Lax Projective Embeddings of Polar Spaces - Springer Link

Milan j. math. 72 (2004), 335–377 DOI 10.1007/s00032-004-0028-3 c 2004 Birkh¨ auser Verlag Basel/Switzerland

Milan Journal of Mathematics

Lax Projective Embeddings of Polar Spaces Eva Ferrara Dentice, Giuseppe Marino and Antonio Pasini Abstract. Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three points. We prove that, if Γ admits a lax embedding e : Γ → Σ in a projective space Σ defined over a skewfield K, then Γ is a classical and defined over a sub-skewfield K0 of K. Accordingly, Γ admits a full embedding e0 in a K0 -projective space Σ0 . We also prove that, under suitable hypotheses on e and e0 , there exists an embedding eˆ : Σ0 → Σ such that eê0 = e and eˆ preserves dimensions. Mathematics Subject Classification (2000). Primary 51A45; Secondary 51A50. 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 injective mapping e from the point-set PΓ of Γ to the point-set of a desarguesian projective 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 be full. 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 particular for point-line geometries arising either from buildings of Lie type or from diagram geometries of Petersen and tilde type. The interested reader is referred 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 of a Petersen or tilde geometry have size 3, a projective embedding of such a geometry is a representation in an elementary abelian 2-group. This naturally leads to the investigation of representations in possibly non-abelian groups generated by involutions (Ivanov and Shpectorov [14]), whence to techniques and problems fairly different from those which we usually meet in the literature on projective embeddings of Lie-type geometries. An attempt to fuse both approaches in a unique more general perspective is done in [20], but we will not expose it here. Turning to geometries of Lie type, full embeddings of polar spaces are well understood since a long time (Tits [35, chapter 8], Buekenhout and Lefevre [2], Dienst [9]; see also Van Maldeghem [36, section 8.5] for a survey of the case of generalized quadrangles, and Johnson [15] for the general case). Less is known on full embeddings of generalized n-gons with n > 4, in particular generalized hexagons and generalized octagons, even in the finite case. We refer to Van Maldeghem [36, section 8.6] for informations on the cases of n = 6 and n = 8. Not so much has lately been added to what one can find in [36]; we only quote a few papers by Thas and Van Maldeghem, namely [33] (a series of three papers, devoted to hexagons of order (q, 1)) and [34], devoted to finite dual Cayley hexagons. All full projective embeddings of the (embeddable) finite dual polar spaces 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 survey of this topic. For dual polar spaces of spin type in characteristic = 2, all full embeddings are known even in the infinite case (Wells [37]); similarly for 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 for a class C of geometries, we mean that every geometry Γ ∈ C admits the universal embedding, namely a full embedding euniv from which all full embeddings 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, the existence of the universal embedding is not necessary for the classification of full embeddings of a given geometry Γ, but in practice, if Γ does not admit any universal embedding, that classification can be very difficult, if not hopeless. So, it is very important to have some conditions sufficient for the existence of universal embeddings. Conditions of this kind have been found by Kasikova and Shult [16] (see also Blok and Pasini [1] for an addition to [16], suited for a few cases where the theory of [16] cannot be applied). Once we know that a given geometry Γ admits the universal embedding (for instance, Γ satisfies the conditions of [16]), we might still lack a description of that embedding. The following is the usual way to provide such a description: suppose we also know that Γ admits a finite spanning 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 embeddings leads to the investigation of spanning sets. We refer to Cooperstein [6] for the state of the art in this latter trend. So far for full embeddings. The literature on non-full embeddings is less rich. However, non-full embeddings are met quite often, in various contexts. For instance, every affine embedding can be regarded as a non-full projective embedding; when we draw a (q + 1) × (q + 1) grid on a sheet of paper, we give a non-full projective embedding of the hyperbolic quadric Q+ (3, q) in the projective plane PG(2, R); also, every Baer subgeometry of PG(d, q 2 ) entails a non-full projective embedding of PG(d, q) in PG(d, q 2 ). More significant examples arise in connection with twisted subgeometries of 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, q 2 ). The natural embedding of the grassmannian of n-spaces of V = V (2n, q 2 ) in PG(V ∧ V ) induces a lax embedding e : Γ → PG(V ∧ V ). The embedding e entails a full embedding e0 ofΓ in a suitable subgeometry Σ of PG(V ∧V ), where Σ ∼ = PG(N −1, q), N = 2n n (see Garosi [10, section 4.4] for an explicit description of a basis of Σ). Actually, when q > 2, e0 is the universal full embedding of Γ (see Cooperstein and Shult [7, 5.2]). Example 1.2. Let ∆ be the building of type D4 over GF(q 3 ), Γ be its metasymplectic space and τ a triality of ∆. It is well known that Γ admits a full embedding e in PG(N − 1, q 3 ), where N = 28 or 27 according to whether q is odd or even. The subgeometry Γτ formed by the points and lines of Γ fixed by τ is a generalized hexagon of order (q, q 3 ), denoted by T (q, q 3 ) in [36]. The full embedding e induces a lax embedding eτ of Γτ

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in a subgeometry of PG(N − 1, q 3 ). The above construction can also be rephrased as follows: Let e∗ be the ordinary full embedding of the dual Γ∗τ ∼ = T (q 3 , q) of Γτ in Σ = PG(7, q 3 ). 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 of the line-grassmannian of Σ in PG(27, q 3 ), thus obtaining eτ . Regretfully, we have 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: Suppose we want to classify lax embeddings, or at least some of them, for a class of geometries for which all full embeddings are known; if we can prove that the considered lax embeddings embody full embeddings, then most of our job is done. Admittedly, the word “embody” is too vague for the reader can 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 a skewfield K, a subskewfield K0 of K and a vector space V0 defined over K0 , we say that a lax embedding e : Γ → PG(V ) embodies an embedding e0 : Γ → PG(V0 ) if, regarded V as a K0 -vector space, there is a semilinear mapping f : V0 → V such that e = PG(f ) · e0 , where PG(f ) stands for the mapping from PG(V0 ) \ Ker(f ) to PG(V ) induced by f . Note that f is uniquely determined modulo the choice of a scalar and an embedding of K0 in K (Granai [11, section 2.4]; compare [22, proposition 9]). We call f the 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 , then we say that f is a scalar extension, also that f is faithful and e faithfully embodies e0 . (We warn that, in general, even if f is injective, it might happen that f (B) is K-dependent for some basis B of V0 .) On the other hand, when K0 = K then f is surjective, by property (E1) of embeddings and the equality e = PG(f )·e0 . In this case we call f a projection, also saying that 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 of embeddings is a bijective projection. Every morphism f : e0 → e splits as the composition f = fproj fext 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 isomorphisms. Note that, when f is injective, fproj is an isomorphism if and only if f is faithful. By [20], for every lax embedding e of Γ there exists a lax embedding e˜ of Γ such that e is a projection of e˜ and, for every embedding e of Γ, if e is a projection of e then e is a projection of e˜. The embedding e˜ is uniquely determined 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, referring the 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 point p of Γ, let p∼ be the set of points of Γ collinar with p or equal to p. We recall 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 look so ‘weak’ after all; words like “firm” might be more appropriate. However, as in the literature on lax embeddings the word “weak” is currently used with 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, we can obtain them by the following trick, essentially due to Thas and Van Maldeghem [31], [32]. Suppose that an embedding e0 : Γ → PG(d, K0 ) is given, where K0 is a field. If K0 admits a simple extension K1 of degree h > d, let e be the embedding of PG(d, K0 ) in PG(d − 1, K1 ) constructed in Remark 5.1 of Section 5. Then the composition e1 = ee0 is an embedding of Γ in PG(d − 1, K1 ). If every finite extension of K0 admits simple extensions of arbitrarily large degrees (as when K0 is finite), then we can repeat the above construction d − 2 times, thus obtaining a sequence of embeddings ek : Γ → PG(d − k, Kk ). When Γ is a polar space of rank n, then ek is non-weak for every k ≥ d − 2(n − 1) (compare Theorem 5.11). Note also that d − k can be smaller than n. In particular, ed−2 (which is the last term of this series of embeddings) embeds Γ in a plane.

