namely [33] (a series of three papers, devoted to hexagons oforder (q, 1)) and [34], devoted to finite dual Cayley hexagons.

All full projective embeddings of the (embeddable) finite dual polarspaces are known except for the dual of the hermitian variety H(2n−1, 22)of PG(2n− 1, 4), n ≥ 4. We refer to Cooperstein and Shult [7] for a surveyof this topic. For dual polar spaces of spin type in characteristic �= 2, allfull embeddings are known even in the infinite case (Wells [37]); similarlyfor half-spin geometries and grassmannian of PG(n, K), K a field (Wells[37]).

In the above, when we say that all full embeddings are known fora class C of geometries, we mean that every geometry Γ ∈ C admits theuniversal embedding, namely a full embedding euniv from which all fullembeddings of Γ can be obtained as projections, and we know what euniv

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is. (See Subsection 1.2 for a definition of projections.) In principle, theexistence of the universal embedding is not necessary for the classificationof full embeddings of a given geometry Γ, but in practice, if Γ does notadmit any universal embedding, that classification can be very difficult, ifnot hopeless. So, it is very important to have some conditions sufficientfor the existence of universal embeddings. Conditions of this kind havebeen found by Kasikova and Shult [16] (see also Blok and Pasini [1] foran addition to [16], suited for a few cases where the theory of [16] cannotbe applied). Once we know that a given geometry Γ admits the universalembedding (for instance, Γ satisfies the conditions of [16]), we might stilllack a description of that embedding. The following is the usual way toprovide such a description: suppose we also know that Γ admits a finitespanning set of size d+1 and we are aware of a particular embedding e : Γ →PG(d, K). Then e is universal. Thus, the search for universal embeddingsleads to the investigation of spanning sets. We refer to Cooperstein [6] forthe state of the art in this latter trend.

So far for full embeddings. The literature on non-full embeddings isless rich. However, non-full embeddings are met quite often, in variouscontexts. For instance, every affine embedding can be regarded as a non-fullprojective embedding; when we draw a (q + 1) × (q + 1) grid on a sheet ofpaper, we give a non-full projective embedding of the hyperbolic quadricQ+(3, q) in the projective plane PG(2, R); also, every Baer subgeometry ofPG(d, q2) entails a non-full projective embedding of PG(d, q) in PG(d, q2).More significant examples arise in connection with twisted subgeometriesof Lie-type geometries. Two examples of this kind are described below.

Example 1.1. Let Γ be the dual of the hermitian polar space H(2n− 1, q2).The natural embedding of the grassmannian of n-spaces of V = V (2n, q2)in PG(V ∧V ) induces a lax embedding e : Γ → PG(V ∧V ). The embeddinge entails a full embedding e0 of Γ in a suitable subgeometry Σ of PG(V ∧V ),where Σ ∼= PG(N−1, q), N =

(2nn

)(see Garosi [10, section 4.4] for an explicit

description of a basis of Σ). Actually, when q > 2, e0 is the universal fullembedding of Γ (see Cooperstein and Shult [7, 5.2]).

Example 1.2. Let ∆ be the building of type D4 over GF(q3), Γ be itsmetasymplectic space and τ a triality of ∆. It is well known that Γ admitsa full embedding e in PG(N − 1, q3), where N = 28 or 27 according towhether q is odd or even. The subgeometry Γτ formed by the points andlines of Γ fixed by τ is a generalized hexagon of order (q, q3), denoted byT (q, q3) in [36]. The full embedding e induces a lax embedding eτ of Γτ

338 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

in a subgeometry of PG(N − 1, q3). The above construction can also berephrased as follows: Let e∗ be the ordinary full embedding of the dualΓ∗

τ∼= T (q3, q) of Γτ in Σ = PG(7, q3). As e∗ is ideal in the sense of Van

Maldeghem [36, 8.5.1], the e∗-images of the points of Γ∗τ can be regarded as

point-plane flags of Σ. So, we can compose e∗ with the natural embedding ofthe line-grassmannian of Σ in PG(27, q3), thus obtaining eτ . Regretfully, wehave not been able to find any explicit information on eτ in the literature.So, we cannot say so much on it, here.

Lax embeddings often embody full embeddings (as in Example 1.1).This fact is crucial for the classification of families of lax embeddings: Sup-pose we want to classify lax embeddings, or at least some of them, for aclass of geometries for which all full embeddings are known; if we can provethat the considered lax embeddings embody full embeddings, then most ofour job is done. Admittedly, the word “embody” is too vague for the readercan be happy with it. We will fix its meaning in the next subsection.

1.2. A few definitions

Given a connected point-line geometry Γ, a vector space V defined over askewfield K, a subskewfield K0 of K and a vector space V0 defined overK0, we say that a lax embedding e : Γ → PG(V ) embodies an embeddinge0 : Γ → PG(V0) if, regarded V as a K0-vector space, there is a semilinearmapping f : V0 → V such that e = PG(f) · e0, where PG(f) stands forthe mapping from PG(V0) \ Ker(f) to PG(V ) induced by f . Note that fis uniquely determined modulo the choice of a scalar and an embedding ofK0 in K (Granai [11, section 2.4]; compare [22, proposition 9]). We call fthe morphism from e0 to e and we adopt the writing f : e0 → e.

If f : e0 → e is injective and sends K0-bases of V0 to K-bases of V , thenwe say that f is a scalar extension, also that f is faithful and e faithfullyembodies e0. (We warn that, in general, even if f is injective, it mighthappen that f(B) is K-dependent for some basis B of V0.) On the otherhand, when K0 = K then f is surjective, by property (E1) of embeddingsand the equality e = PG(f)·e0. In this case we call f a projection, also sayingthat e is a projection of e0, avoiding the word “embodies” in this context.(We warn that we are giving the word “projection” a very broad meaning,allowing a projection to involve automorphisms of K.) An isomorphism ofembeddings is a bijective projection.

Every morphism f : e0 → e splits as the composition f = fprojfext

of a scalar extension fext : e0 → e1 and a projection fproj : e1 → e. The

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intermediate embedding e1 is uniquely determined up to isomorphisms.Accordingly, both fext and fproj are uniquely determined, modulo isomor-phisms. Note that, when f is injective, fproj is an isomorphism if and onlyif f is faithful.

By [20], for every lax embedding e of Γ there exists a lax embedding eof Γ such that e is a projection of e and, for every embedding e′ of Γ, if e isa projection of e′ then e′ is a projection of e. The embedding e is uniquelydetermined up to isomorphisms and is called the hull of e. Following Tits[35, 8.5.2], we say that e is dominant if it is its own hull. (In the literature,dominant embeddings are sometimes called relatively universal; see Shult[23], for instance.)

We need one more definition. We only state it for polar spaces, referringthe reader to [36, 8.5] for a formulation suited to a larger class of geometries.Suppose that Γ is a non-degenerate polar space of rank n ≥ 2. For a pointp of Γ, let p∼ be the set of points of Γ collinar with p or equal to p. Werecall that p∼ is a geometric hyperplane of Γ. Following Van Maldeghem[36], we say that a lax projective embedding e : Γ → Σ is weak if

(W) e(p∼) spans a hyperplane of Σ, for every point p of Γ.

(Admittedly, the word “weak” is disguising, as weak embeddings don’t lookso ‘weak’ after all; words like “firm” might be more appropriate. However,as in the literature on lax embeddings the word “weak” is currently usedwith the above meaning, we have preferred to keep it.)

Remark 1.1. All full embeddings of non-degenerate polar spaces are weak.(See Subsection 3.2, properties (P2’) and (P3’).)

Remark 1.2. Non-weak embeddings are easy to produce. For instance, wecan obtain them by the following trick, essentially due to Thas and VanMaldeghem [31], [32]. Suppose that an embedding e0 : Γ → PG(d, K0) isgiven, where K0 is a field. If K0 admits a simple extension K1 of degreeh > d, let e be the embedding of PG(d, K0) in PG(d−1, K1) constructed inRemark 5.1 of Section 5. Then the composition e1 = ee0 is an embedding ofΓ in PG(d−1, K1). If every finite extension of K0 admits simple extensionsof arbitrarily large degrees (as when K0 is finite), then we can repeat theabove construction d − 2 times, thus obtaining a sequence of embeddingsek : Γ → PG(d − k, Kk). When Γ is a polar space of rank n, then ek isnon-weak for every k ≥ d − 2(n − 1) (compare Theorem 5.11). Note alsothat d−k can be smaller than n. In particular, ed−2 (which is the last termof this series of embeddings) embeds Γ in a plane.

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Remark 1.3. Weak embeddings of polar spaces have been introduced byLefevre ([17], [18], [19]), who however defined them by axioms differentfrom (W). She assumed the followings:(W’) For any two points p, x of Γ, if e(x) belongs to the span of e(p∼), thenx ∈ p∼;(W”) For any two lines L, M of Γ, if the lines of Σ spanned by e(L) ande(M) meet in a point y, then y ∈ e(PΓ).

When Γ is a non-degenerate polar space, (W) and (W’) are equivalent(recall that, if x �∈ p∼, then p∼∪{x} spans Γ), and (W”) follows from (W’)(Thas and Van Maldeghem [29, Lemma 2]). However, if geometries differentfrom polar spaces are considered, then the property corresponding to (W)(see [36, 8.5]) might not be equivalent to (W’) and the latter might haveno relation at all with (W”). In the literature prior to 1998 embeddingssatisfying (W’) but possibly not (W”) are usually called sub-weak (as in[29] and [25], for instance).

1.3. A selection of results on lax embeddings of polar spaces

We shall now focus on lax embeddings of polar spaces of finite rank ≥ 3,exposing the main results obtained so far on this topic. We shall not considergeneralized quadrangles here, since a complete exposition of the state of theart for lax embeddings of generalized quadrangles would take too long. Werefer the interested reader to Thas and Van Maldeghem [30] and [31] forthe finite case and Steinbach and Van Maldeghem [26] and [27] for thegeneral case. A survey of the main results of the above papers is also givenby Van Maldeghem [36, section 8.6]. We only warn that, contrary to whathappens for polar spaces of rank at least 3 (see Theorem 1.6), neither allgeneralized quadrangles that admit a lax embedding are classical, nor everylax embedding of generalized quadrangles embodies a full embedding. Werefer the interested reader to the final part of this subsection for a few moredetails.

In what follows Γ is a non-degenerate polar space of finite rank n ≥ 3,with at least three points on every line, V is a vector space of (possiblyinfinite) dimension d over a skewfield K, and e : Γ → PG(V ) is a laxprojective embedding. When Γ is classical, we denote by euniv and KΓ theuniversal full embedding and the underlying skewfield of Γ. It is easily seenthat KΓ is a sub-skewfield of K (see also Lemma 6.1 of this paper). Wealso recall that, if char(KΓ) �= 2, then euniv is the unique full embedding ofΓ. (Needless to say, “unique” means ‘unique up to isomorphism’.)

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Theorem 1.1 (Thas and Van Maldeghem [29]). Assume the following:(1) e is weak.(2) d is finite and K is a field (namely, it is commutative; accordingly, KΓ

is also a field).(3) Γ is classical. Moreover, if char(KΓ) = 2 and the image euniv(Γ) of Γby euniv is a quadric, then the associated quadratic form has defect δ ≤ 1and, if δ = 1 and KΓ is perfect, then every line of PG(V ) that meets e(Γ)in more than two points is fully contained in e(Γ).

Then e faithfully embodies euniv.

When Γ is finite the previous theorem can be stated in the followingmore concise way:

Theorem 1.2 (Thas and Van Maldeghem [29]). Assume that e is weak, Kis a field and Γ is finite. Then e faithfully embodies a full embedding of Γ.

The first part of the next theorem is taken from Steinbach and VanMaldeghem [26, Theorem 5.1.1]. The second claim is a rephrasing of themain theorem of Steinbach [25].

Theorem 1.3 (Steinbach and Van Maldeghem). Assume that e is weak andΓ is classical. Then e embodies euniv. Moreover, if euniv is the uniquefull embedding of Γ (as when euniv(Γ) arises from a sesquilinear form, forinstance), then the morphism f : euniv → e is injective.

Theorem 1.4 (Thas and Van Maldeghem [31]). Suppose that 4 ≤ d < ∞and K is finite (whence Γ is finite). Moreover, when Γ is of symplectic typeand char(KΓ) �= 2, assume that d ≥ 5. Then e embodies euniv.

Finally, we mention a theorem of [21], where dominant embeddingsof polar spaces of rank n ≥ 3 are characterized by means of a propertyslightly stronger than (W). This characterization will be crucial for theproof of Theorem 1.6 of this paper.

Theorem 1.5. The embedding e is dominant if and only if(D) for every geometric hyperplane H of Γ, the image e(H) of H by e spans

a hyperplane of PG(V ).

More on embeddings of generalized quadrangles. We shall now give someinformation on non-natural embeddings of classical quadrangles and pro-jective embeddings of non-classical quadrangles. We begin with a trivial

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remark: Q+(q) can be laxly embedded in every projective space Σ of di-mension d = 2 or 3 and order > q. Clearly, these embeddings have norelation with the natural embedding of Q+(q) in PG(3, q), except possiblywhen Σ is defined over an extension of GF(q).

All known finite generalized quadrangles of order (q − 1, q + 1) (typesT ∗

2 (O) and AS(q)) are subgeometries of AG(3, q), whence they also admita lax projective embedding in PG(3, q), induced by the natural embeddingof AG(3, q) in PG(3, q). The quadrangle AS(3) ∼= Q−(5, 2) is the onlyclassical example in this family. However, the natural projective embeddingof Q−(5, 2) in PG(5, 2) has nothing to do with the embedding of AS(3)in PG(3, 3) ⊃ AG(3, 3). As Q(4, 2) is a full subquadrangle of Q−(5, 2) ∼=AS(3), the embedding of AS(3) in PG(3, 3) entails an embedding of Q(4, 2)in PG(3, 3), which has no relation with the natural embeddings of Q(4, 2) ∼=W (2) in PG(4, 2) or PG(3, 2).