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Remark 1.3. Weak embeddings of polar spaces have been introduced by Lefevre ([17], [18], [19]), who however defined them by axioms different from (W). She assumed the followings: (W’) For any two points p, x of Γ, if e(x) belongs to the span of e(p∼ ), then x ∈ p∼ ; (W”) For any two lines L, M of Γ, if the lines of Σ spanned by e(L) and e(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 different from polar spaces are considered, then the property corresponding to (W) (see [36, 8.5]) might not be equivalent to (W’) and the latter might have no relation at all with (W”). In the literature prior to 1998 embeddings satisfying (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 consider generalized quadrangles here, since a complete exposition of the state of the art for lax embeddings of generalized quadrangles would take too long. We refer the interested reader to Thas and Van Maldeghem [30] and [31] for the finite case and Steinbach and Van Maldeghem [26] and [27] for the general case. A survey of the main results of the above papers is also given by Van Maldeghem [36, section 8.6]. We only warn that, contrary to what happens for polar spaces of rank at least 3 (see Theorem 1.6), neither all generalized quadrangles that admit a lax embedding are classical, nor every lax embedding of generalized quadrangles embodies a full embedding. We refer the interested reader to the final part of this subsection for a few more details. 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 (possibly infinite) dimension d over a skewfield K, and e : Γ → PG(V ) is a lax projective embedding. When Γ is classical, we denote by euniv and KΓ the universal full embedding and the underlying skewfield of Γ. It is easily seen that KΓ is a sub-skewfield of K (see also Lemma 6.1 of this paper). We also recall that, if char(KΓ ) = 2, then euniv is the unique full embedding of Γ. (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 Γ by euniv is a quadric, then the associated quadratic form has defect δ ≤ 1 and, 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 following more concise way: Theorem 1.2 (Thas and Van Maldeghem [29]). Assume that e is weak, K is 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 Van Maldeghem [26, Theorem 5.1.1]. The second claim is a rephrasing of the main theorem of Steinbach [25]. Theorem 1.3 (Steinbach and Van Maldeghem). Assume that e is weak and Γ is classical. Then e embodies euniv . Moreover, if euniv is the unique full embedding of Γ (as when euniv (Γ) arises from a sesquilinear form, for instance), 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 type and char(KΓ ) = 2, assume that d ≥ 5. Then e embodies euniv . Finally, we mention a theorem of [21], where dominant embeddings of polar spaces of rank n ≥ 3 are characterized by means of a property slightly stronger than (W). This characterization will be crucial for the proof 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 some information on non-natural embeddings of classical quadrangles and projective 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 dimension d = 2 or 3 and order > q. Clearly, these embeddings have no relation with the natural embedding of Q+ (q) in PG(3, q), except possibly when Σ is defined over an extension of GF(q). All known finite generalized quadrangles of order (q − 1, q + 1) (types ∗ T2 (O) and AS(q)) are subgeometries of AG(3, q), whence they also admit a lax projective embedding in PG(3, q), induced by the natural embedding of AG(3, q) in PG(3, q). The quadrangle AS(3) ∼ = Q− (5, 2) is the only classical example in this family. However, the natural projective embedding of 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 above are 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. Indeed Q− (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 field K = GF(2) (see Van Maldeghem [36, section 8.6]). The generalized quadrangles Q(4, 3) and H(3, 22 ) also admit exceptional (but non-weak) embeddings, in PG(4, q) for q ≡ 1 mod 3 and, respectively, PG(3, p) for any prime p > 5 (see [36, 8.6.5] for Q(4, 3) and Thas and Van Maldeghem [28] for H(3, 22 )). However, if e is a weak projective embedding of a finite thick generalized quadrangle Γ, then Γ is classical and either e embodies the universal full projective embedding of Γ or Γ ∼ = Q(4, 2) and e is one of the exceptional embeddings of Q(4, 2) mentioned above (Thas and Van Maldeghem [30]). Weak embeddings of infinite generalized quadrangles have been classified, too (Steinbach and Van Maldeghem [26], [27]). If Γ is an infinite thick generalized quadrangle and e is a weak embedding of Γ, then Γ is Moufang and e can be explicitly described. In particular, if Γ is classical then 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 for generalized quadrangles. Indeed, the above mentioned exceptional embeddings 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. On the contrary, a few questions naturally arise from them. For instance: (1) Only finite polar spaces are considered in Theorem 1.4. What about the infinite case? Moreover, it would be nice to get rid of the restriction d ≥ 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 interesting to have a geometric proof of the latter, too. (3) In the second part of Theorem 1.3, where it is assumed that euniv is the unique full embedding of Γ, one might wonder if e faithfully embodies euniv . We shall fully answer questions (1) and (3). As for the program of (2), we make a first but far reaching step towards its accomplishment. As in the previous subsection, in the sequel Γ is a non-degenerate polar space of finite rank n ≥ 3, where all lines have at least three points. When Γ is classical, euniv is its universal full embedding and KΓ is the underlying skewfield of Γ. The next theorem fully answers question (1). We shall prove it in Section 6. Theorem 1.6. Suppose that Γ admits a lax projective embedding. Then all the 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 subsection we assume that Γ is classical. Our second main result is Theorem 1.8, to be stated below. In view of it, we need the following definition. Let e0 : Γ → Σ0 and e : Γ → Σ be two lax projective embeddings of Γ. We say that e matches with e0 if the following holds: (M) For any three maximal singular subspaces M1 , M2 , M3 of Γ such that M1 ∩ M2 ∩ M3 has rank equal to n − 1, if the span of e0 (M1 ∪ M2 ) in Σ0 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 hypotheses 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 (which can happen only when char(KΓ ) = 2), then it admits a unique terminal full embedding eterm , which embodies all full embeddings of Γ (see also Subsection 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 admits more 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, combined with Theorem 1.6(3), it immediately implies the following: Corollary 1.9. If euniv is the unique full projective embedding of Γ, then all weak 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 entirely geometric; so, the program sketched in (2) is half-done, too. That proof amounts to construct an embedding eˆ : Σ0 → Σ such that e = eê0 . The hypothesis that if euniv = eterm then e0 = eterm , is essential for that construction. To finish the program of (2) we should remove that hypothesis, allowing eˆ to be a morphism from Σ0 \ R to Σ for a suitable subspace R of Σ0 , namely proving that e faithfully embodies a quotient of e0 . Organization of the paper. Section 2 contains a number of definitions to be exploited in this paper and Section 3 contains a survey of well known properties of polar spaces. Sections 4 and 5 are preliminary to the proofs of theorems 1.6 and 1.8. In particular, Section 4 is devoted to subgeometries of projective geometries whereas in Section 5 we collect a number of results on embeddings of projective spaces and polar spaces. Sections 6 and 7 contain the proof of Theorem 1.6 and of Fact 1.7, respectively. Section 8 is devoted to 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 (the set of points) is a nonempty set, LΓ is a collection of subsets of PΓ , called lines, any two distinct points belong to at most one common line, every line contains 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 are regarded 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 containing both of them. If two points p, q are collinear, then we write p ∼ q. The graph (PΓ , ∼) is the collinearity graph of Γ. We say that Γ is connected if its collinearity graph is connected. For a point p, we denote by p∼ the set of points collinear with p or equal to p and, for X ⊆ PΓ , we put X ∼ = ∩p∈X p∼ . (We warn that many authors write p⊥ and X ⊥ instead of p∼ and X ∼ ; however, as in this paper the symbol ⊥ will be used to denote the ortogonality relation associated to a sesquilinear form of a vector space, we have replaced p⊥ and X ⊥ with p∼ and X ∼ .) If PΓ∼ = 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 such that, for any two collinear points of S, the line joining them is contained in S. Any intersection of subspaces is a subspace and PΓ is a subspace of Γ (the improper one). So, we can define the span XΓ of a subset X ⊆ PΓ as the intersection of all subspaces containing X. In particular, for two distinct points p, q, their span p, qΓ 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 are collinear, namely S ⊆ S ∼ . Every singular subspace of Γ is contained in a maximal one. A (geometric) hyperplane of Γ is a proper subspace meeting every line of Γ non-trivially. Note that, in general, not every maximal subspace is a hyperplane 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 ⊂ XΓ for every proper subset Y ⊂ X. An independent spanning set of a subspace S is called a basis of S. Clearly, every line L ∈ LΓ admits a basis: any pair of distinct points of L is a basis of L. If the improper subspace PΓ admits a basis, 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 as the minimal cardinality of a spanning set of S. In particular, if S admits a basis, 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 singular subspace of Γ} in the class of cardinal numbers. As is customary, if all bases of a singular subspace S have the same cardinality (as when S is a projective space, for instance), the number dim(S) := rk(S) − 1 is called the dimension of S. In the sequel, we will often switch from ranks to dimensions, but only when S is a projective space.

2.3. Conventions for projective spaces A projective space of dimension at least 2 where all lines are long is said to be ordinary (also, irreducible). Given a skewfield K and a K-vector space V , let Σ = PG(V ), the projective space of linear subspaces of V . For a subset X ⊆ V , we denote by [X] the set of 1-dimensional linear subspaces of V (namely, points of Σ) spanned by non-zero vectors of X. Clearly, [{0}] = ∅. The skewfield KΣ := 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 by PG(d, q) when K = GF(q).

2.4. More on embeddings Almost everthing we needed to say on embeddings and their morphisms has already been said in Introduction. We shall only fix a few more details here and add a few things, which will be exploited later in this paper. Given a projective embedding e : Γ → Σ, we call Σ the codomain of e. As stated in Introduction, Γ is supposed to be connected and Σ is an ordinary projective space, with the additional assumption that Σ is desarguesian when dim(Σ) = 2. So, Σ = PG(V ) for a vector space V . The underlying 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, henceforth we simply call them embeddings, omitting the word “projective”.

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Faithful embeddings. Let e : Γ → Σ be an embedding and assume that rk(Γ) ≥ 4. Suppose first that Γ is a linear space. Let {a, b, c} be an independent triple of points of Γ. Then a, bΓ = a, cΓ . 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 singular subspace S of Γ of rank at least 4, e induces on S a faithful embedding in the subspace e(S)Σ . When rk(Γ) ≤ 3, the above definitions are empty. In that case, all embeddings of Γ may be regarded as faithful, by convention. Remark 2.1. It is easy to see (and well known) that all full embeddings of projective spaces are isomorphisms. As a consequence, if all singular subspaces of a point-line geometry Γ are projective spaces, then every full embedding of Γ is faithful. Remark 2.2. In Introduction we have defined faithful morphisms of embeddings. Clearly, if f : e0 → e is a such a morphism, then PG(f ) is a faithful embedding of the codomain of e0 in the codomain of e. Planar embeddings. We say that an embedding e : Γ → Σ is planar if Σ is a plane. Planar embeddings are not so rare (see Remark 1.2; also Cossidente, Ferrara Dentice, Marino and Siciliano [8]), but they are nonfull, in general. For instance, if rk(Γ) ≥ 3 then no full embedding of Γ is planar, 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 then the class of planar embeddings. On the other hand, when every line of Γ belongs to a larger singular subspace, then an embedding of Γ is flat if and only if it is planar.

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3. Basics on polar spaces According to Buekenhout and Shult [3], a polar space is a point-line geometry Γ = (PΓ , LΓ ) satisfying the so-called one-all axiom: for every point p and 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 every X ⊆ PΓ . In particular, the set Rad(Γ) := PΓ∼ is a singular subspace of Γ, called the radical of Γ. The polar space Γ is said to be non-degenerate if Rad(Γ) = ∅. If Γ is non-degenerate, 2 ≤ rk(Γ) < ∞ and all lines of Γ are long, 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 subspaces of Γ are projective spaces (Buekenhout and Shult [3]). In particular, the singular subspaces of rank 3 are projective planes. They are called planes of Γ, for short. Moreover, the following properties hold (Tits [35], Buekenhout and 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 hyperplane of the projective space S and p, p∼ ∩ SΓ is a singular subspace with the same rank as S. (2) For every maximal singular subspace M and every singular subspace S of Γ (possibly, S = ∅) there exists a maximal singular subspace M ⊇ 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 given skewfield KΓ , called the underlying skewfield of Γ. Moreover, either every line 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 are pencils of lines incident to point-plane flags of PG(3, KΓ ). The planes of Gr(PG(3, KΓ )) are the sets of lines of PG(3, KΓ ) either containing a point of PG(3, KΓ ) or contained in a plane of PG(3, KΓ ).

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Non-classical grassmannians. n = 3 and Γ = Gr(PG(3, K)) for a noncommutative skewfield K. A plane of Γ is coordinatized by either K or its opposite K op , according to whether it arises from either a point or a plane of PG(3, K). Tits polar spaces. These polar spaces have rank n = 3 and their planes are Moufang 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 ≥ 3 is classical if and only if it admits a full embedding. Clearly, if Γ is classical of rank n ≥ 3, then all full embeddings of Γ are KΓ -embeddings. Moreover every classical polar space Γ admits a universal full embedding euniv : Γ → Σ = PG(V ) (Tits [35]). The image euniv (Γ) of Γ arises from a non-degenerate trace-valued sesquilinear form f or a non-singular pseudoquadratic form q of V . More explicitly, denoting by Sk (Γ) the family of singular subspaces of Γ of rank k, (P1) euniv (Sk (Γ)) is the set of totally f -isotropic (totally q-singular) kdimensional linear subspaces of V . In particular, euniv (PΓ ) is the set of 1-dimensional linear subspaces of V spanned by f -isotropic (q-singular) vectors and euniv (LΓ ) is the family of totally f -isotropic (totally q-singular) 2-dimensional linear subspaces of V . Moreover, all the following hold, where ⊥ stands for the ortogonality relation associated to the sequilinear form f or to the sesquilinearization of the pseudoquadratic form q: (P2) for any two distinct points p, q ∈ PΓ , we have p ∼ q if and only if euniv (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 the image euniv (Γ) of Γ by euniv arises from a non-singular pseudoquadratic form 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 subspaces of 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.

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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). The codomain 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 ) = 0 and 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 can still 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 than in the case of euniv . More explicitly, let f /R be the form induced by f on V /R. (Note that Rad(f /R) = Rad(f )/R; thus, f /R is non-degenerate if and 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 of Σ/R := PG(V /R). (We warn that, when K is infinite, e(PΓ ) might be a proper subset of S(f /R), even when R = Rad(f ).) As said above, no pseudoquadratic form q/R can be found on V /R such that the points of e(PΓ ) are precisely the q/R-singular points of Σ/R. Nevertheless, denoting by ⊥ 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 if e(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 be used for euniv . Remark 3.1. We have assumed n ≥ 3 in this subsection, but everything we have 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 over a 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.

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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 the point-set and L(X) as the set of lines. We call it the subgeometry induced by Σ on X. Needless to say, Σ(X) is not a projective space, in general. If Σ(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 called a subline of Σ. We say that a projective subgeometry of Σ is ordinary if it has no short lines. 4.1. Firm projective subgeometries We say that an ordinary subgeometry Σ(X) of Σ is firm if any basis of the projective space Σ(X) is independent as a set of points of Σ. Clearly, when dim(Σ(X)) is finite, Σ(X) is firm if and only if dim(Σ(X)) = dim(XΣ ). We shall firstly describe a class of firm projective subgeometries. Next, we will prove that all firm subgeometries of dimension n ≥ 2 belong to that class. For a subset U ⊆ V and a sub-skewfield K0 of K, we put U K0 = {

k

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

i=1

Henceforth, we denote by K0 U the set [U K0 ] of points [v] ∈ PΣ for v ∈ U K0 \ {0}. Clearly, the subgeometry Σ0 = Σ(K0 U ) is an ordinary projective space. Suppose that U is independent. Then Σ0 is firm and K0 is the underlying skewfield of Σ0 . We call U a vector basis of Σ0 relative to K0 and we say that K0 is the sub-skewfield of K that coordinatizes Σ0 with respect to U . Clearly, if U is another K0 -basis of V0 = U K0 , then K0 U = K0 U , 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 }ni=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 being taken as the origin, the hyperplane p1 , p2 , . . . , pn Σ0 as the hyperplane at infinity and pn+1 playing the role of the unit point. As Σ0 ∼ = PG(V0 ), the frames of Σ0 bijectively correspond to the classes of mutually proportional ordered bases of V0 . More explicitly, for every ordered basis U = (ui )ni=0 of V0 , the sequence F (U ) = (pi )n+1 i=0 , where pi = [ui ] for i = 0, 1, . . . , n and pn+1 = [u0 + u1 + . . . + un ], is a frame of Σ0 . Conversely, for every frame F = (pi )n+1 i=0 and every choice of a representative vector u0 of p0 , there exists a unique choice of representative vectors u1 , u2 , . . . , un of p1 , p2 , . . . , pn such