The embeddings of Q−(5, 2) and Q(4, 2) in PG(3, 3) mentioned aboveare projections of dominant exceptional embeddings of Q−(5, 2) and Q(4, 2)in PG(5, 3) and PG(4, 3) respectively. But we have more than this. IndeedQ−(5, 2) admits a (non-weak) embedding in PG(5, p) for any odd prime p,and Q(4, 2) admits a (weak) embedding in PG(5, K) for any prime fieldK �= GF(2) (see Van Maldeghem [36, section 8.6]).

The generalized quadrangles Q(4, 3) and H(3, 22) also admit excep-tional (but non-weak) embeddings, in PG(4, q) for q ≡ 1 mod 3 and, re-spectively, PG(3, p) for any prime p > 5 (see [36, 8.6.5] for Q(4, 3) and Thasand Van Maldeghem [28] for H(3, 22)).

However, if e is a weak projective embedding of a finite thick general-ized quadrangle Γ, then Γ is classical and either e embodies the universalfull projective embedding of Γ or Γ ∼= Q(4, 2) and e is one of the exceptionalembeddings of Q(4, 2) mentioned above (Thas and Van Maldeghem [30]).

Weak embeddings of infinite generalized quadrangles have been clas-sified, too (Steinbach and Van Maldeghem [26], [27]). If Γ is an infinitethick generalized quadrangle and e is a weak embedding of Γ, then Γ isMoufang and e can be explicitly described. In particular, if Γ is classicalthen e embodies euniv, except possibly when euniv embeds Γ in PG(3, K0)for a quaternion skewfied K0 (see [26] for more details on this case).

Finally, we warn that the statement of Theorem 1.5 fails to hold forgeneralized quadrangles. Indeed, the above mentioned exceptional embed-dings of Q−(5, 2), Q(4, 3) and H(3, 22) in PG(5, p), PG(4, q) and PG(3, p)respectively, are dominant but non-weak; whence they do not satisfy (D).

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1.4. The main results of this paper

The results of the previous subsection do not yet seal up the subject. Onthe contrary, a few questions naturally arise from them. For instance:

(1) Only finite polar spaces are considered in Theorem 1.4. What aboutthe infinite case? Moreover, it would be nice to get rid of the restrictiond ≥ 4 (d ≥ 5 in the symplectic case).

(2) Differently from the proof of Theorem 1.1, which is mainly geometric,the proof of Theorem 1.3 is essentially algebraic. It would be interestingto have a geometric proof of the latter, too.

(3) In the second part of Theorem 1.3, where it is assumed that euniv is theunique full embedding of Γ, one might wonder if e faithfully embodieseuniv.

We shall fully answer questions (1) and (3). As for the program of (2), wemake a first but far reaching step towards its accomplishment.

As in the previous subsection, in the sequel Γ is a non-degenerate polarspace of finite rank n ≥ 3, where all lines have at least three points. WhenΓ is classical, euniv is its universal full embedding and KΓ is the underlyingskewfield of Γ. The next theorem fully answers question (1). We shall proveit in Section 6.

Theorem 1.6. Suppose that Γ admits a lax projective embedding. Then allthe following hold:

(1) Γ is classical;(2) every lax projective embedding of Γ embodies euniv;(3) a lax projective embedding of Γ is dominant if and only if it faithfully

embodies euniv.

In the light of (1) of the previous theorem, for the rest of this subsectionwe assume that Γ is classical.

Our second main result is Theorem 1.8, to be stated below. In viewof it, we need the following definition. Let e0 : Γ → Σ0 and e : Γ → Σ betwo lax projective embeddings of Γ. We say that e matches with e0 if thefollowing holds:

(M) For any three maximal singular subspaces M1, M2, M3 of Γ such thatM1 ∩ M2 ∩M3 has rank equal to n − 1, if the span of e0(M1 ∪ M2) inΣ0 contains e0(M3), then the span of e(M1∪M2) in Σ contains e(M3).

The next fact, to be proved in Section 7, gives a motivation for the hy-potheses of Theorem 1.8.

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Fact 1.7. Every lax embedding of Γ matches with euniv.

We also recall that, if Γ admits at least two full embeddings (whichcan happen only when char(KΓ) = 2), then it admits a unique terminalfull embedding eterm, which embodies all full embeddings of Γ (see alsoSubsection 3). For, instance, if Γ ∼= Q(4, 2), then eterm embeds Γ in PG(3, 2)as W (2).

Theorem 1.8. Let e0 : Γ → Σ0 and e : Γ → Σ be projective embeddings of Γ,where e0 is full and e is weak. Moreover, when char(KΓ) = 2 and Γ admitsmore than one full projective embedding, suppose that e matches with e0

and e0 = eterm. Then e faithfully embodies e0.

This theorem answers question (3) in the affirmative and, combinedwith Theorem 1.6(3), it immediately implies the following:

Corollary 1.9. If euniv is the unique full projective embedding of Γ, then allweak embeddings of Γ are dominant and they are scalar extensions of euniv.

We shall prove Theorem 1.8 in Section 8. Our proof will be entirelygeometric; so, the program sketched in (2) is half-done, too. That proofamounts to construct an embedding e : Σ0 → Σ such that e = ee0. Thehypothesis that if euniv �= eterm then e0 = eterm, is essential for that con-struction. To finish the program of (2) we should remove that hypothesis,allowing e to be a morphism from Σ0 \R to Σ for a suitable subspace R ofΣ0, namely proving that e faithfully embodies a quotient of e0.

Organization of the paper. Section 2 contains a number of definitions tobe exploited in this paper and Section 3 contains a survey of well knownproperties of polar spaces. Sections 4 and 5 are preliminary to the proofs oftheorems 1.6 and 1.8. In particular, Section 4 is devoted to subgeometries ofprojective geometries whereas in Section 5 we collect a number of results onembeddings of projective spaces and polar spaces. Sections 6 and 7 containthe proof of Theorem 1.6 and of Fact 1.7, respectively. Section 8 is devotedto the proof of Theorem 1.8.

2. Basics on point-line geometries

In this paper, a point-line geometry is a pair Γ = (PΓ,LΓ) where PΓ (theset of points) is a nonempty set, LΓ is a collection of subsets of PΓ, calledlines, any two distinct points belong to at most one common line, every linecontains at least two points and every point belongs to at least one line.

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In other words, a point-line geometry is a semi-linear space where lines areregarded as sets of points.

2.1. Long and short lines, collinearity, connectedness

A line is said to be long (short) if it contains at least three (exactly two)points. Two distinct points are collinear if there exists a line containingboth of them. If two points p, q are collinear, then we write p ∼ q. Thegraph (PΓ,∼) is the collinearity graph of Γ. We say that Γ is connected ifits collinearity graph is connected.

For a point p, we denote by p∼ the set of points collinear with p orequal to p and, for X ⊆ PΓ, we put X∼ = ∩p∈Xp∼. (We warn that manyauthors write p⊥ and X⊥ instead of p∼ and X∼; however, as in this paperthe symbol ⊥ will be used to denote the ortogonality relation associated toa sesquilinear form of a vector space, we have replaced p⊥ and X⊥ with p∼

and X∼.) If P∼Γ = PΓ, namely all points of Γ are mutually collinear, then

Γ is called a linear space.

2.2. Subspaces, hyperplanes, bases and ranks

A subspace of a point-line geometry Γ = (PΓ,LΓ) is a subset S ⊆ PΓ suchthat, for any two collinear points of S, the line joining them is containedin S. Any intersection of subspaces is a subspace and PΓ is a subspace of Γ(the improper one). So, we can define the span 〈X〉Γ of a subset X ⊆ PΓ asthe intersection of all subspaces containing X. In particular, for two distinctpoints p, q, their span 〈p, q〉Γ is either {p, q} (if p �∼ q) or the line containing{p, q} (if p ∼ q).

A subspace S of Γ is said to be singular if any two points of S arecollinear, namely S ⊆ S∼. Every singular subspace of Γ is contained in amaximal one.

A (geometric) hyperplane of Γ is a proper subspace meeting every lineof Γ non-trivially. Note that, in general, not every maximal subspace is ahyperplane and not every hyperplane is maximal as a subspace. However,for a hyperplane H of Γ, if the graph induced by ∼ on PΓ \H is connected,then H is a maximal subspace of Γ.

A subset X ⊆ PΓ is said to be independent if 〈Y 〉Γ ⊂ 〈X〉Γ for everyproper subset Y ⊂ X. An independent spanning set of a subspace S iscalled a basis of S. Clearly, every line L ∈ LΓ admits a basis: any pair ofdistinct points of L is a basis of L. If the improper subspace PΓ admits abasis, its bases are called bases of Γ.

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Let S be a singular subspace of Γ. We define the rank rk(S) of S asthe minimal cardinality of a spanning set of S. In particular, if S admits abasis, then rk(S) is the minimal cardinality of a basis of S. The (singular)rank rk(Γ) of Γ is the least upper bound of the set {rk(S) | S singularsubspace of Γ} in the class of cardinal numbers.

As is customary, if all bases of a singular subspace S have the samecardinality (as when S is a projective space, for instance), the numberdim(S) := rk(S) − 1 is called the dimension of S. In the sequel, we willoften switch from ranks to dimensions, but only when S is a projectivespace.

2.3. Conventions for projective spaces

A projective space of dimension at least 2 where all lines are long is said tobe ordinary (also, irreducible).

Given a skewfield K and a K-vector space V , let Σ = PG(V ), theprojective space of linear subspaces of V . For a subset X ⊆ V , we denoteby [X] the set of 1-dimensional linear subspaces of V (namely, points ofΣ) spanned by non-zero vectors of X. Clearly, [{0}] = ∅. The skewfieldKΣ := K is called the underlying skewfield of Σ. Following the custom,when V = V (d + 1, K) we denote PG(V ) by the symbol PG(d, K), also byPG(d, q) when K = GF(q).

2.4. More on embeddings

Almost everthing we needed to say on embeddings and their morphismshas already been said in Introduction. We shall only fix a few more detailshere and add a few things, which will be exploited later in this paper.

Given a projective embedding e : Γ → Σ, we call Σ the codomainof e. As stated in Introduction, Γ is supposed to be connected and Σ isan ordinary projective space, with the additional assumption that Σ isdesarguesian when dim(Σ) = 2. So, Σ = PG(V ) for a vector space V . Theunderlying skewfield KΣ of Σ will be called the underlying skewfield of e.We also say that e is defined over K := KΣ, also that e is a K-embedding,for short.

As all embeddings considered in this paper are projective, henceforthwe simply call them embeddings, omitting the word “projective”.

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Faithful embeddings. Let e : Γ → Σ be an embedding and assume thatrk(Γ) ≥ 4. Suppose first that Γ is a linear space. Let {a, b, c} be an inde-pendent triple of points of Γ. Then 〈a, b〉Γ �= 〈a, c〉Γ. Hence 〈e(a), e(b)〉Σ �=〈e(a), e(c)〉Σ, as e maps distinct lines of Γ into distinct lines of Σ; namely,{e(a), e(b), e(c)} is independent. So,

(F3) e(X) is independent in Σ for every independent subset X ⊆ PΓ with|X| ≤ 3.

We say that e is k-faithful for an integer k ≥ 4 if:

(Fk) e(X) is independent in Σ for every independent subset X ⊆ PΓ with|X| ≤ k.

If e is k-faithful for all k ≥ 4, then we say that e is faithful.If Γ is not a linear space, then we say that e is faithful if, for every sin-

gular subspace S of Γ of rank at least 4, e induces on S a faithful embeddingin the subspace 〈e(S)〉Σ.

When rk(Γ) ≤ 3, the above definitions are empty. In that case, allembeddings of Γ may be regarded as faithful, by convention.

Remark 2.1. It is easy to see (and well known) that all full embeddingsof projective spaces are isomorphisms. As a consequence, if all singularsubspaces of a point-line geometry Γ are projective spaces, then every fullembedding of Γ is faithful.

Remark 2.2. In Introduction we have defined faithful morphisms of embed-dings. Clearly, if f : e0 → e is a such a morphism, then PG(f) is a faithfulembedding of the codomain of e0 in the codomain of e.

Planar embeddings. We say that an embedding e : Γ → Σ is planar ifΣ is a plane. Planar embeddings are not so rare (see Remark 1.2; alsoCossidente, Ferrara Dentice, Marino and Siciliano [8]), but they are non-full, in general. For instance, if rk(Γ) ≥ 3 then no full embedding of Γ isplanar, except trivially when Γ is a projective plane.

Remark 2.3. Planar embeddings are flat in the sense of Van Maldeghem[36]. When rk(Γ) = 2, the class of flat embeddings of Γ is far larger thenthe class of planar embeddings. On the other hand, when every line of Γbelongs to a larger singular subspace, then an embedding of Γ is flat if andonly if it is planar.

348 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

3. Basics on polar spaces

According to Buekenhout and Shult [3], a polar space is a point-line geom-etry Γ = (PΓ,LΓ) satisfying the so-called one-all axiom: for every point pand every line L, the set p∼ contains either exactly one or all points of L.In view of this property, Γ is connected and X∼ is a subspace for everyX ⊆ PΓ. In particular, the set Rad(Γ) := P∼

Γ is a singular subspace of Γ,called the radical of Γ. The polar space Γ is said to be non-degenerate ifRad(Γ) = ∅. If Γ is non-degenerate, 2 ≤ rk(Γ) < ∞ and all lines of Γ arelong, then we say that Γ is an ordinary polar space.