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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 must accordingly replace ui with λui for i = 1, 2, . . . , n, thus replacing U with λU = (λui )ni=0 . Moreover, if F (U ) = F (U ) for two ordered bases U, U of V0 , 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 also want to replace U with U = λU for a scalar λ ∈ K \ K0 . Then λU K0 = U K0 where K0 = λK0 λ−1 . Clearly, [U K0 ] = [λU K0 ] = [U K0 ], namely K0 U = K0 U . Hence, we can also take U instead of U as a vector basis of Σ0 but, when doing so, we must replace K0 with K0 . The latter is isomorphic to K0 , but possibly different from it. Every vector basis of Σ0 can be obtained from a given vector basis U of Σ0 by combining the two operations considered above, namely replacing U with another K0 -basis U of U K0 , next replacing U with λU for a scalar λ ∈ K \ K0 . So, the sub-skewfields of K that coordinatize Σ0 bijectively correspond to the conjugates of K0 \ {0} in the multiplicative group of K. Notation. Henceforth, given a vector basis U = (u0 , u1 , . . . , un ) of Σ0 , we denote by KΣ0 (u0 , u1 , . . . , un ) (also KΣ0 (U ), for short) the sub-skewfield of K 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 projective 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 = K0 U for an independent subset U of V and a sub-skewfield K0 of K. (2) Assuming that Σ(X) is firm, let U = (ui )ni=0 and U = (ui )ni=0 be two vector bases of Σ(X). Suppose that ui = 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 ) is a 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 basis.

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Lemma 4.3. Let Σ(X) be a firm projective subgeometry of Σ of finite dimension n ≥ 2, and (ui )ni=0 be a vector basis of Σ(X). Then all the following hold: (1) For a subset Y ⊆ PΣ containing X, suppose that Σ(Y ) is a firm projective subgeometry of Σ of dimension n + 1 and X is a subspace of Σ(Y ). Then, for every point p ∈ Y \ X we can choose a representative vector un+1 of p in such a way that (u0 , u1 , . . . , un , un+1 ) is a vector basis of Σ(Y ). (2) Let Σ(Y1 ), Σ(Y2 ) and Σ(Z) be finite dimensional firm projective subgeometries 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 subspace of Σ(Z) for i = 1, 2. Suppose the vectors un+1 and un+2 are such that (u0 , u1 , . . . , un , un+1 ) and (u0 , u1 , . . . , un , un+2 ) are vector bases of Σ(Y1 ) and Σ(Y2 ), respectively. Then (u0 , u1 , . . . , un , un+1 , un+2 ) is a vector basis of Σ(Z). Proof. Let q0 be the unit point of the frame associated to the vector basis (ui )ni=0 of Σ(X). Given Y and p as in (1), let v be a representative vector of p. For t ∈ K, the point p(t) = [u0 +u1 . . .+un +tv] belongs to q0 , pΣ \{p}, and all points of q0 , pΣ \ {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 point p := [u1 + . . . + un + un+1 + un+2 ]. However, those two lines are coplanar in Σ(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 (possibly infinite) dimension dim(Σ0 ) ≥ 2 is firm if and only if it admits a vector basis. Proof. The ‘if’ claim is obvious. In order to prove the ‘only if’ part, we choose a line L of Σ0 , a vector basis (u0 , u1 ) of L (which exists, by Lemma 4.1(1)) and we put K0 = KL (u0 , u1 ). Let P(L, Σ0 ) be the set of planes of Σ0 on L. In view of claim (1) of Lemma 4.3, for every X ∈ P(L, Σ0 ) we can choose 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 },

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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 and L 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, K0 U1 is contained in the point-set of Σ1 . On the other hand, [U1 ] = B is a basis of Σ1 . Hence U1 is a vector basis of Σ1 , by Lemma 4.1 and the firmness of Σ1 . We shall prove that U1 = W1 . Suppose the contrary and let w ∈ 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 M the line of X containing the points [u0 + u1 ] and p = [uX ], and recalling that 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 . This is 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-independent. Hence W1 is K-independent. It follows that U is K-independent. As every finite subset of U is a vector basis of a subgeometry of Σ0 , we also have K0 U ⊆ Σ0 . On the other hand, Σ0 ⊆ [V0 ] = K0 U , as X ⊆ [V0 ] for every X ∈ P(L, Σ0 ). Hence Σ0 = Σ(K0 U ). Therefore, U is a K0 -basis of Σ0 . Remark 4.1. Ordinary projective subgeometries of dimension n > 2 are non-firm, in general. For instance, given a field K0 and integers n, h with h > n and n > 2, let K = K0 (ω) be a simple extension of K0 of degree h. Put Y = {(xi + x0 ω i )ni=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 interested in such a broad notion here. We will only consider lines L0 ∈ L(X), for an ordinary projective subplane Σ(X) = (X, L(X)). We call these sublines

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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 a vector basis. Lemma 4.6. For a truly projective subline L0 of Σ, let (u1 , u2 ) be a vector basis of L0 and Σ(X) = (X, L(X)) an ordinary projective subplane such that L0 ∈ L(X). Given a vector basis U = (v1 , v2 , v3 ) of Σ(X), suppose that v1 = λu1 for a scalar λ ∈ K \ {0}. Then KL0 (u1 , u2 ) = λ−1 KΣ(X) (U )λ. In particular, if v1 = u1 then KL0 (u1 , u2 ) = KΣ(X) (U ). Proof. Put K1 = KL0 (u1 , u2 ) and K2 = KΣ(X) (U ), for short. Modulo replacing 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 a line 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 = K2 µ.

(4.5)

By (4.5), λ = λ·1 ∈ K2 µ. Therefore λµ−1 ∈ K2 . Accordingly, (λµ−1 )−1 ∈ K2 . Hence: (4.6) µλ−1 ∈ K2 . Condition (4.5) is equivalent to the following: K1 = λ−1 K2 µ = λ−1 K2 µλ−1 λ. However, K2 µλ−1 = K2 by (4.6). Hence K1 = λ−1 K2 λ.

(4.7)

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 a point p, let p1 and p2 be points of L1 and L2 different from p. Suppose there 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 representative vectors v1 and v2 of p1 and p2 in such a way that KL1 (v, v1 ) = KL2 (v, v2 ).

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Proof. Choose a point q = p in L3 = X1 ∩ X2 and, for i = 1, 2, choose a point qi ∈ Xi such that Fi = (p, pi , q, qi ) is a frame of Σ(Xi ). Given a representative vector v of p, let Ui = (v, vi , wi , ui ) be the vector basis of 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 to Ui . 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) of Lemma 4.1. The equality KL1 (v, v1 ) = KL2 (v, v2 ) follows.

5. A few results on embeddings of projective and polar spaces 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 line L ∈ LΓ , we put e(L) = e(PΓ )∩e(L)Σ . For every subspace S of Γ, we denote by L(S) the in S and we put Γ(S) = (S,L(S)), set of lines of Γ contained 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 mapping 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), the structures 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 line L ∈ 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, yΓ . 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 satisfied by every embedding of Γ. Hence e(L) = e(L) for every line L ∈ LΓ and e(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. In particular, 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 obvious 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 fact that Σ is desarguesian (by assumption when dim(Σ) = 2, according to the conventions 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 Γ onto a basis of Σ. As a consequence, dim(Γ) = dim(Σ). Proof. The first claim of the lemma follows from the definition of faithfulness and the fact that a set of points of a projective geometry is independent if and only if all of its finite subsets are independent. The second claim follows 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 -vector space 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.