3.1. Ordinary polar spaces of rank n ≥ 3

Let Γ be an ordinary polar space of rank n ≥ 3. Then the singular subspacesof Γ are projective spaces (Buekenhout and Shult [3]). In particular, thesingular subspaces of rank 3 are projective planes. They are called planes ofΓ, for short. Moreover, the following properties hold (Tits [35], Buekenhoutand Shult [3]; see also Cohen [4]). We will freely use them in this paper,with no explicit reference.

(1) For a singular subspace S and a point p �∈ S∼, p∼ ∩ S is a hyperplaneof the projective space S and 〈p, p∼ ∩ S〉Γ is a singular subspace withthe same rank as S.

(2) For every maximal singular subspace M and every singular subspaceS of Γ (possibly, S = ∅) there exists a maximal singular subspaceM ′ ⊇ S such that M ∩ M ′ = M ∩ S.

(3) All maximal singular subspaces of Γ have the same rank n = rk(Γ).(4) Every set of pairwise collinear points spans a singular subspace.(5) For any three points p1, p2, p3 ∈ PΓ such that p1 ∼ p2 ∼ p3 but

p1 �∼ p3, there exists a point p4 ∈ PΓ such that the quadruple Q ={p1, p2, p3, p4} is a proper quadrangle, namely p3 ∼ p4 ∼ p1 but p2 �∼ p4.

There exist three classes of ordinary polar spaces of rank n ≥ 3 (Tits [35]):

Classical polar spaces. The planes of Γ are desarguesian, defined over a givenskewfield KΓ, called the underlying skewfield of Γ. Moreover, either everyline belongs to at least three planes (which is always the case when n > 3),or n = 3, KΓ is commutative and Γ is the line-grassmannian Gr(PG(3, KΓ))of PG(3, KΓ), where the points are the lines of PG(3, KΓ) and the lines arepencils of lines incident to point-plane flags of PG(3, KΓ). The planes ofGr(PG(3, KΓ)) are the sets of lines of PG(3, KΓ) either containing a pointof PG(3, KΓ) or contained in a plane of PG(3, KΓ).

Vol. 72 (2004) Lax Projective Embeddings of Polar Spaces 349

Non-classical grassmannians. n = 3 and Γ = Gr(PG(3, K)) for a non-commutative skewfield K. A plane of Γ is coordinatized by either K or itsopposite Kop, according to whether it arises from either a point or a planeof PG(3, K).

Tits polar spaces. These polar spaces have rank n = 3 and their planes areMoufang but non-desarguesian. We refer to Tits [35, Chapter 9] for details.

3.2. Properties of classical polar spaces

It is well known (Tits [35]) that an ordinary polar space Γ of rank n ≥ 3is classical if and only if it admits a full embedding. Clearly, if Γ is clas-sical of rank n ≥ 3, then all full embeddings of Γ are KΓ-embeddings.Moreover every classical polar space Γ admits a universal full embeddingeuniv : Γ → Σ = PG(V ) (Tits [35]). The image euniv(Γ) of Γ arises froma non-degenerate trace-valued sesquilinear form f or a non-singular pseu-doquadratic form q of V . More explicitly, denoting by Sk(Γ) the family ofsingular subspaces of Γ of rank k,

(P1) euniv(Sk(Γ)) is the set of totally f -isotropic (totally q-singular) k-dimensional linear subspaces of V .

In particular, euniv(PΓ) is the set of 1-dimensional linear subspaces of Vspanned by f -isotropic (q-singular) vectors and euniv(LΓ) is the family oftotally f -isotropic (totally q-singular) 2-dimensional linear subspaces of V .Moreover, all the following hold, where ⊥ stands for the ortogonality rela-tion associated to the sequilinear form f or to the sesquilinearization of thepseudoquadratic form q:

(P2) for any two distinct points p, q ∈ PΓ, we have p ∼ q if and only ifeuniv(p) ⊥ euniv(q);

(P3) 〈euniv(X∼)〉Σ = euniv(X)⊥ for every subset X ⊆ PΓ;(P4) If a line L of Σ meets euniv(PΓ) in exactly one point, say euniv(p), then

L ⊆ euniv(p)⊥.When char(K) �= 2 all full embeddings of Γ are isomorphic to euniv. Namely,the full embedding of Γ is unique (up to isomorphisms). When char(KΓ) =2, Γ might admit more than one full embedding. This happens only if theimage euniv(Γ) of Γ by euniv arises from a non-singular pseudoquadraticform q. Suppose this is the case and let f be the sesquilinearization of q.Then all full embeddings of Γ are obtained by factorizing V by subspacesof Rad(f). So, if Rad(f) = 0 then euniv is the unique embedding of Γ;otherwise, Γ admits at least one embedding different from the initial one.

350 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

In particular, if π is the projection of V onto W = V/Rad(f), then eterm =PG(π)·euniv is the terminal full embedding of Γ (see Subsection 1.4). Thecodomain of eterm is the so-called Veldkamp space of Γ (Shult [23]).

With q and f as in the previous paragraph, suppose that Rad(f) �= 0and let R be a non-trivial subspace of Rad(f). According to Tits [35, 8.6(II),8.7] (see also Johnson [15]), the (non-universal) full embedding e : Γ →PG(V/R) cannot arise from any pseudoquadratic form. However we canstill associate e(Γ) with a (possibly degenerate) sesquilinear form on V/R,although, in general, the relation between e(Γ) and that form is laxer thanin the case of euniv. More explicitly, let f/R be the form induced by f onV/R. (Note that Rad(f/R) = Rad(f)/R; thus, f/R is non-degenerate ifand only if e = eterm.) The following holds:

(P1’) for every singular subspace X of Γ, e(X) is totally isotropic for f/R.In particular, e(PΓ) is contained in the set S(f/R) of the f/R-points ofΣ/R := PG(V/R). (We warn that, when K is infinite, e(PΓ) might bea proper subset of S(f/R), even when R = Rad(f).) As said above, nopseudoquadratic form q/R can be found on V/R such that the points ofe(PΓ) are precisely the q/R-singular points of Σ/R. Nevertheless, denotingby ⊥ the orthogonality relation associated to f/R, properties analogous of(P2)-(P4) still hold:

(P2’) for any two distinct points p, q ∈ PΓ, we have p ∼ q if and only ife(p) ⊥ e(q);

(P3’) 〈e(X∼)〉Σ/R = e(X)⊥ for every subset X ⊆ PΓ;(P4’) If a line L of Σ/R meets e(PΓ) in exactly one point e(p), then L ⊆

e(p)⊥.We call ⊥ the perp-relation associated to e. A similar terminology will beused for euniv.

Remark 3.1. We have assumed n ≥ 3 in this subsection, but everything wehave said above can be repeated for n = 2, with the only difference that,when Γ is a classical generalized quadrangle, its underlying skewfield KΓ

can only be defined as the underlying skewfield of euniv.

4. Subgeometries of a projective space

In this section Σ = PG(V ) for a vector space V of dimension at least 3 overa given skewfield K. According to the conventions stated in Section 2, PΣ

and LΣ stand for the point-set and the set of lines of Σ, respectively.

Vol. 72 (2004) Lax Projective Embeddings of Polar Spaces 351

For a subset X ⊆ PΣ, we put L(X) = {L ∩ X | L ∈ LΣ, |L ∩ X| ≥ 2}and Σ(X) = (X,L(X)). Clearly, Σ(X) is a linear space, with X as thepoint-set and L(X) as the set of lines. We call it the subgeometry inducedby Σ on X. Needless to say, Σ(X) is not a projective space, in general. IfΣ(X) is a projective space (plane) then we call it a projective subgeometry(subplane) of Σ. If X ⊆ L for a line L ∈ LΣ and |X| ≥ 2, then X is calleda subline of Σ. We say that a projective subgeometry of Σ is ordinary if ithas no short lines.

4.1. Firm projective subgeometries

We say that an ordinary subgeometry Σ(X) of Σ is firm if any basis of theprojective space Σ(X) is independent as a set of points of Σ. Clearly, whendim(Σ(X)) is finite, Σ(X) is firm if and only if dim(Σ(X)) = dim(〈X〉Σ).

We shall firstly describe a class of firm projective subgeometries. Next,we will prove that all firm subgeometries of dimension n ≥ 2 belong to thatclass. For a subset U ⊆ V and a sub-skewfield K0 of K, we put

〈U〉K0 = {k∑

i=1

tiui | t1, t2, . . . , tk ∈ K0, u1, u2, . . . , uk ∈ U, k = 1, 2, 3, . . .}.

Henceforth, we denote by K0U the set [〈U〉K0] of points [v] ∈ PΣ forv ∈ 〈U〉K0 \ {0}. Clearly, the subgeometry Σ0 = Σ(K0U) is an ordinaryprojective space.

Suppose that U is independent. Then Σ0 is firm and K0 is the under-lying skewfield of Σ0. We call U a vector basis of Σ0 relative to K0 and wesay that K0 is the sub-skewfield of K that coordinatizes Σ0 with respectto U . Clearly, if U ′ is another K0-basis of V0 = 〈U〉K0 , then K0U

′ = K0U ,namely U ′ is a vector basis of Σ0.

Suppose that Σ0 has finite dimension n. Following Hughes and Piper[12], we say that a sequence F = (pi)n+1

i=0 of points of Σ0 is a frame of Σ0 if{pi}n

i=0 is a basis of Σ0 and pn+1 �∈ 〈p0, . . . , pi−1, pi+1, . . . , pn〉Σ0 for any i =0, 1, . . . , n. Every frame defines a coordinatization of Σ0, the point p0 beingtaken as the origin, the hyperplane 〈p1, p2, . . . , pn〉Σ0 as the hyperplane atinfinity and pn+1 playing the role of the unit point. As Σ0

∼= PG(V0), theframes of Σ0 bijectively correspond to the classes of mutually proportionalordered bases of V0. More explicitly, for every ordered basis U = (ui)n

i=0 ofV0, the sequence F (U) = (pi)n+1

i=0 , where pi = [ui] for i = 0, 1, . . . , n andpn+1 = [u0 + u1 + . . . + un], is a frame of Σ0. Conversely, for every frameF = (pi)n+1

i=0 and every choice of a representative vector u0 of p0, there existsa unique choice of representative vectors u1, u2, . . . , un of p1, p2, . . . , pn such

352 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

that U = (ui)ni=0 is a basis of V0 and pn+1 = [u0 + u1 + . . . + un], namely

F (U) = F . If we replace u0 with λu0 for a scalar λ ∈ K0\{0}, then we mustaccordingly replace ui with λui for i = 1, 2, . . . , n, thus replacing U withλU = (λui)n

i=0. Moreover, if F (U) = F (U ′) for two ordered bases U, U ′ ofV0, then U ′ = λU for some λ ∈ K0 \ {0}.

However, given an ordered basis U of V0, the points of the frame F (U)are 1-dimensional subspaces of V rather than of V0. Thus, we might alsowant to replace U with U ′ = λU for a scalar λ ∈ K \ K0. Then λ〈U〉K0 =〈U ′〉K′

0where K ′

0 = λK0λ−1. Clearly, [〈U ′〉K′

0] = [λ〈U〉K0] = [〈U〉K0 ],

namely K ′0U

′ = K0U . Hence, we can also take U ′ instead of U as a vectorbasis of Σ0 but, when doing so, we must replace K0 with K ′

0. The latter isisomorphic to K0, but possibly different from it.

Every vector basis of Σ0 can be obtained from a given vector basis U ofΣ0 by combining the two operations considered above, namely replacing Uwith another K0-basis U ′ of 〈U〉K0 , next replacing U ′ with λU ′ for a scalarλ ∈ K \ K0. So, the sub-skewfields of K that coordinatize Σ0 bijectivelycorrespond to the conjugates of K0 \ {0} in the multiplicative group of K.

Notation. Henceforth, given a vector basis U = (u0, u1, . . . , un) of Σ0, wedenote by KΣ0(u0, u1, . . . , un) (also KΣ0(U), for short) the sub-skewfield ofK that coordinatizes Σ0 with respect to U .

We summarize the previous remarks in the next lemma:

Lemma 4.1. For a subset X ⊂ PΣ, suppose that Σ(X) is an ordinary pro-jective space of finite dimension dim(Σ(X)) = n ≥ 2. Then:(1) The subgeometry Σ(X) is firm if and only if it admits a vector basis,

namely X = K0U for an independent subset U of V and a sub-skewfieldK0 of K.

(2) Assuming that Σ(X) is firm, let U = (ui)ni=0 and U ′ = (u′

i)ni=0 be

two vector bases of Σ(X). Suppose that u′i = λuj for a scalar λ ∈

K \ {0} and at least one pair of indices 0 ≤ i, j ≤ n. Then KΣ0(U′) =

λKΣ0(U)λ−1.(3) Given a basis p0, p1, . . . , pn of Σ(X) and representative vectors u0,

u1, . . . , un of the points p0, p1, . . . , pn, the sequence (u0, u1, . . . , un) isa vector basis of Σ(X) if and only if [u0 + u1 + . . . + un] ∈ X.

All ordinary projective subplanes of Σ are firm. Therefore,

Corollary 4.2. Every ordinary projective subplane of Σ admits a vector ba-sis.

Vol. 72 (2004) Lax Projective Embeddings of Polar Spaces 353

Lemma 4.3. Let Σ(X) be a firm projective subgeometry of Σ of finite dimen-sion n ≥ 2, and (ui)n

i=0 be a vector basis of Σ(X). Then all the followinghold:(1) For a subset Y ⊆ PΣ containing X, suppose that Σ(Y ) is a firm pro-

jective subgeometry of Σ of dimension n + 1 and X is a subspace ofΣ(Y ). Then, for every point p ∈ Y \X we can choose a representativevector un+1 of p in such a way that (u0, u1, . . . , un, un+1) is a vectorbasis of Σ(Y ).

(2) Let Σ(Y1), Σ(Y2) and Σ(Z) be finite dimensional firm projective sub-geometries of Σ such that dim(Σ(Y1)) = dim(Σ(Y2)) = n + 1,dim(Σ(Z)) = n + 2, X = Y1 ∩ Y2, Y1 ∪ Y2 ⊂ Z and Yi is a subspaceof Σ(Z) for i = 1, 2. Suppose the vectors un+1 and un+2 are suchthat (u0, u1, . . . , un, un+1) and (u0, u1, . . . , un, un+2) are vector basesof Σ(Y1) and Σ(Y2), respectively. Then (u0, u1, . . . , un, un+1, un+2) isa vector basis of Σ(Z).

Proof. Let q0 be the unit point of the frame associated to the vector basis(ui)n

i=0 of Σ(X). Given Y and p as in (1), let v be a representative vector ofp. For t ∈ K, the point p(t) = [u0+u1 . . .+un +tv] belongs to 〈q0, p〉Σ\{p},and all points of 〈q0, p〉Σ \{p} are obtained in this way. In particular, p(t) ∈Y \(X∪{p}) for some choices of t ∈ K\{0}. For such a scalar t, the sequence(u0, u1, . . . , un, tv) is a vector basis of Σ(Y ), by claim (3) of Lemma 4.1.

Given Y1, Y2, Z, un+1 and un+2 as in (2), let pi = [un+i] and qi =[u0 + u1 + . . . + un + un+i] be the unit point of the frame associated to(u1, . . . , un, un+i). Then the lines 〈q1, p2〉Σ and 〈q2, p1〉Σ meet in the pointp := [u1 + . . .+un +un+1 +un+2]. However, those two lines are coplanar inΣ(Z), as they belong to the plane of Σ(Z) spanned by the lines 〈p1, q0〉Σ(Y1)

and 〈p2, q0〉Σ(Y2), which are lines of Σ(Z) too. Hence p ∈ Z. By claim (3)of Lemma 4.1, (ui)n+2

i=0 is a vector basis of Σ(Z). �

Theorem 4.4. An ordinary projective subgeometry Σ0 of Σ of (possiblyinfinite) dimension dim(Σ0) ≥ 2 is firm if and only if it admits a vectorbasis.

Proof. The ‘if’ claim is obvious. In order to prove the ‘only if’ part, wechoose a line L of Σ0, a vector basis (u0, u1) of L (which exists, by Lemma4.1(1)) and we put K0 = KL(u0, u1). Let P(L, Σ0) be the set of planes ofΣ0 on L. In view of claim (1) of Lemma 4.3, for every X ∈ P(L, Σ0) we canchoose a vector uX such that (u0, u1, uX) is a vector basis of X. By claim (2)of Lemma 4.1, KX(u0, u1, uX) = K0. Put W := {uX}X∈P(L,Σ0) ∪ {u0, u1},

354 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

V0 := 〈W 〉K0 and let U be a K0-basis of the K0-vector space V0, with{u0, u1} ⊂ U ⊆ W . Given a finite subset W1 of U containing {u0, u1},〈[W1]〉Σ0 is the point-set of a finite dimensional subgeometry Σ1 of Σ0 andL is a line of Σ1. As Σ0 is firm, Σ1 is also firm and, as dim(Σ1) is finite, [W1]contains a basis B of Σ1. With no loss, we may assume that B contains[u0] and [u1]. Let U1 = {w ∈ W | [w] ∈ B}. By claim (2) of Lemma 4.3,K0U1 is contained in the point-set of Σ1. On the other hand, [U1] = Bis a basis of Σ1. Hence U1 is a vector basis of Σ1, by Lemma 4.1 and thefirmness of Σ1. We shall prove that U1 = W1. Suppose the contrary and letw ∈ W1\U1. Then w = λu for a vector u ∈ 〈U1〉K0 and a scalar λ ∈ K \{0}.However, w = uX for some plane X ∈ P(L, Σ0). Therefore, denoting by Mthe line of X containing the points [u0 + u1] and p = [uX ], and recallingthat K0 = KL(u0, u1) = KX(u0, u1, uX), we have

M \ {p} = {[u0 + u1 + tw] | t ∈ K0} = {[u0 + u1 + tλu] | t ∈ K0}. (4.1)

However K0 = KΣ1(U1). Hence we also have

M \ {p} = {[u0 + u1 + tu] | t ∈ K0}. (4.2)

By comparing (4.1) and (4.2) we obtain that K0 = λK0, namely λ ∈ K0.Thus, w is a linear combination of vectors of U1 with coefficents in K0. Thisis a contradiction, as W1 is K0-independent. The equality W1 = U1 follows.

So, W1 is a vector basis of Σ1. However, vector bases are K-indepen-dent. Hence W1 is K-independent. It follows that U is K-independent. Asevery finite subset of U is a vector basis of a subgeometry of Σ0, we alsohave K0U ⊆ Σ0. On the other hand, Σ0 ⊆ [V0] = K0U , as X ⊆ [V0] forevery X ∈ P(L, Σ0). Hence Σ0 = Σ(K0U). Therefore, U is a K0-basis ofΣ0. �

Remark 4.1. Ordinary projective subgeometries of dimension n > 2 arenon-firm, in general. For instance, given a field K0 and integers n, h withh > n and n > 2, let K = K0(ω) be a simple extension of K0 of degree h.Put

Y = {(xi + x0ωi)n

i=1 | x0, x1, . . . , xn ∈ K0}.In Σ = PG(n−1, K), consider the set X = [Y ]. We have Σ(X) ∼= PG(n, K0).Thus, dim(Σ(X)) = n > n − 1 = dim(Σ). Hence Σ(X) is non-firm.

4.2. Truly projective sublines

All sublines of Σ might be called ‘projective’, but we are not interestedin such a broad notion here. We will only consider lines L0 ∈ L(X), foran ordinary projective subplane Σ(X) = (X,L(X)). We call these sublines

Vol. 72 (2004) Lax Projective Embeddings of Polar Spaces 355

truly projective sublines. Lemma 4.1 applies to truly projective sublines,too. Thus,

Corollary 4.5. A subline L0 is truly projective if and only if it admits avector basis.

Lemma 4.6. For a truly projective subline L0 of Σ, let (u1, u2) be a vectorbasis of L0 and Σ(X) = (X,L(X)) an ordinary projective subplane suchthat L0 ∈ L(X). Given a vector basis U = (v1, v2, v3) of Σ(X), suppose thatv1 = λu1 for a scalar λ ∈ K \ {0}. Then KL0(u1, u2) = λ−1KΣ(X)(U)λ. Inparticular, if v1 = u1 then KL0(u1, u2) = KΣ(X)(U).

Proof. Put K1 = KL0(u1, u2) and K2 = KΣ(X)(U), for short. Modulo re-placing U with another K2-basis of 〈U〉K2 , we may assume that [v2] = [u2],namely v2 = µu2 for a scalar µ ∈ K \ {0}. Put p = [v2] = [u2]. As L0 is aline of Σ(X), we have

L0 \ {p} = {[v1 + sv2] | s ∈ K2} = {[λu1 + sµu2] | s ∈ K2}. (4.3)

On the other hand,

L0 \ {p} = {[u1 + tu2] | t ∈ K1}. (4.4)

By comparing (4.3) with (4.4) we obtain that

λK1 = K2µ. (4.5)

By (4.5), λ = λ·1 ∈ K2µ. Therefore λµ−1 ∈ K2. Accordingly, (λµ−1)−1 ∈K2. Hence:

µλ−1 ∈ K2. (4.6)

Condition (4.5) is equivalent to the following:

K1 = λ−1K2µ = λ−1K2µλ−1λ. (4.7)

However, K2µλ−1 = K2 by (4.6). Hence K1 = λ−1K2λ. �

The next lemma is crucial for the proof of Theorem 1.6.

Lemma 4.7. Given two truly projective sublines L1 and L2 meeting in apoint p, let p1 and p2 be points of L1 and L2 different from p. Supposethere are ordinary projective subplanes Σ(X1) = (X1,L(X1)) and Σ(X2) =(X2,L(X2)) such that L1 ∈ L(X1), L2 ∈ L(X2) but L1 �= X1 ∩ X2 �= L2.Then, for every representative vector v of p, we can choose representativevectors v1 and v2 of p1 and p2 in such a way that KL1(v, v1) = KL2(v, v2).

356 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

Proof. Choose a point q �= p in L3 = X1 ∩ X2 and, for i = 1, 2, choosea point qi ∈ Xi such that Fi = (p, pi, q, qi) is a frame of Σ(Xi). Givena representative vector v of p, let Ui = (v, vi, wi, ui) be the vector basisof Σ(Xi) such that F (Ui) = Fi. In particular, [vi] = pi, [ui] = qi and[w1] = [w2] = q. By Lemma 4.6, for i = 1, 2 the sub-skewfield KLi(v, vi)is equal to the sub-skewfield Ki that coordinatizes Σ(Xi) with respect toUi. Similarly, Ki = KL3(v, wi) for i = 1, 2. Hence KLi(v, vi) = KL3(v, wi)for i = 1, 2. On the other hand, KL3(v, w1) = KL3(v, w2) by claim (2) ofLemma 4.1. The equality KL1(v, v1) = KL2(v, v2) follows. �

5. A few results on embeddings of projective and polarspaces

5.1. Preliminaries on embeddings of arbitrary point-line geometries

In this subsection Γ = (PΓ,LΓ) is an arbirary point-line geometry and e :Γ → Σ is an embedding of Γ. We state the following conventions: For a lineL ∈ LΓ, we put e(L) = e(PΓ)∩〈e(L)〉Σ. For every subspace S of Γ, we denoteby L(S) the set of lines of Γ contained in S and we put Γ(S) = (S,L(S)),e(Γ(S)) =

(e(S), {e(L)}L∈L(S)

)and e(Γ(S)) =

(e(S), {e(L)}L∈L(S)

). Note

that Γ = Γ(PΓ). Accordingly, we put e(Γ) := (e(PΓ), {e(L)}L∈LΓ) and

e(Γ) := (e(PΓ), {e(L)}L∈LΓ). By the injectivity of e and properties (E2)

and (E3) (see Introduction), we immediately obtain the following:

Lemma 5.1. Γ(S) ∼= e(Γ(S)) for every subspace S of Γ,and the identity map-ping on e(S) induces a bijection between {e(L)}L∈L(S) and {e(L)}L∈L(S).

However, as e(L) might properly contain e(L) for some L ∈ L(S), thestructures e(Γ(S)) and e(Γ(S)) might be non-isomorphic.

Lemma 5.2. Assume the following:

(W’) 〈e(p∼)〉Σ ∩ e(PΓ) = e(p∼) for every point p ∈ PΓ.

(Compare Remark 1.3, condition (W’).) Then e(L) = e(L) for every lineL ∈ LΓ.

Proof. Let x ∈ PΓ be such that e(x) ∈ e(L). We shall prove that x ∈ L.Suppose to the contrary that x �∈ L. Choose a point y ∈ L. Clearly, e(y) �=e(x) by the injectivity of e and the assumption that x �∈ L. Moreover,L ⊆ y∼. Hence 〈e(L)〉Σ ⊆ 〈e(y∼)〉Σ. Consequently, e(x) ∈ 〈e(y∼)〉Σ∩e(PΓ).So, e(x) ∈ e(y∼) by (W’). As e is injective, we have x ∈ y∼. Thus, we

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can consider the line M = 〈x, y〉Γ. Clearly, M �= L. However, 〈e(M)〉Σ =〈e(x), e(y)〉Σ = 〈e(L)〉Σ, which is a contradiction. �

By Lemma 5.2 we obtain the following:

Corollary 5.3. If e satisfies condition (W’) of Lemma 5.2, then e(Γ(S)) =e(Γ(S)) for every subspace S of Γ. In particular, e(Γ) = e(Γ).

5.2. Embeddings of projective spaces

In this subsection Γ = (PΓ,LΓ) is an ordinary projective space and e : Γ →Σ is an embedding of Γ. Condition (W’) of Lemma 5.2 is trivially satisfiedby every embedding of Γ. Hence e(L) = e(L) for every line L ∈ LΓ ande(Γ(S)) = e(Γ(S)) for every subspace S of Γ. Note also that e(Γ(S)) =Σ(e(S)). Therefore,

Lemma 5.4. Γ(S) ∼= e(Γ(S)) = Σ(e(S)) for every subspace S of Γ. Inparticular, Γ ∼= e(Γ) = Σ(e(PΓ)).

Lemma 5.5.

(1) For every plane S of Γ, e(S) spans a plane of Σ.(2) The embedding e maps every spanning set of Γ onto a spanning set of

Σ.(3) dim(Γ) ≥ dim(Σ).(4) Γ is desarguesian, its underlying skewfield KΓ is a sub-skewfield of KΣ

and e(Γ) is a projective subgeometry of Σ.

Proof. (1) follows from property (F3) of Subsection 2.4. Claim (2) is ob-vious and (3) immediately follows from (2). The claims collected in (4),when non-trivial, follow from (1), Lemma 5.4, Corollary 4.2 and the factthat Σ is desarguesian (by assumption when dim(Σ) = 2, according to theconventions stated in Subsection 2.4). �Lemma 5.6. Let e be faithful. Then e maps every independent subset of PΓ

onto an independent subset of PΣ. In particular, e maps every basis of Γonto a basis of Σ. As a consequence, dim(Γ) = dim(Σ).

Proof. The first claim of the lemma follows from the definition of faithful-ness and the fact that a set of points of a projective geometry is independentif and only if all of its finite subsets are independent. The second claim fol-lows from the first one and claims (2) and (3) of Lemma 5.5. �

By the first claim of (4) of Lemma 5.5, Γ = PG(V0) for a KΓ-vectorspace V0. Also, Σ = PG(V ) for a KΣ-vector space V . As KΓ ≤ KΣ by (4)of Lemma 5.5, we may also regard V as a KΓ-vector space.