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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 if dim(Γ) = dim(Σ). Proof. The equivalence of (1) and (2) follows from Lemma 5.4. Suppose that (1) and (2) hold. By Theorem 4.4, (2) implies the existence of a vector basis U of e(Γ). So, e is an isomorphism from Γ = PG(V0 ) to PG(U KΓ ). It is well known that every isomorphism of projective spaces defined over the same skewfield arises from a semilinear mapping. Conversely, suppose that e([v]) = [ϕ(v)] for a semilinear mapping ϕ : V0 → V , with V regarded as a KΓ -vector space. As e in injective, ϕ is injective. Hence ϕ(U ) is independent for 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 )ni=0 ∈ V (n + 1, K0 ) onto the vector (xi + x0 ω i )ni=1 of V (n, K) induces an embedding 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 finite rank n ≥ 2 and e : Γ → Σ is a weak embedding of Γ. We recall that, as Γ is a non-degenerate polar space, condition (W’) of Lemma 5.2 is equivalent to property (W), chosen in Introduction to define weak embeddings of polar spaces (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 singular subspace M of Γ such that M ∩ S = p2 , . . . , pk Γ . For every point p ∈ M we 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 , contrary k to the choice of M . We are forced to admit that {e(pi )}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 ⊆ S2∼ by (W’) and since S2 ⊆ S2∼ . 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 , namely S1 ⊆ S2 . We shall now prove that dim(Σ) ≥ 2n − 1. We will do that by an inductive argument, but we must preliminarily show that induction can be applied here. We state some notation first. Given a point p of Γ, we put Γ/p := resΓ (p). Namely, Γ/p is the polar space of rank n − 1 formed by the singular subspaces of Γ that properly contain p, the lines and the planes of Γ on p being the points and the lines of Γ/p. Let LeΣ (p) (respectively, PΣe (p)) be the set of lines (planes) of Σ that contain e(p) and are contained in e(p∼ )Σ . Regarding the planes of PΣe (p) as bundles of lines through e(p), the pair e(p∼ )Σ /p := (LeΣ (p), PΣe (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 Corollary 5.9, 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. Let n > 2 and choose a point p ∈ PΓ . By Lemma 5.10 we can apply induction to e/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.

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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 is a given non-planar embedding of Γ. Lemma 5.12. We have e(p) ∈ e(S)Σ for at least one point p and one singular plane S of Γ. Proof. Indeed, if otherwise, e(S)Σ ⊇ e(PΓ ) for every singular plane S of Γ, and e would be planar. Lemma 5.13. Let p and S be a point and a singular plane of Γ such that e(p) ∈ e(S)Σ and p ∈ S ∼ . Then there exists at least one proper quadrangle Q 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 plane S1 on p disjoint from S and a point q ∈ L. For a point p1 ∈ S \ L, put L1 = 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 contained in 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 obtain that e(p), which belongs to e(S1 )Σ , also belongs to e(S)Σ , contradicting our choice of S and p. Therefore, e(L1 ) ⊆ e(p), e(q), e(p1 )Σ for at least one choice of p1 ∈ S \ L. With p1 chosen in that way, the line L1 contains at most one point x such that e(x) ∈ e(p), e(q), e(p1 )Σ and exactly one point y ∈ q ∼ . As, by assumption, Γ has no short lines, we can take a point q1 ∈ L1 different from y and from x (if x exists). Then Q = {p, q, p1 , q1 } is a proper quadrangle and dim(e(Q)Σ ) = 3. Theorem 5.14. dim(e(Q)Σ ) = 3 for at least one proper quadrangle Q of Γ. 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 that e(p) ∈ e(S)Σ . In particular, p ∈ S. However, S = S ∼ , as n = 3. Hence p ∈ 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 admits 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(∪4i=1 e(Li )Σ ) = 4. For i = 1, 2, 3, 4, choose a point pi ∈ Li \ {p}. Then Q = {p1 , p2 , p3 , p4 } is a 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 from claim (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 the cases to examine. Eventually, we will show that the non-classical surviving case 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 and K op ∼ = 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 singular plane S of Γ. The restriction of e to S is an isomorphism from S to a projective subplane Σ(e(S)) of Σ. By Corollary 4.2, S is desarguesian and its underlying skewfield KS is a sub-skewfield of KΣ . In view of the classification of ordinary polar spaces of rank n ≥ 3 by Tits [35], either Γ is classical or 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 K op . Lemma 6.2. Suppose that Γ ∼ = Gr(PG(3, K)) for a skewfield K. Then QΓ is 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 and meets each of pi−1 and pi+1 in a point, where i = 0, 1, 2, 3 and the indices i − 1, i + 1, i + 2 are computed modulo 4. For i = 0, 1, 2, 3, let ai be the meet-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 are the 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 triple of points of αi , namely ai , ai−1 = pi−1 ∩ pi−2 and ai+1 = pi ∩ pi+1 . This forces 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 the line αi ∩ αi+2 in a point a(p) = ai+2 . Put πii+2 (p) = a(p), ai+2 ∆ . Then πii+2 (p), regarded as a point of Γ, is the unique point of Li+2 collinear with p. For q0 ∈ L0 and q1 ∈ L1 , let M0 = q0 , π02 (q0 )Γ and M1 = q1 , π13 (q1 )Γ . Then M0 and M1 correspond to the point-plane flags (b0 , β0 ) and (b1 , β1 )

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of ∆, where bi = qi ∩ πii+2 (qi ) and βi = qi , πii+2 (qi )∆ for i = 0, 1. Clearly, b0 = b1 and β0 = β1 . As b0 belongs to the line α0 ∩ α2 , which belongs to β1 , we have b0 ∈ β1 . Similarly, b1 ∈ β0 . Therefore, b0 , b1 ∆ = β0 ∩ β1 . It follows that the line r = β0 ∩ β1 , regarded as a point of Γ, belongs to either of M0 and M1 . Namely, M0 ∩ M1 = r. Thus, QΓ is a grid. Lemma 6.3. Suppose that Γ ∼ = Gr(PG(3, K)). Then K ∼ = K op and K is isomorphic to a sub-skewfield of KΣ . If moreover e is non-planar, then K is commutative. Proof. We have Σ = PG(V ) for a vector space V over KΣ and, by Lemma 6.1, KΣ contains copies of both K and K op . Moreover, every line of Γ is contained in a plane (in fact, in exactly two planes). Hence e(L) is a truly projective subline of Σ, for every line L ∈ LΓ . By Corollary 4.5, e(L) admits a vector basis. Let (v, w) be a vector basis of e(L). By Lemma 4.6, the subskewfield Ke(L) (v, w) ≤ KΣ is isomorphic to either of K and K op . (Recall that L is contained in two planes of Γ, one of which is coordinatized by K and the other one by K op .) So far, K ∼ = K op (but K might be noncommutative 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 plane S1 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 ). Put K1 := Ke(L0 ) (v1 , v0 ) = Ke(L1 ) (v1 , v2 ). Next we choose representative vectors v3 and v3 of e(p3 ) such that (v2 , v3 ) is a vector basis of e(L2 ) and (v0 , v3 ) is a vector basis of e(L3 ). Clearly, v3 = λv3 for a scalar λ ∈ KΣ \ {0}. Put K2 := Ke(L2 ) (v2 , v3 ) and K3 := Ke(L3 ) (v0 , v3 ). Thus, e(L0 ) \ {e(p0 )} = {[v1 + tv0 ] e(L1 ) \ {e(p2 )} = {[v1 + tv2 ] e(L2 ) \ {e(p3 )} = {[v2 + tv3 ] e(L3 ) \ {e(p3 )} = {[v0 + tv3 ]

| | | |

t ∈ K1 }, t ∈ K1 }, t ∈ K2 }, 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 and e(p) = [v1 + tv2 ] (resp. p ∈ L2 and e(p) = [v2 + tv3 ], or p ∈ L3 and e(p) = [v0 + tv3 ]). 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, QΓ 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. Rephrasing this fact for Σ = PG(V ), and recalling that v3 = λv3 , we obtain the the following equation admits a solution (x, y, z) ∈ KΣ3 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(QΣ ) = 3. Namely, the vectors v0 , v1 , v2 , v3 are independent. So, (6.1) is equivalent to the 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 as follows: (6.3) sα0 (t) = tα1 (s)λ. As s, t are arbitrary elements of K1 , (6.3) holds for all s, t ∈ K1 . Putting s = 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 get α0 (0) = α1 (0) = 0. As α0 and α1 are bijections, α0 (t) = 0 = α1 (s) for t, s = 0. In particular, α0 (1) = 0 = α1 (1) and the third equality of (6.4) can be rewritten as follows: λ = α1 (1)−1 α0 (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 is also commutative. End of the proof. As every embedding admits a hull, we may assume that e is 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 = fproj fext where fext : euniv → eˆ is a scalar extension and fproj : eˆ → e is a projection, for a suitable KΣ -embedding eˆ of Γ. Clearly, if e is dominant then eˆ ∼ = 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 the above decomposition f = fext fproj 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 the canonical 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 of Γ. 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 semilinear mapping from V to the underlying vector space V of Σ, induces a bijection from 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 embedding of Γ. Suppose by contradiction that e does not match with euniv . Thus, there are three distinct maximal singular singular subspaces M1 , M2 , M3 of Γ such that (¬M) rk(M1 ∩ M2 ∩ M3 ) = n − 1, euniv (M3 ) ⊆ euniv (M1 ∪ M2 )Σ0 but e(M3 ) ⊆ e(M1 ∪ M2 )Σ . (where Σ0 stands for the codomain of euniv ). For an embedding e of Γ, if e 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 may assume 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 subset B ⊂ PΓ \ {p} in such a way that A ⊂ B, e(p) ∈ e(B) and (e(B ∪ {p}) is a 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, as S contains M1 and M2 . Every proper subspace of Γ of rank at least 2 is contained in a hyperplane of Γ (see [21] for a proof of this claim). Hence S ⊆ H for a hyperplane H of Γ. Since e is assumed to be dominant, e(H) spans a hyperplane of Σ by Theorem 1.5. Hence p ∈ H, as e(H) contains e(B) and e(B) ∪ {e(p)} spans Σ. On the other hand, every hyperplane of Γ is the preimage by euniv of a hyperplane 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). A contradiction has been reached.

8. Proof of Theorem 1.8 Throughout this section Γ = (PΓ , LΓ ) is a classical polar space of finite rank n ≥ 3, e : Γ → Σ = PG(V ) is a weak K-embedding of Γ for a skewfield K ≥ KΓ and e0 : Γ → Σ0 = (PΣ0 , LΣ0 ) is a full embedding of Γ in Σ0 = PG(V0 ), for a KΓ -vector space V0 . As e is weak, it satisfies (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 M of Γ. According to the hypotheses of Theorem 1.8, when Γ admits at least two 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 embedding, 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-degenerate sesquilinear 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 rephrased as follows: (B’) PΣ⊥0 = ∅. According to the above, we may regard e as an embedding of a subgeometry Γ = e0 (Γ) of Σ0 in Σ and the thesis of Theorem 1.8 amounts to say that we can extend e to a faithful embedding eˆ : Σ0 → Σ.

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The first two lemmas of this section enable us to use induction, when profitable. Before to state them, we need a few preliminary remarks. We refer to Subsection 5.3 for the definition of Γ/p, e/p and e(p∼ )Σ /p. We denote by p⊥ /p the projective space formed by the lines and the planes of Σ0 that contain p and are contained in p⊥ . The inclusion mapping of Γ/p in p⊥ /p is a full embedding and Γ/p is classical. Explicitly, regarding p and p⊥ as linear subspaces of V0 and given a complement W of p in p⊥ , the projection πp,W of p⊥ = p ⊕ W onto W induces an isomorphism from p⊥ /p to PG(W ), which maps every line L of Σ0 containing p and contained in p⊥ onto the point L ∩ [W ]. The restriction of f to W × W defines the perprelation 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 ] (see Tits [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, put RW = [W ]⊥ ∩ [W ]. Then p⊥⊥ = {p} if and only if RW = ∅. Suppose to the contrary that RW = ∅. Choose r ∈ RW , p1 ∈ PΣ0 \ p⊥ and put L = p, rΣ0 . Then r⊥ = p⊥ = L⊥ . The hyperplane p⊥ 1 meets the line L in a point r1 . As r1⊥ contains both L⊥ and p1 , and since L⊥ = p⊥ , which is a hyperplane of Σ0 , we have r1⊥ = 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 with the inclusion mapping e0 of Γ in Σ0 . We now turn to the problem of extending e to a faithful embedding eˆ of Σ0 in Σ. The first step of this construction requires a few preliminary results. Lemma 8.3. Let M1 and M2 be two maximal singular subspaces of Γ with dim(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 is faithful. If dim(e(M1 ∪ M2 )Σ ) = n − 1, then e(M2 ) ⊆ e(M1 )Σ . However, M1∼ = M1 , as M1 is maximal. As e is weak, the inclusion e(M2 ) ⊆ e(M1∼ )Σ forces M2 ⊆ M1∼ , 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 unique faithful embedding eM1 ,M2 of M1 ∪ M2 Σ0 in e(M1 ∪ M2 )Σ such that eM1 ,M2 and e induce the same mapping on M1 ∪ M2 . Proof. Put S0 = M1 ∪ M2 Σ0 and S = e(M1 ∪ M2 )Σ . Given a basis B = {pi }n−1 i=1 of M1 ∩M2 , for i = 1, 2 let qi ∈ Mi \(M1 ∩M2 ). Then {qi }∪B is a 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 obtain a frame F = (q1 , q2 , p1 , . . . , pn−1 , a) of S0 , with a as the unit point. For {i, j} = {1, 2}, the line qi , aΣ0 meets Mj in a point aj , which can be taken as the unit point of a frame Fj = (qj , p1 , . . . , pn−1 , aj ) of Mj . The sequence e(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 the plane e(q1 ), e(q2 ), e(b)Σ of Σ. That plane also contains the points e(a1 ) and e(a2 ). Hence the lines e(a1 ), e(q2 )Σ and e(a2 ), e(q1 )Σ meet in a point, say a . 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 maps F onto F . Clearly, for i = 1, 2 we have eM1 ,M2 (Fi ) = e(Fi ). As e(Fi ) is a frame 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 for uniqueness, 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 a singular subspace S of Γ, the upper residue res+ Γ (S) of S is the family of singular 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 Γ containing a given singular subspace S of rank rk(S) = n − 2 and forming 3 a quadrangle in res+ Γ (S). Then dim(e(∪i=0 Mi )Σ ) = n + 1. Proof. Put X0 = ∪3i=0 Mi Σ0 and X = e(∪3i=0 Mi )Σ . Clearly, dim(X0 ) = n + 1 ≥ dim(X). On the other hand, dim(X) ≥ n by Lemma 8.3. By contradiction, suppose that dim(X) = n. Then X = e(M0 ∪ M1 )Σ by Lemma 8.3. On the other hand, e(M0 ∪ M1 )Σ ⊆ e((M0 ∩ M1 )∼ )Σ . Therefore e(M2 ∪ M3 ) ⊆ e((M0 ∪ M1 )∼ )Σ . As e is weak, the previous inclusion implies 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 singular subspaces satisfying the hypotheses of Lemma 8.5. Then, for 0 ≤ i < j < k ≤ 3, there exists a unique faithful embedding eijk Q of Mi ∪ Mj ∪ Mk Σ0

in e(Mi ∪ Mj ∪ Mk )Σ such that eijk Q 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 of Lemma 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. 012 Lemma 8.7. With Q = {M0 , M1 , M2 , M3 } and e012 Q as in Lemma 8.6, eQ and e induce the same mapping on M3 .

Proof. For i = 0, 2, let Si = Mi ∩ M3 . As e012 Q and e induce the same mappings on M0 and M2 , they induce the same mapping on S0 ∪ S2 . Hence the composition g of the restriction of e012 Q to M3 with the inverse of the restriction of e to M3 is an automorphism of M3 which fixes all points of two distinct hyperplanes M3 ∩ M0 and M3 ∩ M2 of M3 . This property forces g to be the identity. Hence e012 Q and e coincide on M3 . ijk Corollary 8.8. e012 Q = eQ for 0 ≤ i < j < k ≤ 3.