358 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

Theorem 5.7. The following are equivalent:(1) The embedding e is faithful.(2) Σ(e(PΓ)) is a firm subgeometry of Σ.(3) Regarding V as a vector space over KΓ ≤ KΣ, the embedding e is

induced by a semilinear mapping from V0 to V .In particular, when Γ has finite dimension, e is faithful if and only ifdim(Γ) = dim(Σ).

Proof. The equivalence of (1) and (2) follows from Lemma 5.4. Supposethat (1) and (2) hold. By Theorem 4.4, (2) implies the existence of a vectorbasis U of e(Γ). So, e is an isomorphism from Γ = PG(V0) to PG(〈U〉KΓ

). Itis well known that every isomorphism of projective spaces defined over thesame skewfield arises from a semilinear mapping. Conversely, suppose thate([v]) = [ϕ(v)] for a semilinear mapping ϕ : V0 → V , with V regarded as aKΓ-vector space. As e in injective, ϕ is injective. Hence ϕ(U) is independentfor every independent subset U of V0. Namely, e is faithful. �Remark 5.1. In general, a projective space admits non-faithful embeddings.For instance, given n, h, K0 and K = K0(ω) as in Remark 4.1, let Γ =PG(n, K0) and Σ = PG(n − 1, K). The function mapping every vector(xi)n

i=0 ∈ V (n+1, K0) onto the vector (xi +x0ωi)n

i=1 of V (n, K) induces anembedding e of Γ in Σ. As the codomain of e has dimension n − 1 < n =dim(Γ), the embedding e is non-faithful (but it is n-faithful). In particular,if n = 3 then e is planar.

5.3. Weak embeddings of polar spaces

In this subsection Γ = (PΓ,LΓ) is a non-degenerate polar space of finiterank n ≥ 2 and e : Γ → Σ is a weak embedding of Γ. We recall that, as Γ isa non-degenerate polar space, condition (W’) of Lemma 5.2 is equivalent toproperty (W), chosen in Introduction to define weak embeddings of polarspaces (see Remark 1.3). So, e satisfies (W’), too.

Theorem 5.8. The embedding e is faithful.

Proof. Let {pi}ki=1 be a finite independent subset of a singular subspace

S of Γ. We shall prove that {e(pi)}ki=1 is independent in Σ. Suppose to

the contrary that e(p1) ∈ 〈e(p2), . . . , e(pk)〉Σ. Choose a maximal singularsubspace M of Γ such that M ∩ S = 〈p2, . . . , pk〉Γ. For every point p ∈ Mwe have p∼ ∩ S ⊇ 〈p2, . . . , pk〉Γ. Therefore 〈e(p2), . . . , e(pk)〉Σ ⊆ 〈e(p∼)〉Σ.As 〈e(p1), e(p2), . . . , e(pk)〉Σ = 〈e(p2), . . . , e(pk)〉Σ we get e(p1) ∈ 〈e(p∼)〉Σ.Condition (W’) now forces e(p1) ∈ e(p∼). Hence p1 ∈ p∼ by the injectivity

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of e. However, this holds for every point p ∈ M . Therefore M ⊆ p∼1 , contraryto the choice of M . We are forced to admit that {e(pi)}k

i=1 is independent.�

Corollary 5.9. For any two singular subspaces S1 and S2 of Γ, we have〈e(S1)〉Σ ⊆ 〈e(S2)〉Σ if and only if S1 ⊆ S2. In particular, 〈e(S1)〉Σ =〈e(S2)〉Σ if and only if S1 = S2.

Proof. Suppose that 〈e(S1)〉Σ ⊆ 〈e(S2)〉Σ. Then S1 ⊆ S∼2 by (W’) and

since S2 ⊆ S∼2 . Therefore, S := 〈S1 ∪ S2〉Γ is a singular subspace of Γ. On

the other hand, the inclusion 〈e(S1)〉Σ ⊆ 〈e(S2)〉Σ implies that 〈e(S)〉Σ =〈e(S2)〉Σ. As e is faithful (Theorem 5.8), this equality forces S = S2, namelyS1 ⊆ S2. �

We shall now prove that dim(Σ) ≥ 2n − 1. We will do that by aninductive argument, but we must preliminarily show that induction can beapplied here. We state some notation first. Given a point p of Γ, we putΓ/p := resΓ(p). Namely, Γ/p is the polar space of rank n− 1 formed by thesingular subspaces of Γ that properly contain p, the lines and the planesof Γ on p being the points and the lines of Γ/p. Let Le

Σ(p) (respectively,Pe

Σ(p)) be the set of lines (planes) of Σ that contain e(p) and are containedin 〈e(p∼)〉Σ. Regarding the planes of Pe

Σ(p) as bundles of lines through e(p),the pair 〈e(p∼)〉Σ/p := (Le

Σ(p),PeΣ(p)) is a projective space. The embedding

e induces a mapping e/p from the point-set of Γ/p to the point-set LeΣ(p)

of 〈e(p∼)〉Σ/p.

Lemma 5.10. The mapping e/p is a weak embedding of Γ/p in 〈e(p∼)〉Σ/p.

Proof. The injectivity of e/p follows from property (E3) on e. By Corollary5.9, e/p is an embedding. Clearly, e/p inherits conditions (W) and (W’)from e. �Theorem 5.11. dim(Σ) ≥ 2n − 1.

Proof. We prove first that dim(Σ) > 2. By contradiction, let dim(Σ) = 2,namely e is planar. Given a point p ∈ PΓ and two distinct lines L1 and L2

of Γ on p, 〈e(L1), e(L2)〉Σ = PΣ as e is planar. However, L1 ∪ L2 ⊆ p∼.Hence e(x) ∈ 〈e(p∼)〉Σ for every point x ∈ PΓ, contrary to (W). Therefore,dim(Σ) > 2.

By the above, the statement of the lemma holds true when n = 2. Letn > 2 and choose a point p ∈ PΓ. By Lemma 5.10 we can apply induction toe/p. Hence dim(〈e(p∼)〉Σ/p) ≥ 2(n− 1)− 1. Accordingly, dim(〈e(p∼)〉Σ) ≥2n−2. However, 〈e(p∼)〉Σ �= PΣ, by (W) on e. Hence dim(Σ) ≥ 2n−1. �

360 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

5.4. Non-planar embeddings of polar spaces

In the sequel, Γ is an ordinary polar space of rank n ≥ 3 and e : Γ → Σ isa given non-planar embedding of Γ.

Lemma 5.12. We have e(p) �∈ 〈e(S)〉Σ for at least one point p and onesingular plane S of Γ.

Proof. Indeed, if otherwise, 〈e(S)〉Σ ⊇ e(PΓ) for every singular plane S ofΓ, and e would be planar. �Lemma 5.13. Let p and S be a point and a singular plane of Γ such thate(p) �∈ 〈e(S)〉Σ and p �∈ S∼. Then there exists at least one proper quadrangleQ of Γ such that p ∈ Q and dim(〈e(Q)〉Σ) = 3.

Proof. As p �∈ S∼, p∼ meets S in a line, say L. Choose a singular planeS1 on p disjoint from S and a point q ∈ L. For a point p1 ∈ S \ L, putL1 = p∼1 ∩ S1. Note that L1 is a line and p �∈ L1, as p �∼ p1. Suppose that〈e(L1)〉Σ ⊆ 〈e(p), e(q), e(p1)〉Σ. Then 〈e({p1} ∪ S1)〉Σ ⊆ 〈e(p), e(q), e(p1)〉Σ,as 〈p, L1〉Γ = S1. On the other hand, 〈e(S1)〉Σ is a plane and is containedin 〈e({p1} ∪S1)〉Σ. Hence 〈e(S1)〉Σ = 〈e({p1} ∪S1)〉Σ = 〈e(p), e(q), e(p1)〉Σ.Therefore, e(p1) ∈ 〈e(S1)〉Σ. Suppose now that this happens for every p1 ∈S \ L. Then e(S) ⊆ 〈e(S1)〉Σ. Namely, 〈e(S)〉Σ = 〈e(S1)〉Σ and we obtainthat e(p), which belongs to 〈e(S1)〉Σ, also belongs to 〈e(S)〉Σ, contradictingour choice of S and p. Therefore, e(L1) �⊆ 〈e(p), e(q), e(p1)〉Σ for at leastone choice of p1 ∈ S \ L. With p1 chosen in that way, the line L1 containsat most one point x such that e(x) ∈ 〈e(p), e(q), e(p1)〉Σ and exactly onepoint y ∈ q∼. As, by assumption, Γ has no short lines, we can take a pointq1 ∈ L1 different from y and from x (if x exists). Then Q = {p, q, p1, q1} isa proper quadrangle and dim(〈e(Q)〉Σ) = 3. �Theorem 5.14. dim(〈e(Q)〉Σ) = 3 for at least one proper quadrangle Q ofΓ.

Proof. We shall argue by induction on n = rk(Γ). Suppose first that n = 3.By Lemma 5.12, there exist a point p and a singular plane S such thate(p) �∈ 〈e(S)〉Σ. In particular, p �∈ S. However, S = S∼, as n = 3. Hencep �∈ S∼. We can now apply Lemma 5.13, obtaining the conclusion.

Let n > 3. By the inductive hypothesis, chosen a point p ∈ PΓ, Γ/p ad-mits a proper quadrangle Qp = {L1, L2, L3, L4} such that dim(〈e/p(Qp)〉Σp)= 3, where Σp stands for 〈e(p∼)〉Σ/p. Accordingly, dim(〈∪4

i=1e(Li)〉Σ) = 4.For i = 1, 2, 3, 4, choose a point pi ∈ Li \ {p}. Then Q = {p1, p2, p3, p4} isa proper quadrangle and dim(〈e(Q)〉Σ = 3. �

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6. Proof of Theorem 1.6

In this section Γ is an ordinary polar space of finite rank n ≥ 3 and e : Γ →Σ is a given embedding of Γ. Since claim (2) of Theorem 1.6 follows fromclaim (1) via Theorem 1.3, we shall only prove claims (1) and (3).

6.1. Proof of claim (1)

In this subsection we shall prove that Γ is classical. We first restrict thecases to examine. Eventually, we will show that the non-classical survivingcase is impossible.

Lemma 6.1. Either Γ is classical and KΓ is a sub-skewfield of KΣ, or Γ ∼=Gr(PG(3, K)) for a non-commutative skewfield K such that K ∼= K1 andKop ∼= K2 for suitable sub-skewfields K1, K2 of KΣ.

Proof. By (1) of Lemma 5.5, e(S) spans a plane of Σ for every singularplane S of Γ. The restriction of e to S is an isomorphism from S to a pro-jective subplane Σ(e(S)) of Σ. By Corollary 4.2, S is desarguesian and itsunderlying skewfield KS is a sub-skewfield of KΣ. In view of the classifica-tion of ordinary polar spaces of rank n ≥ 3 by Tits [35], either Γ is classicalor Γ ∼= Gr(PG(3, K)) for a non-commutative skewfield K. In the first case,KS

∼= KΓ. In the latter case, KS is isomorphic to either K or Kop. �

Lemma 6.2. Suppose that Γ ∼= Gr(PG(3, K)) for a skewfield K. Then 〈Q〉Γis a grid, for every proper quadrangle Q of Γ.

Proof. Let Q = {p0, p1, p2, p3} be a proper quadrangle of Γ. Then p0, p1, p2,p3 are distinct lines of ∆ = PG(3, K) such that pi is skew with pi+2 andmeets each of pi−1 and pi+1 in a point, where i = 0, 1, 2, 3 and the indicesi − 1, i + 1, i + 2 are computed modulo 4. For i = 0, 1, 2, 3, let ai be themeet-point of pi−1 and pi in ∆, αi := 〈pi−1, pi〉∆ and Li := 〈pi−1, pi〉Γ.Then (ai, αi) is a point-plane flag of ∆ and the lines of αi through ai arethe points of the line Li of Γ. As pi is skew with pi+2, we have αi �= αi+2

and ai �= ai+2. Suppose ai ∈ αi+2. Then αi+2 contains a non-collinear tripleof points of αi, namely ai, ai−1 = pi−1 ∩ pi−2 and ai+1 = pi ∩ pi+1. Thisforces αi+2 = αi, a contradiction. Therefore, ai �∈ αi+2 for i = 0, 1, 2, 3.

Let now p be any point of Li. Regarded as a line of αi, p meets theline αi ∩ αi+2 in a point a(p) �= ai+2. Put πi+2

i (p) = 〈a(p), ai+2〉∆. Thenπi+2

i (p), regarded as a point of Γ, is the unique point of Li+2 collinear withp. For q0 ∈ L0 and q1 ∈ L1, let M0 = 〈q0, π

20(q0)〉Γ and M1 = 〈q1, π

31(q1)〉Γ.

Then M0 and M1 correspond to the point-plane flags (b0, β0) and (b1, β1)

362 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

of ∆, where bi = qi ∩ πi+2i (qi) and βi = 〈qi, π

i+2i (qi)〉∆ for i = 0, 1. Clearly,

b0 �= b1 and β0 �= β1. As b0 belongs to the line α0 ∩ α2, which belongs toβ1, we have b0 ∈ β1. Similarly, b1 ∈ β0. Therefore, 〈b0, b1〉∆ = β0 ∩ β1. Itfollows that the line r = β0 ∩β1, regarded as a point of Γ, belongs to eitherof M0 and M1. Namely, M0 ∩ M1 = r. Thus, 〈Q〉Γ is a grid. �

Lemma 6.3. Suppose that Γ ∼= Gr(PG(3, K)). Then K ∼= Kop and K isisomorphic to a sub-skewfield of KΣ. If moreover e is non-planar, then Kis commutative.