Proof. This follows from Lemma 8.7 and the uniqueness claim of Lemma 8.6. In view of Corollary 8.8, the embedding eQ := eijk Q 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 singular subspaces of Γ as in the hypotheses of Lemma 8.5, there exists a faithful embedding eQ of the (n + 1)-dimensional subspace QΣ0 := ∪3i=0 Mi Σ0 of Σ0 in the (n + 1)-dimensional subspace e(Q)Σ := e(∪3i=0 Mi )Σ of Σ. The embedding eQ induces e on ∪3i=0 Mi and, for any embedding e : QΣ0 → 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 Γ such that dim(M1 ∩ M2 ) = n − 2 and M1 ∩ M2 ⊂ M3 ⊂ M1 ∪ M2 Σ0 . Let eM1 ,M2 be the embedding of M1 ∪ M2 Σ0 in e(M1 ∪ M2 )Σ considered in Lemma 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 are planes. Put X0 := M1 ∪ M2 Σ0 and X = e(M1 ∪ M2 )Σ . We recall that dim(X0 ) = dim(X) = 3 (Lemma 8.3) and eM1 ,M2 is the unique embedding of X0 in X that induces e on M1 ∪ M2 (Lemma 8.4). Given a point p in the line M1 ∩ M2 , let L be a line of M3 containing p and different from M1 ∩ M2 and M be a singular plane of Γ on L and different from M3 . Let H = L be another line of M on p. For i = 1, 2, put Li,H = Mi ∩ H ∼ and Mi,H = H ∪ Li,H Γ . Then Li,H is a line of Mi and Mi,H is a singular plane of Γ. Put X0 (H) := M1,H ∪ M2,H Σ0 and Y0 (H) := L1,H ∪ L2,H Σ0 . Then X0 (H) is a 3-space, Y0 (H) is a plane of Σ0 (but not a plane of Γ) and X0 (H) ∩ X0 = Y0 (H). Also, Y0 (H) ∩ M3 is a line of M3 on p. Note that H ⊥ ∩ 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 ∪ LΓ = H ∪ LΣ0 , 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 H and all contained in X0 (H). By Lemma 8.4, there exists a unique faithful embedding 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 same mapping 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 ) and e(M3 ), we obtain that e(L) ⊂ X(H) ∩ X. Moreover M1 , M2 , M1,H and M2,H form a proper quadrangle in resΓ (p). By Lemma 8.5, the subspace Z := 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 intersect in a plane Y (H) = X ∩ X(H). However, X0 ∩ X0 (H) = Y0 (H). Hence Y (H) = eH (Y0 (H))Σ = eM1 ,M2 (Y0 (H))Σ . It follows that Y (H) also contains 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 p different from L, the space ∩H∈H Y (H) contains both e(L) and eM1 ,M2 (L). We shall prove that this fact implies that e(L) = eM1 ,M2 (L). Suppose the contrary. Then, as Y (H) is a plane for every line H ∈ H, we obtain that Y (H) = Y (H ) for any two lines H, H ∈ H. By the faithfulness of the embedding 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

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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 the same mapping on the set of lines of M3 on p. In its turn, p is an arbitrary point of the line M1 ∩ M2 . Consequently, e and eM1 ,M2 induce the same mapping on the set of lines of M3 , whence they also induce the same mapping 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 containing e(p). By the inductive hypothesis, eM1 ,M2 /p and e/p : Γ/p → e(p∼ )Σ /p induce the same mapping on the set of lines of M3 through p. However, p is an arbitrary point of M1 ∩ M2 , which is a hyperplane of M3 . It follows that eM1 ,M2 and e induce the same mapping on the set of lines of M3 , whence they 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 maximal singular subspaces of Γ containing a given singular subspace S of rank rk(S) = n − 2 and forming a proper quadrangle in res+ Γ (S). Let eQ : QΣ0 → 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 QΣ0 , the embeddings eQ and e induce the same mapping on M . Proof. As Q is a proper quadrangle in res+ Γ (S), the subspace X := (M0 ∩ M1 )⊥ ∩QΣ0 is a hyperplane of QΣ0 . Hence dim(X) = n, as dim(QΣ0 ) = n + 1. Therefore X = M0 ∪ M1 Σ0 , as M1 ∪ M2 Σ0 also has dimension n and M0 ∪ M1 ⊂ (M0 ∩ M1 )∼ . Also, M ⊂ (M0 ∩ M1 )∼ . Hence M ⊂ X. By Lemma 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 eˆ : Σ0 → Σ. Hypothesis (B) comes into play now. Let p ∈ PΣ0 \ PΓ . According to (B’) (which is equivalent to (B)), p⊥ is a hyperplane of Σ0 . Hence PΓ ⊆ p⊥ . So, there is at least one maximal singular subspace M of Γ such that S := M ∩ p⊥ is an (n − 2)-dimensional subspace of M . Put XM,p = M, pΣ0 and let MM,p be the family of maximal singular subspaces M of Γ, different from M and such 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 such that rk(M ∩ M ) = n − 1, then M ∈ MM,p . Proof. Claim (1) is obvious and (2) follows from (1) and the fact that S ⊂ M and p ∈ M (as p is not a point of Γ). We shall prove (3). In view of property (P4’) of Subsection 3.2, for every point x ∈ M \ S, the line x, pΣ0 contains at least one point y of Γ different from x. Clearly, y ∈ M ⊥ , the space y, S and, as S ⊆ XM,p 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, they meet 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 − 2 and M, M and M1 form a proper triangle in the generalized quadrangle res+ 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 the extension of e from M ∪ M to XM,p (Lemma 8.4). Lemma 8.13. The point eM (p) ∈ PΣ does not depend on the particular choice of M ∈ MM,p . Proof. This follows from Lemma 8.10 and the uniqueness claim of Lemma 8.4. We put eˆ(p) := eM (p), where M is as above. Proposition 8.14. The point eˆ(p) ∈ PΣ does not depend on the particular choice of M . Proof. We must prove that eM (p) = eM (p) for any two maximal singular subspaces M and M of Γ not contained in p⊥ . The set H = p⊥ ∩ PΓ is a geometric hyperplane of Γ and the collinearity of Γ induces a connected graph on PΓ \ H (Shult [24, Lemma 5.2]). It is not difficult to see that this fact implies the following: For any two maximal singular subspaces M and M of Γ not contained in H, there exists a sequence of maximal singular subspaces of Γ, say M0 , M1 , . . . , Mk , such that Mi ⊂ H for every 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 the case where rk(M ∩ M ) = n − 1. If M ⊆ XM,p , then Lemma 8.13 yields the conclusion. Otherwise, we consider the (n + 1)-dimensional subspace X := M, M , pΣ0 . 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 + 1 and dim(XM ,p ) = n, XM ,p is a hyperplane of X. Accordingly, the intersection S1 := XM ,p ∩ M1 has dimension n − 2. Put S2 := S1∼ ∩ M . As R ⊂ S1 , S2 is an (n − 2)-dimensional subspace of M and M2 := S1 ∪ S2 Γ is a maximal singular subspace of Γ. On the other hand, M2 ⊂ XM ,p as XM ,p contains both S1 and M . By claim (4) of Lemma 8.12 we obtain that M2 ∈ MM ,p . The quadruple Q = {M, M1 , M2 , M } forms a proper quadrangle in res+ Γ (R). Clearly, X = QΣ0 (notation as in Proposition 8.9), XM,p = M ∪ M1 Σ0 and XM ,p = M ∪ M2 Σ0 . So, with eQ : X → e(Q)Σ as 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 subspace 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 ∩ S2∼ and M2 = S2 , S3 Γ . We have S3 = S, otherwise M, M1 and M2 would form a 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 eˆ(p) := e(p). By Proposition 8.14, the mapping eˆ : PΣ0 → PΣ , sending p ∈ PΣ0 to eˆ(p), is well defined. Lemma 8.15. The mapping eˆ is injective, eˆ(PΣ0 ) spans Σ and eˆ maps every line of Σ0 into a line of Σ. Proof. As eˆ induces e on PΓ and e(PΓ )Σ = PΣ , eˆ(PΣ0 ) spans Σ. We shall prove that, if p1 , p2 and p3 are distinct points on a line L of Σ0 , then the points eˆ(p1 ), eˆ(p2 ) and eˆ(p3 ) are mutually distinct and belong to a common line of Σ, thus proving both that eˆ is injective and that it maps lines into lines.

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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 singular subspace 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 singular subspaces of Γ. Clearly S2 = S3 = L⊥ ∩ M1 . Thus, M1 , M2 , M3 are as in 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 . Hence e(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 to Γ. Let p1 ∈ PΓ , to fix ideas. Assume first that L contains a point p ∈ PΓ . By hypothesis (B’), L⊥ is a hyperplane of p⊥ . However, p⊥ is spanned by the union of the maximal singular subspaces of Γ that contain p. (Recall that the 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 can consider the embedding eM = eM,M of XM,p1 = M, LΣ0 in e(M ∪M )Σ . So eˆ(pi ) = eM (pi ) for i = 1, 2, 3. Therefore eˆ(p1 ), eˆ(p2 ) and eˆ(p3 ) are distinct points of the line eM (L) of Σ. Finally, let L ∩ PΓ = ∅. By hypothesis (B’), the subspace L⊥ of Σ0 has codimension 2. Hence dim(M ∩L⊥ ) ≥ n−3 for every maximal singular subspace M of Γ. Suppose that dim(M ∩L⊥ ) ≥ n−2 for every maximal singular subspace M of Γ. Then L⊥ ∩PΓ is a geometric hyperplane of Γ. Hence L⊥ is a hyperplane of Σ0 , as the geometric hyperplanes of Γ are maximal subspaces of 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 at least one maximal singular subspace M of Γ. Given such a singular subspace M0 , 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 ∪ LΣ0 has dimension n + 1. Moreover, X ⊥ ∩ X = S. Let Γ(S, X) be the set of singular subspaces 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 of Γ. By the above mentioned properties of X, Γ(S, X) is non-degenerate generalized quadrangle and M ⊥ ∩ X = M for every M ∈ M(S, X). For ⊥ = S, which has codimension i = 1, 2, put Hi = p⊥ i ∩ M0 . As M0 ∩ L 2 in M0 , H1 and H2 are distinct hyperplanes of M0 and we can choose M1 , 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 different from H2 . Clearly, S ⊂ H3 . Hence H3∼ ∩ M1 is a hyperplane of M1 , different