Proof. We have Σ = PG(V ) for a vector space V over KΣ and, by Lemma6.1, KΣ contains copies of both K and Kop. Moreover, every line of Γ iscontained in a plane (in fact, in exactly two planes). Hence e(L) is a trulyprojective subline of Σ, for every line L ∈ LΓ. By Corollary 4.5, e(L) admitsa vector basis. Let (v, w) be a vector basis of e(L). By Lemma 4.6, the sub-skewfield Ke(L)(v, w) ≤ KΣ is isomorphic to either of K and Kop. (Recallthat L is contained in two planes of Γ, one of which is coordinatized byK and the other one by Kop.) So far, K ∼= Kop (but K might be non-commutative as well).

Let Q = {p0, p1, p2, p3} be a proper quadrangle of Γ. For i = 0, 1, 2, 3,put Li = 〈pi, pi+1〉Γ, indices being computed modulo 4. Given any planeS1 of Γ on L1, let S0 be the unique plane on L0 that meets S1 in a line.Then we can apply Lemma 4.7 to e(L0) and e(L1): Given any vector v1 ∈V \ {0} such that [v1] = e(p1), we can choose representative vectors v0

and v2 of e(p0) and e(p2) such that (v1, v0) is a vector basis of e(L0),(v1, v2) is a vector basis of e(L1) and Ke(L0)(v1, v0) = Ke(L1)(v1, v2). PutK1 := Ke(L0)(v1, v0) = Ke(L1)(v1, v2).

Next we choose representative vectors v3 and v′3 of e(p3) such that(v2, v3) is a vector basis of e(L2) and (v0, v

′3) is a vector basis of e(L3).

Clearly, v′3 = λv3 for a scalar λ ∈ KΣ \ {0}. Put K2 := Ke(L2)(v2, v3) andK3 := Ke(L3)(v0, v

′3). Thus,

e(L0) \ {e(p0)} = {[v1 + tv0] | t ∈ K1},e(L1) \ {e(p2)} = {[v1 + tv2] | t ∈ K1},e(L2) \ {e(p3)} = {[v2 + tv3] | t ∈ K2},e(L3) \ {e(p3)} = {[v0 + tv′3] | t ∈ K3}.

For a point p ∈ L0 \ {p0}, we write p = p0(t) if e(p) = [v1 + tv0]. Similarly,p = p1(t) (respectively, p = p2(t) or p = p3(t)) means that p ∈ L1 ande(p) = [v1 + tv2] (resp. p ∈ L2 and e(p) = [v2 + tv3], or p ∈ L3 ande(p) = [v0 + tv′3]). For i = 0, 1, the function mapping p ∈ Li onto the

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point p∼∩Li+2 induces a bijection αi : K1 → Ki+2 such that pi+2(αi(t)) =pi(t)∼ ∩ Li+2. However, 〈Q〉Γ is a grid by Lemma 6.2. Therefore, the line〈p0(t), p2(α0(t))〉Γ meets the line 〈p1(t), p3(α1(t))〉Γ in a point. Rephrasingthis fact for Σ = PG(V ), and recalling that v′3 = λv3, we obtain the thefollowing equation admits a solution (x, y, z) ∈ K3

Σ for any choice of s, t ∈K1:

(v1 + tv0) + x(v2 + α0(t)v3) = z ((v1 + sv2) + y(v0 + α1(s)λv3)) . (6.1)

Henceforth we assume that e is non-planar. Accordingly, by Theorem 5.14,we may assume to have chosen Q in such a way that dim(〈Q〉Σ) = 3.Namely, the vectors v0, v1, v2, v3 are independent. So, (6.1) is equivalent tothe following quadruple of equalities:

z = 1, zy = t, x = zs, xα0(t) = zyα1(s)λ. (6.2)

Hence x = s, y = t and the fourth equality of (6.2) can be rewritten asfollows:

sα0(t) = tα1(s)λ. (6.3)

As s, t are arbitrary elements of K1, (6.3) holds for all s, t ∈ K1. Puttings = 1 in (6.3) we obtain α0(t) = tα1(1)λ. Also, for t = 1, sα0(1) = α1(s)λ.For t = s = 1 we get α0(1) = α1(1)λ. Hence (6.1) implies that

α0(t) = tα1(1)λ, α1(s) = sα0(1)λ−1, α0(1) = α1(1)λ. (6.4)

From the first and second equality of (6.4) with t = 0 and s = 0 we getα0(0) = α1(0) = 0. As α0 and α1 are bijections, α0(t) �= 0 �= α1(s) fort, s �= 0. In particular, α0(1) �= 0 �= α1(1) and the third equality of (6.4)can be rewritten as follows:

λ = α1(1)−1α0(1), (6.5)

By (6.5) and (6.4) we obtain the following:

α0(t) = tα0(1), α1(s) = sα1(1). (6.6)

By comparing (6.3) with (6.5) and (6.6) we obtain that stα0(1) = tsα0(1),namely st = ts, as α0(1) �= 0. So, K1 is commutative. As K ∼= K1, K isalso commutative. �

End of the proof. As every embedding admits a hull, we may assume that eis dominant, possibly replacing it with its hull. By Theorem 1.5, e is weak.On the other hand, planar embeddings are non-weak. Hence Γ is classical,by Lemma 6.3. �

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6.2. Proof of claim (3)

By claims (1) and (2), Γ is classical and there is a morphism f : euniv → e,where euniv : Γ → PG(Vuniv) is the universal full embedding of Γ. Moreover,f splits as f = fprojfext where fext : euniv → e is a scalar extension andfproj : e → e is a projection, for a suitable KΣ-embedding e of Γ. Clearly, ife is dominant then e ∼= e and, consequently, e is a scalar extension of euniv.The “only if” part of (3) is proved.

On the other hand, suppose that e faithfully embodies euniv. So, in theabove decomposition f = fextfproj we may assume that f = fext and fproj

is the identity. Let e : Γ → PG(V ) be the hull of e and g : e → e be thecanonical projection from e to e. By the first paragraph of this subsection,there is a scalar extension f : euniv → e. Let S ⊂ PΓ be a generating set ofΓ. Then euniv(S) contains a basis B of PG(Vuniv). Put BΓ := e−1

univ(B). As f

is a scalar extension, e(BΓ) = f(B) is a basis of Σ. Similarly, e(BΓ) = f(B)is a basis of PG(V ). However g induces a bijection from e(BΓ) to e(BΓ),as it is a morphism from e to e. In other words g, regarded as a semilinearmapping from V to the underlying vector space V of Σ, induces a bijectionfrom a basis of V to a basis of V . Hence g is an isomorphism. Namely,e ∼= e. �

7. Proof of Fact 1.7

Let Γ be classical polar space of rank n ≥ 3 and e : Γ → Σ be an embeddingof Γ. Suppose by contradiction that e does not match with euniv. Thus,there are three distinct maximal singular singular subspaces M1, M2, M3 ofΓ such that

(¬M) rk(M1 ∩ M2 ∩ M3) = n − 1, euniv(M3) ⊆ 〈euniv(M1 ∪ M2)〉Σ0 bute(M3) �⊆ 〈e(M1 ∪ M2)〉Σ.

(where Σ0 stands for the codomain of euniv). For an embedding e′ of Γ, ife is a projection of e′, then e′ inherits (¬M) from e. In particular, (¬M)is inherited by the hull of e. So, modulo replacing e with its hull, we mayassume that e is dominant.

We can choose a subset A ⊂ M1 ∪M2 and a point p ∈ M3 \ (M1 ∩M2)in such a way that e(A ∪ {p}) is a basis of 〈e(M1 ∪ M2 ∪ M3)〉Σ and e(A)is a basis of 〈e(M1 ∪M2)〉Σ. As e(PΓ) spans Σ, we can also choose a subsetB ⊂ PΓ \ {p} in such a way that A ⊂ B, e(p) �∈ e(B) and (e(B ∪ {p}) isa basis of Σ. Put HΣ := 〈e(B)〉Σ and S := e−1(HΣ ∩ e(PΓ)). As HΣ is a

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proper subspace of Σ, S is a proper subspace of Γ. Moreover rk(S) ≥ 2, asS contains M1 and M2. Every proper subspace of Γ of rank at least 2 iscontained in a hyperplane of Γ (see [21] for a proof of this claim). HenceS ⊆ H for a hyperplane H of Γ.

Since e is assumed to be dominant, e(H) spans a hyperplane of Σ byTheorem 1.5. Hence p �∈ H, as e(H) contains e(B) and e(B)∪{e(p)} spansΣ. On the other hand, every hyperplane of Γ is the preimage by euniv of a hy-perplane of Σ0 (Cohen and Shult [5]). Therefore H = e−1

univ(〈euniv(H)〉Σ0 ∩euniv(PΓ)). Hence p ∈ H, as euniv(p) ∈ 〈euniv(M1 ∪ M2)〉Σ0 by (¬M). Acontradiction has been reached. �

8. Proof of Theorem 1.8

Throughout this section Γ = (PΓ,LΓ) is a classical polar space of finiterank n ≥ 3, e : Γ → Σ = PG(V ) is a weak K-embedding of Γ for askewfield K ≥ KΓ and e0 : Γ → Σ0 = (PΣ0 ,LΣ0) is a full embeddingof Γ in Σ0 = PG(V0), for a KΓ-vector space V0. As e is weak, it satis-fies (W’) of Lemma 5.2, which, in this context, is equivalent to property(W) of Introduction. Moreover, e is faithful, by Theorem 5.8. In particular,dim(〈e(M)〉Σ) = dim(M) = n − 1 for every maximal singular subspace Mof Γ. According to the hypotheses of Theorem 1.8, when Γ admits at leasttwo full embeddings, we assume that

(A) e matches with e0, and(B) e0 = eterm.

We recall that both (A) and (B) hold when Γ admits a unique full embed-ding, too: in that case, (B) is trivial and (A) is just Fact 1.7.

In the sequel we regard Γ as the same thing as its image e0(Γ) in Σ0.So, the points of Γ are points of Σ0 isotropic for a given non-degeneratesesquilinear form f and the lines of Γ are lines of Σ0 totally isotropic for f .As stated in Subsection 3.2, ⊥ is the perp-relation associated to f . Thus,X⊥ = 〈X∼〉Σ0 for every subset X ⊆ PΓ. Hypothesis (B) can be rephrasedas follows:(B’) P⊥

Σ0= ∅.

According to the above, we may regard e as an embedding of a subgeometryΓ = e0(Γ) of Σ0 in Σ and the thesis of Theorem 1.8 amounts to say thatwe can extend e to a faithful embedding e : Σ0 → Σ.

366 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

The first two lemmas of this section enable us to use induction, whenprofitable. Before to state them, we need a few preliminary remarks. Werefer to Subsection 5.3 for the definition of Γ/p, e/p and 〈e(p∼)〉Σ/p. Wedenote by p⊥/p the projective space formed by the lines and the planes ofΣ0 that contain p and are contained in p⊥. The inclusion mapping of Γ/pin p⊥/p is a full embedding and Γ/p is classical. Explicitly, regarding p andp⊥ as linear subspaces of V0 and given a complement W of p in p⊥, theprojection πp,W of p⊥ = p⊕W onto W induces an isomorphism from p⊥/pto PG(W ), which maps every line L of Σ0 containing p and contained inp⊥ onto the point L∩ [W ]. The restriction of f to W ×W defines the perp-relation associated to the image πp,W (Γ/p) of Γ/p. In particular, PΓ ∩ [W ]is contained in the set of f -isotropic points of [W ] and it spans [W ] (seeTits [35, Chapter 8]).

Lemma 8.1. The inclusion mapping ιp : Γ/p → p⊥/p satisfies (B’).

Proof. ιp satisfies (B’) if and only if p⊥⊥ = {p}. With W as above, putRW = [W ]⊥ ∩ [W ]. Then p⊥⊥ = {p} if and only if RW = ∅. Suppose to thecontrary that RW �= ∅. Choose r ∈ RW , p1 ∈ PΣ0 \p⊥ and put L = 〈p, r〉Σ0 .Then r⊥ = p⊥ = L⊥. The hyperplane p⊥1 meets the line L in a point r1. Asr⊥1 contains both L⊥ and p1, and since L⊥ = p⊥, which is a hyperplane ofΣ0, we have r⊥1 = PΣ0 , contrary to hypothesis (B’) on Γ. �

Lemma 8.2. The mapping e/p is a weak embedding of Γ/p in 〈e(p∼)〉Σ/p

and it matches with ιp.

Proof. The first claim of the lemma has already been proved (Lemma 5.10).The second claim follows from the fact that, by assumption, e matches withthe inclusion mapping e0 of Γ in Σ0. �

We now turn to the problem of extending e to a faithful embeddinge of Σ0 in Σ. The first step of this construction requires a few preliminaryresults.

Lemma 8.3. Let M1 and M2 be two maximal singular subspaces of Γ withdim(M1 ∩ M2) = n − 2. Then dim(〈e(M1 ∪ M2)〉Σ) = n.

Proof. Obviously, n − 1 ≤ dim(〈e(M1 ∪ M2)〉Σ) ≤ n. Also, dim(〈e(M1 ∪M2)〉Σ) ≥ n − 1, as dim(〈e(M1)〉Σ) = dim(〈e(M2)〉Σ) = n − 1, since e isfaithful. If dim(〈e(M1 ∪M2)〉Σ) = n− 1, then e(M2) ⊆ 〈e(M1)〉Σ. However,M∼

1 = M1, as M1 is maximal. As e is weak, the inclusion e(M2) ⊆ 〈e(M∼1 )〉Σ

forces M2 ⊆ M∼1 , hence M2 = M1, contrary to the choice of M1 and M2. �

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Lemma 8.4. Given M1 and M2 as in Lemma 8.3, there exists a uniquefaithful embedding eM1,M2 of 〈M1 ∪ M2〉Σ0 in 〈e(M1 ∪ M2)〉Σ such thateM1,M2 and e induce the same mapping on M1 ∪ M2.

Proof. Put S0 = 〈M1 ∪ M2〉Σ0 and S = 〈e(M1 ∪ M2)〉Σ. Given a basisB = {pi}n−1

i=1 of M1∩M2, for i = 1, 2 let qi ∈ Mi\(M1∩M2). Then {qi}∪B isa basis of Mi, B′ = B∪{q1, q2} is a basis of S0 and, since dim(S0) = dim(S)by Lemma 8.3, e(B′) is a basis of S. Choose a point a ∈ S0 so as to obtaina frame F = (q1, q2, p1, . . . , pn−1, a) of S0, with a as the unit point. For{i, j} = {1, 2}, the line 〈qi, a〉Σ0 meets Mj in a point aj , which can be takenas the unit point of a frame Fj = (qj , p1, . . . , pn−1, aj) of Mj . The sequencee(Fj) = (e(qj), e(p1), . . . , e(pn−1), e(aj)) is a frame of 〈e(Mj)〉Σ. The lines〈q1, a1〉Σ0 and 〈q2, a2〉Σ0 meet in a point b ∈ M1 ∩M2. We now consider theplane 〈e(q1), e(q2), e(b)〉Σ of Σ. That plane also contains the points e(a1) ande(a2). Hence the lines 〈e(a1), e(q2)〉Σ and 〈e(a2), e(q1)〉Σ meet in a point, saya′. The sequence F ′ = (e(q1), e(q2), e(p1), . . . , e(pn−1), a′) is a frame of S.Accordingly, there exists a unique embedding eM1,M2 : S0 → S that mapsF onto F ′. Clearly, for i = 1, 2 we have eM1,M2(Fi) = e(Fi). As e(Fi) is aframe of 〈e(Mi)〉Σ, eM1,M2 coincides with e on Mi. As dim(S0) = dim(S) =n, eM1,M2 is faithful. The existential part of the lemma is proved. As foruniqueness, let e′ : S0 → S be an embedding such that e′|M1∪M2

= e|M1∪M2.

Then e′(Fi) = e(Fi) for i = 1, 2. If follows that e′(F ) = F ′ = eM1,M2(F ).Hence e′ = eM1,M2 . �

In view of the next lemma, we need the following definition: given asingular subspace S of Γ, the upper residue res+Γ (S) of S is the family ofsingular subspaces of Γ that properly contain S. Note that, if rk(S) ≤ n−2,then res+Γ (S) is a polar space of rank n−rk(S). In particular, if rk(S) = n−2,then res+Γ (S) is a generalized quadrangle.

Lemma 8.5. Let M0, M1, M2, M3 be four maximal singular subspaces of Γcontaining a given singular subspace S of rank rk(S) = n − 2 and forminga quadrangle in res+Γ (S). Then dim(〈e(∪3

i=0Mi)〉Σ) = n + 1.

Proof. Put X0 = 〈∪3i=0Mi〉Σ0 and X = 〈e(∪3

i=0Mi)〉Σ. Clearly, dim(X0) =n + 1 ≥ dim(X). On the other hand, dim(X) ≥ n by Lemma 8.3. By con-tradiction, suppose that dim(X) = n. Then X = 〈e(M0∪M1)〉Σ by Lemma8.3. On the other hand, 〈e(M0 ∪ M1)〉Σ ⊆ 〈e((M0 ∩ M1)∼)〉Σ. Thereforee(M2 ∪ M3) ⊆ 〈e((M0 ∪ M1)∼)〉Σ. As e is weak, the previous inclusionimplies that M2 ∪ M3 ⊆ (M0 ∪ M1)∼, contrary to the hypothesis that{M0, M1, M2, M3} is a proper quadrangle of res+Γ (S). �

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Lemma 8.6. Let Q = {M0, M1, M2, M3} be a quadruple of maximal singularsubspaces satisfying the hypotheses of Lemma 8.5. Then, for 0 ≤ i < j <

k ≤ 3, there exists a unique faithful embedding eijkQ of 〈Mi ∪ Mj ∪ Mk〉Σ0

in 〈e(Mi ∪ Mj ∪ Mk)〉Σ such that eijkQ and e induce that same mapping on

Mi ∪ Mj ∪ Mk.

Proof. We only give a hint of the proof, leaving details for the reader.Assume that (i, j, k) = (0, 1, 2), to fix ideas. We can argue as in the proof ofLemma 8.4, but replacing M1 with 〈M0∪M1〉Σ0 and M2 with 〈M1∪M2〉Σ0 ,and constructing e012

Q in such a way as to extend the embeddings eM0,M1

and eM1,M2 supplied by Lemma 8.4. �Lemma 8.7. With Q = {M0, M1, M2, M3} and e012

Q as in Lemma 8.6, e012Q

and e induce the same mapping on M3.

Proof. For i = 0, 2, let Si = Mi ∩ M3. As e012Q and e induce the same

mappings on M0 and M2, they induce the same mapping on S0∪S2. Hencethe composition g of the restriction of e012

Q to M3 with the inverse of therestriction of e to M3 is an automorphism of M3 which fixes all points oftwo distinct hyperplanes M3∩M0 and M3∩M2 of M3. This property forcesg to be the identity. Hence e012

Q and e coincide on M3. �

Corollary 8.8. e012Q = eijk

Q for 0 ≤ i < j < k ≤ 3.

Proof. This follows from Lemma 8.7 and the uniqueness claim of Lemma8.6. �

In view of Corollary 8.8, the embedding eQ := eijkQ does not depend

on the particular choice of the triple {i, j, k} ⊂ {0, 1, 2, 3}. Thus:

Proposition 8.9. For a quadruple Q = {M0, M1, M2, M3} of maximal singu-lar subspaces of Γ as in the hypotheses of Lemma 8.5, there exists a faithfulembedding eQ of the (n + 1)-dimensional subspace 〈Q〉Σ0 := 〈∪3

i=0Mi〉Σ0 ofΣ0 in the (n+1)-dimensional subspace 〈e(Q)〉Σ := 〈e(∪3

i=0Mi)〉Σ of Σ. Theembedding eQ induces e on ∪3

i=0Mi and, for any embedding e′ : 〈Q〉Σ0 →〈e(Q)〉Σ, if e′ induces e on at least three of M0, M1, M2, M3, then e′ = eQ.

Note that, so far, neither of the hypotheses (A) or (B) has been used.Condition (A) will be exploited in the proof of the next lemma.

Lemma 8.10. Let M1, M2, M3 be three maximal singular subspaces of Γsuch that dim(M1 ∩ M2) = n − 2 and M1 ∩ M2 ⊂ M3 ⊂ 〈M1 ∪ M2〉Σ0. LeteM1,M2 be the embedding of 〈M1 ∪ M2〉Σ0 in 〈e(M1 ∪ M2)〉Σ considered inLemma 8.4. Then eM1,M2 induces e on M3.

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Proof. We will prove this lemma by induction on n = rk(Γ).

Starting the induction. Let n = 3. So, the maximal subspaces of Γ areplanes. Put X0 := 〈M1 ∪ M2〉Σ0 and X = 〈e(M1 ∪ M2)〉Σ. We recall thatdim(X0) = dim(X) = 3 (Lemma 8.3) and eM1,M2 is the unique embeddingof X0 in X that induces e on M1 ∪ M2 (Lemma 8.4). Given a point p inthe line M1 ∩ M2, let L be a line of M3 containing p and different fromM1 ∩ M2 and M be a singular plane of Γ on L and different from M3. LetH �= L be another line of M on p. For i = 1, 2, put Li,H = Mi ∩ H∼ andMi,H = 〈H ∪Li,H〉Γ. Then Li,H is a line of Mi and Mi,H is a singular planeof Γ. Put X0(H) := 〈M1,H ∪M2,H〉Σ0 and Y0(H) := 〈L1,H ∪L2,H〉Σ0 . ThenX0(H) is a 3-space, Y0(H) is a plane of Σ0 (but not a plane of Γ) andX0(H) ∩ X0 = Y0(H). Also, Y0(H) ∩ M3 is a line of M3 on p. Note thatH⊥ ∩M3 = H∼ ∩M3 = L and Y0(H) ⊂ H⊥. It follows that Y0(H)∩M3 ⊆H⊥ ∩ M3 = L. Hence L = Y0(H) ∩ M3. As M = 〈H ∪ L〉Γ = 〈H ∪ L〉Σ0 ,H = M1,H ∩ M2,H and L ⊂ Y0(H), we have M ⊂ X0(H). Thus, M , M1,H

and M2,H are three maximal singular subspaces of Γ, intersecting in Hand all contained in X0(H). By Lemma 8.4, there exists a unique faithfulembedding eH : X0(H) → X(H) := 〈e(M1,H ∪ M2,H)〉Σ such that eH

induces e on M1,H∪M2,H . As both eH and eM1,M2 induce e on L1,H∪L2,H =(M1,H ∪M2,H)∩(M1∪M2), the embedding eH and eM1,M2 induce the samemapping on Y0(H). In particular, as L ⊂ Y0(H), eM1,M2(L) = eH(L).

By hypothesis (A), e(M) ⊂ X(H) and e(M3) ⊂ X. Note that dim(X(H)) = 3, by Lemma 8.3. As e(L) is contained in either of e(M) ande(M3), we obtain that e(L) ⊂ X(H) ∩ X. Moreover M1, M2, M1,H andM2,H form a proper quadrangle in resΓ(p). By Lemma 8.5, the subspaceZ := 〈e(M1 ∪ M2 ∪ M1,H ∪ M2,H)〉Σ = 〈X ∪ X(H)〉Σ has dimension 4.As X and X(H) are distinct 3-dimensional subspaces of Z, they intersectin a plane Y (H) = X ∩ X(H). However, X0 ∩ X0(H) = Y0(H). HenceY (H) = 〈eH(Y0(H))〉Σ = 〈eM1,M2(Y0(H))〉Σ. It follows that Y (H) alsocontains eM1,M2(L) = eH(L). However, H is an arbitrary line of M on p,different from L. Accordingly, denoted by H the set of lines of M on pdifferent from L, the space ∩H∈HY (H) contains both e(L) and eM1,M2(L).We shall prove that this fact implies that e(L) = eM1,M2(L). Suppose thecontrary. Then, as Y (H) is a plane for every line H ∈ H, we obtain thatY (H) = Y (H ′) for any two lines H, H ′ ∈ H. By the faithfulness of the em-bedding eM1,M2 : X0 → X, and since X0 contains both Y0(H) and Y0(H ′),we obtain that Y0(H) = Y0(H ′). Hence the lines Li,H = Y0(H) ∩ Mi and

370 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

Li,H′ = Y0(H ′) ∩ Mi are equal. Namely, L∼i,H contains both H and H ′.

Hence M ⊆ L∼i,H , but this is a contradiction. Therefore, e(L) = eM1,M2(L).

However, L is an arbitrary line of M3 on p. Hence eM1,M2 and e induce thesame mapping on the set of lines of M3 on p. In its turn, p is an arbitrarypoint of the line M1 ∩ M2. Consequently, e and eM1,M2 induce the samemapping on the set of lines of M3, whence they also induce the same map-ping on the set of points of M3, as we wanted to prove.

The inductive step. Let n > 3. Put X0 = 〈M1 ∪ M2〉Σ0 and X = 〈e(M1 ∪M2)〉Σ. For p ∈ M1 ∩ M2, let eM1,M2/p be the mapping induced by eM1,M2

from the set of lines of X0 containing p to the set of lines of X containinge(p). By the inductive hypothesis, eM1,M2/p and e/p : Γ/p → 〈e(p∼)〉Σ/pinduce the same mapping on the set of lines of M3 through p. However, p isan arbitrary point of M1 ∩M2, which is a hyperplane of M3. It follows thateM1,M2 and e induce the same mapping on the set of lines of M3, whencethey also induce the same mapping on the set of points of M3. �

Proposition 8.11. Let Q = {M0, M1, M2, M3} be a quadruple of maxi-mal singular subspaces of Γ containing a given singular subspace S ofrank rk(S) = n − 2 and forming a proper quadrangle in res+Γ (S). LeteQ : 〈Q〉Σ0 → 〈e(Q)〉Σ be the embedding considered in Proposition 8.9.Then, for every maximal singular subspace M of Γ containing M0 ∩ M1

and contained in 〈Q〉Σ0, the embeddings eQ and e induce the same mappingon M .

Proof. As Q is a proper quadrangle in res+Γ (S), the subspace X := (M0 ∩M1)⊥∩〈Q〉Σ0 is a hyperplane of 〈Q〉Σ0 . Hence dim(X) = n, as dim(〈Q〉Σ0) =n + 1. Therefore X = 〈M0 ∪ M1〉Σ0 , as 〈M1 ∪ M2〉Σ0 also has dimension nand M0 ∪ M1 ⊂ (M0 ∩ M1)∼. Also, M ⊂ (M0 ∩ M1)∼. Hence M ⊂ X. ByLemma 8.10, eM0,M1 induces on M the same mapping as e. However, eQ

induces eM0,M1 on X, by Proposition 8.9. Hence eQ induces e on M . �

We shall now extend e to an embedding e : Σ0 → Σ. Hypothesis(B) comes into play now. Let p ∈ PΣ0 \ PΓ. According to (B’) (which isequivalent to (B)), p⊥ is a hyperplane of Σ0. Hence PΓ �⊆ p⊥. So, there isat least one maximal singular subspace M of Γ such that S := M ∩ p⊥ isan (n−2)-dimensional subspace of M . Put XM,p = 〈M, p〉Σ0 and let MM,p

be the family of maximal singular subspaces M ′ of Γ, different from M andsuch that S ⊂ M ′ ⊂ XM,p.

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Lemma 8.12. (1) dim(XM,p) = n.(2) XM,p = 〈M ∪ M ′〉Σ0 = 〈p, M ′〉Σ0 for every M ′ ∈ MM,p.(3) MM,p �= ∅.(4) If M ′ is a maximal singular subspace of Γ contained in XM,p and suchthat rk(M ∩ M ′) = n − 1, then M ′ ∈ MM,p.

Proof. Claim (1) is obvious and (2) follows from (1) and the fact thatS ⊂ M ′ and p �∈ M ′ (as p is not a point of Γ). We shall prove (3). In viewof property (P4’) of Subsection 3.2, for every point x ∈ M \ S, the line〈x, p〉Σ0 contains at least one point y of Γ different from x. Clearly, y �∈ Mand, as S ⊆ X⊥

M,p, the space 〈y, S〉Σ0 is a maximal singular subspace of Γ,different from M and contained in XM,p.

Finally, let M ′ be as in the hypotheses of (4) and suppose that M ′ �∈MM,p, namely S �⊆ M ′. By (3), we can choose M1 ∈ MM,p. As M1 and M ′

are (n − 1)-dimensional subspaces of XM,p, which is n-dimensional, theymeet in an (n−2)-dimensional subspace S1. We have S1 �= S since S �⊆ M ′.Also, S1 �= M ∩ M ′, since M1 �= M and M1 ∩ M = S �= S1 = M1 ∩ M ′.Moreover, S ∩ M ′ = S1 ∩ M = S1 ∩ S = R, say. Thus, rk(R) = n − 2and M, M ′ and M1 form a proper triangle in the generalized quadrangleres+Γ (R): we have reached a contradiction. Therefore, S ⊆ M1. �

For M ′ ∈ MM,p, let eM = eM,M ′ : XM,p → 〈e(M ∪ M ′)〉Σ be theextension of e from M ∪ M ′ to XM,p (Lemma 8.4).

Lemma 8.13. The point eM (p) ∈ PΣ does not depend on the particularchoice of M ′ ∈ MM,p.

Proof. This follows from Lemma 8.10 and the uniqueness claim of Lemma8.4. �

We put e(p) := eM (p), where M is as above.

Proposition 8.14. The point e(p) ∈ PΣ does not depend on the particularchoice of M .

Proof. We must prove that eM (p) = eM ′(p) for any two maximal singularsubspaces M and M ′ of Γ not contained in p⊥. The set H = p⊥ ∩ PΓ isa geometric hyperplane of Γ and the collinearity of Γ induces a connectedgraph on PΓ \ H (Shult [24, Lemma 5.2]). It is not difficult to see thatthis fact implies the following: For any two maximal singular subspacesM and M ′ of Γ not contained in H, there exists a sequence of maximalsingular subspaces of Γ, say M0, M1, . . . , Mk, such that Mi �⊂ H for ev-ery i ∈ {0, 1, . . . , k}, M0 = M , Mk = M ′ and rk(Mi−1 ∩ Mi) = n − 1

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for all i = 1, . . . , k. Thus, without loss of generality, we can focus on thecase where rk(M ∩ M ′) = n − 1. If M ′ ⊆ XM,p, then Lemma 8.13 yieldsthe conclusion. Otherwise, we consider the (n + 1)-dimensional subspaceX := 〈M, M ′, p〉Σ0 . Two cases must be examined.

Case 1. p⊥ ∩ M �= p⊥ ∩ M ′. Then R := p⊥ ∩ M ∩ M ′ has rank n − 2.Choose a maximal singular subspace M1 ∈ MM,p. As dim(X) = n + 1and dim(XM ′,p) = n, XM ′,p is a hyperplane of X. Accordingly, the inter-section S1 := XM ′,p ∩ M1 has dimension n − 2. Put S2 := S∼

1 ∩ M ′. AsR ⊂ S1, S2 is an (n− 2)-dimensional subspace of M ′ and M2 := 〈S1 ∪S2〉Γis a maximal singular subspace of Γ. On the other hand, M2 ⊂ XM ′,p asXM ′,p contains both S1 and M ′. By claim (4) of Lemma 8.12 we obtainthat M2 ∈ MM ′,p. The quadruple Q = {M, M1, M2, M

′} forms a properquadrangle in res+Γ (R). Clearly, X = 〈Q〉Σ0 (notation as in Proposition 8.9),XM,p = 〈M ∪M1〉Σ0 and XM ′,p = 〈M ′∪M2〉Σ0 . So, with eQ : X → 〈e(Q)〉Σas in Proposition 8.9, eQ induces eM,M1 on XM,p and eM ′,M2 on XM ′,p.Hence eM,M1(p) = eM ′,M2(p) = eQ(p), as p ∈ XM,p ∩ XM ′,p. However,M1 ∈ MM,p and M2 ∈ MM ′,p. Therefore, eM (p) = eM ′(p).

Case 2. M ∩ M ′ = p⊥ ∩ M = p⊥ ∩ M ′ = S, say. We take an (n − 2)-dimensional subspace S1 of M different from S, a maximal singular sub-space M1 on S1 different from M and an (n − 2)-dimensional subspace S2

of M1 containing R := S1 ∩ S but different from either of S1 and p⊥ ∩M1.(Note that M1 �⊆ p⊥, as M1 contains S1 �⊆ p⊥.) Let S3 = M ′ ∩ S∼

2 andM2 = 〈S2, S3〉Γ. We have S3 �= S, otherwise M, M1 and M2 would forma proper triangle in the generalized quadrangle res+Γ (R). Thus, all pairs{M, M1}, {M1, M2} and {M2, M

′} are as in Case 1. Therefore, eM (p) =eM1(p) = eM2(p) = eM ′(p). �

For p ∈ PΓ, we put e(p) := e(p). By Proposition 8.14, the mappinge : PΣ0 → PΣ, sending p ∈ PΣ0 to e(p), is well defined.

Lemma 8.15. The mapping e is injective, e(PΣ0) spans Σ and e maps everyline of Σ0 into a line of Σ.

Proof. As e induces e on PΓ and 〈e(PΓ)〉Σ = PΣ, e(PΣ0) spans Σ. We shallprove that, if p1, p2 and p3 are distinct points on a line L of Σ0, then thepoints e(p1), e(p2) and e(p3) are mutually distinct and belong to a commonline of Σ, thus proving both that e is injective and that it maps lines intolines.

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If L ∈ LΓ then the statement follows from the analogous property of e.Suppose that {p1, p2, p3} ⊂ PΓ but L �∈ LΓ. Then, given a maximal singularsubspace M1 of Γ on p1, for i = 2, 3 let Si = p∼i ∩ M1 and Mi = 〈pi, Si〉Γ.Then rk(S2) = rk(S3) = n − 1 and M2 and M3 are maximal singularsubspaces of Γ. Clearly S2 = S3 = L⊥ ∩ M1. Thus, M1, M2, M3 are asin the hypotheses of Lemma 8.10. By that lemma, the embedding eM1,M2

of 〈M1 ∪ M2〉Σ0 in 〈e(M1 ∪ M2)〉Σ induces e on M1 ∪ M2 ∪ M3. Hencee(p1), e(p2), e(p3) are distinct points of the line eM1,M2(L).

Suppose now that at least one of the points p1, p2, p3 does not belong toΓ. Let p1 �∈ PΓ, to fix ideas. Assume first that L contains a point p ∈ PΓ. Byhypothesis (B’), L⊥ is a hyperplane of p⊥. However, p⊥ is spanned by theunion of the maximal singular subspaces of Γ that contain p. (Recall thatthe inclusion mapping of Γ/p in Σ0/p is a full embedding of Γ/p.) Therefore,there exists a maximal singular subspace M of Γ such that p ∈ M �⊆ L⊥.However, L⊥ = {p, p1}⊥. Hence M �⊆ p⊥1 and, for M ′ ∈ MM,p1 we canconsider the embedding eM = eM,M ′ of XM,p1 = 〈M, L〉Σ0 in 〈e(M∪M ′)〉Σ.So e(pi) = eM (pi) for i = 1, 2, 3. Therefore e(p1), e(p2) and e(p3) are distinctpoints of the line eM (L) of Σ.

Finally, let L∩PΓ = ∅. By hypothesis (B’), the subspace L⊥ of Σ0 hascodimension 2. Hence dim(M∩L⊥) ≥ n−3 for every maximal singular sub-space M of Γ. Suppose that dim(M∩L⊥) ≥ n−2 for every maximal singularsubspace M of Γ. Then L⊥∩PΓ is a geometric hyperplane of Γ. Hence L⊥ is ahyperplane of Σ0, as the geometric hyperplanes of Γ are maximal subspacesof Γ (Shult [24, Lemma 5.2] and Γ spans Σ0. However, this is a contradiction;indeed L⊥ has codimension 2 in Σ0. Therefore, dim(M ∩L⊥) = n−3 for atleast one maximal singular subspace M of Γ. Given such a singular subspaceM0, put S = M0 ∩ L⊥. Thus, dim(S) = n − 3 by the choice of M0. Also,M0 ∩L = ∅, as L∩PΓ = ∅. Accordingly, the subspace X := 〈M0 ∪L〉Σ0 hasdimension n+1. Moreover, X⊥∩X = S. Let Γ(S, X) be the set of singularsubspaces of Γ that properly contain S and are contained in X and M(S, X)be the set of members of Γ(S, X) that are maximal singular subspaces ofΓ. By the above mentioned properties of X, Γ(S, X) is non-degenerategeneralized quadrangle and M⊥ ∩ X = M for every M ∈ M(S, X). Fori = 1, 2, put Hi = p⊥i ∩ M0. As M0 ∩ L⊥ = S, which has codimension2 in M0, H1 and H2 are distinct hyperplanes of M0 and we can chooseM1, M2 ∈ M(S, X) \ {M} in such a way that H1 ⊂ M1 and H2 ⊂ M2.As H2 �⊆ L⊥, the subspace H3 := p⊥3 ∩ M2 is a hyperplane of M2 differentfrom H2. Clearly, S ⊂ H3. Hence H∼

3 ∩M1 is a hyperplane of M1, different

374 E. Ferrara Dentice, G. Marino and A. Pasini Vol. 72 (2004)

from H1 (otherwise we obtain a proper triangle in the generalized quadran-gle Γ(S, X)). Put M3 = 〈H3, H

∼3 ∩ M1〉Γ. Clearly, M3 ∈ M(S, X). So, we

have obtained a quadruple Q = {M0, M1, M2, M3} of members of M(S, X)forming a proper quadrangle in res+Γ (S). Clearly, X = 〈Q〉Σ0 and we canconsider the embedding eQ : X → 〈e(Q)〉Σ (Proposition 8.9). We haveeQ(p1) = eM0,M1(p1), eQ(p2) = eM0,M2(p2) and eQ(p3) = eM2,M3(p3). How-ever, eM0,M1(p1) = eM0(p1) = eM1(p1) = e(p1), eM0,M2(p2) = eM0(p2) =eM2(p2) = e(p2) and eM2,M3(p3) = eM2(p3) = eM3(p3) = e(p3). Hencee(pi) = eQ(pi), for i = 1, 2, 3. Therefore, e(p1), e(p2) and e(p3) are distinctpoints of the line eQ(L) of Σ. �

Lemma 8.16. For every subset I ⊆ PΣ0, if I is independent in Σ0, then e(I)is independent in Σ.

Proof. We recall that, for every subset J ⊆ PΣ, every point of 〈J〉Σ belongsthe the span of a suitable finite subset of J . In view of this, we only need toconsider finite independent subsets I ⊂ PΣ0 . Thus, we can work by induc-tion on k = |I|. The statement of the lemma is trivial when k = 1 and itfollows from the injectivity of e when k = 2. Let I be an independent subsetof PΣ0 of size k > 2 and suppose that the statement of the lemma is validfor every independent subset of PΣ0 of size less than k. By hypothesis (B’)and since PΓ spans Σ0, we have I �⊆ p⊥ for at least one point p of Γ. Hencethe subspace X = 〈I〉Σ0 ∩ p⊥ has dimension k − 2. By induction applied toa basis of X, we have dim(〈e(X)〉Σ) = k−2. Suppose by contradiction thate(I) is dependent. Then dim(〈e(I)〉Σ) < k−1. However, 〈e(I)〉Σ ⊇ 〈e(X)〉Σand the latter is (k − 2)-dimensional. It follows that 〈e(I)〉Σ = 〈e(X)〉Σ.Choose a point q ∈ I \ (I ∩ p⊥). Hence e(q) ∈ 〈e(X)〉Σ. However, X ⊆ p⊥

and 〈q, p⊥〉Σ0 = PΣ0 since, by (B’), p⊥ is a hyperplane of Σ0. Moreover,p⊥ = 〈p∼〉Σ0 . Therefore, {q} ∪ p∼ spans Σ0. Hence 〈e({q} ∪ p∼)〉Σ = PΣ.On the other hand, e(q) ∈ 〈e(X)〉Σ ⊆ 〈e(p⊥)〉Σ = 〈e(〈p∼〉Σ0)〉Σ = 〈e(p∼)〉Σ.It follows that PΣ = 〈e(p∼)〉Σ. As e is weak, the above forces p∼ = PΓ,contrary to the non-degeneracy of Γ. In order to avoid this contradiction,we must admit that e(I) is independent. �

By lemmas 8.14 and 8.15 we immediately obtain the following propo-sition, which finishes the proof of Theorem 1.8:

Proposition 8.17. The mapping e is a faithful embedding of Σ0 in Σ.

Vol. 72 (2004) Lax Projective Embeddings of Polar Spaces 375

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Eva Ferrara DenticeSeconda Universita di NapoliDipartimento di Matematicavia Vivaldi, 4381100 - Caserta - Italye-mail: [email protected]

Giuseppe MarinoUniversita di Napoli “Federico II”Dipartimento di Matematica “R. Caccioppoli”via Cintia,80126 - Napoli - Italye-mail: [email protected]

Antonio PasiniUniversita di SienaDipartimento di MatematicaPian dei Matellini, 2253100 - Siena - Italye-mail: [email protected]

Received: March, 2004

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