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from H1 (otherwise we obtain a proper triangle in the generalized quadrangle Γ(S, X)). Put M3 = H3 , H3∼ ∩ 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 = QΣ0 and we can consider the embedding eQ : X → e(Q)Σ (Proposition 8.9). We have eQ (p1 ) = eM0 ,M1 (p1 ), eQ (p2 ) = eM0 ,M2 (p2 ) and eQ (p3 ) = eM2 ,M3 (p3 ). However, eM0 ,M1 (p1 ) = eM0 (p1 ) = eM1 (p1 ) = eˆ(p1 ), eM0 ,M2 (p2 ) = eM0 (p2 ) = eM2 (p2 ) = eˆ(p2 ) and eM2 ,M3 (p3 ) = eM2 (p3 ) = eM3 (p3 ) = eˆ(p3 ). Hence eˆ(pi ) = eQ (pi ), for i = 1, 2, 3. Therefore, eˆ(p1 ), eˆ(p2 ) and eˆ(p3 ) are distinct points of the line eQ (L) of Σ. Lemma 8.16. For every subset I ⊆ PΣ0 , if I is independent in Σ0 , then eˆ(I) is independent in Σ. Proof. We recall that, for every subset J ⊆ PΣ , every point of JΣ belongs the the span of a suitable finite subset of J. In view of this, we only need to consider finite independent subsets I ⊂ PΣ0 . Thus, we can work by induction on k = |I|. The statement of the lemma is trivial when k = 1 and it follows from the injectivity of eˆ when k = 2. Let I be an independent subset of PΣ0 of size k > 2 and suppose that the statement of the lemma is valid for 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. Hence the subspace X = IΣ0 ∩ p⊥ has dimension k − 2. By induction applied to a basis of X, we have dim(ˆ e(X)Σ ) = k − 2. Suppose by contradiction that e(I)Σ ⊇ ˆ e(X)Σ eˆ(I) is dependent. Then dim(ˆ e(I)Σ ) < k − 1. However, ˆ and the latter is (k − 2)-dimensional. It follows that ˆ e(I)Σ = ˆ e(X)Σ . e(X)Σ . However, X ⊆ p⊥ Choose a point q ∈ I \ (I ∩ p⊥ ). Hence eˆ(q) ∈ ˆ and q, p⊥ Σ0 = PΣ0 since, by (B’), p⊥ is a hyperplane of Σ0 . Moreover, e({q} ∪ p∼ )Σ = PΣ . p⊥ = p∼ Σ0 . Therefore, {q} ∪ p∼ spans Σ0 . Hence ˆ ⊥ e(p )Σ = ˆ e(p∼ Σ0 )Σ = ˆ e(p∼ )Σ . On the other hand, eˆ(q) ∈ ˆ e(X)Σ ⊆ ˆ e(p∼ )Σ . As e is weak, the above forces p∼ = PΓ , It follows that PΣ = ˆ contrary to the non-degeneracy of Γ. In order to avoid this contradiction, we must admit that eˆ(I) is independent. By lemmas 8.14 and 8.15 we immediately obtain the following proposition, which finishes the proof of Theorem 1.8: Proposition 8.17. The mapping eˆ is a faithful embedding of Σ0 in Σ.

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References [1] R.J. Blok and A. Pasini, On absolutely universal embeddings. Discr. Math. 267 (2003), 45–62. [2] F. Buekenhout and C. Lefévre Percsy, Semiquadratic sets in finite projective spaces. J. Geometry 7 (1976), 17–42. [3] F. Buekenhout and E.E. Shult, On the foundations of polar geometry. Geometriae Dedicata 3 (1974), 155–170. [4] A.M. Cohen, Point-line spaces related to buildings. In “Handbook of Incidence Geometry” (ed. F. Buekenhout), North-Holland, 1995, 647–737. [5] A.M. Cohen and E. Shult, Affine Polar Spaces. Geometriae Dedicata 35 (1990), 43–76. [6] B.N. Cooperstein, Generation of embeddable incidence geometries: a survey. In “Topics in Diagram Geometry” (ed. A. Pasini), Quaderni di Matematica 12 (2004), 27–53. [7] B.N. Cooperstein and E.E. Shult, A note on embedding and generating dual polar spaces. Adv. Geometry 1 (2001), 37–48. [8] A. Cossidente, E. Ferrara Dentice, G. Marino and A. Siciliano On planar embeddings. Preprint. [9] K.J. Dienst, Verallgemeinerte Vierecke in projektiven R˝ aumen. Arch. Math. 35 (1980), 177–186. [10] R. Garosi, Estensioni di spazi polari duali. Tesi di Laurea, Università di Siena, a.a. 1997–1998. [11] G. Granai, Immersioni proiettive. Tesi di Laurea, Università di Siena, a.a. 2000–2001. [12] D.R. Hughes, F.C. Piper, Projective Planes. Springer-Verlag, New York, 1973. [13] A.A. Ivanov, Geometries of Sporadic Groups I. Petersen and Tilde Geometries. Cambridge Univ. Press, 1999. [14] A.A. Ivanov and S.V. Shpectorov, Geometries of Sporadic Groups II. Representations and Amalgams. Cambridge Univ. Press, 2002. [15] P.M. Johnson, Semiquadratic sets and embedded polar spaces. J. Geometry 64 (1999), 102–127. [16] A. Kasikova and E. E. Shult, Absolute embeddings of point-line geometries. J. Algebra 238 (2001), 265–291. [17] C. Lefèvre Percsy, Projectivitiés conservant un espace polaire faiblement plongé. Acad. Roy. Belg. Bull. Cl. Sci. (5) 67 (1981), 45–50. [18] C. Lefèvre Percsy, Quadrilatères généralisés faiblement plongés dans P G(3, q). Europ. J. Comb. 2 (1981), 249–255.

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[19] C. Lefèvre Percsy, Espaces polaires faiblement plongés dans un espace projectif. J. Geometry 16 (1982), 126–137. [20] A. Pasini, Embeddings and expansions. Bull. Belgian Math. Soc.-Simon Stevin 10 (2003), 585–626. [21] A. Pasini, Dominant lax embeddings of polar spaces. Preprint. [22] A. Pasini and H. Van Maldeghem, Some constructions and embeddings of the Tilde Geometry. Note di Matematica 21 (2002–2003), 13–45. [23] E. E. Shult, Embeddings and hyperplanes of Lie incidence gemetries. In “Groups of Lie Types and Their Geometries” (eds. L. Di Martino and W. Kantor), Cambridge Univ. Press, 1995, 315–232. [24] E. E. Shult, On Veldkamp lines. Bull. Belgian Math. Soc. 4 (1997), 299–316. [25] A. Steinbach, Classical polar spaces (sub-)weakly embedded in projective spaces. Bull. Belgian Math. Soc.-Simon Stevin 3 (1996), 477–490. [26] A. Steinbach and H. Van Maldeghem, Generalized quadrangles weakly embedded of degree > 2 in projective space. Forum Math. 11 (1999), 139–176. [27] A. Steinbach and H. Van Maldeghem, Generalized quadrangles weakly embedded of degree 2 in projective space. Pacific J. Math. 193 (2000), 227–248. [28] J.A. Thas and H. Van Maldeghem, Embeddings of small generalized polygons. To appear. [29] J.A. Thas and H. Van Maldeghem, Orthogonal, symplectic and unitary polar spaces sub-weakly embedded in projective space. Compositio Math. 103 (1996), 75–93. [30] J.A. Thas and H. Van Maldeghem, Generalized quadrangles weakly embedded in finite projective spaces. J. Statistical Planning and Inference 73 (1998), 353–361. [31] J.A. Thas and H. Van Maldeghem, Lax embeddings of polar spaces in finite projective spaces. Forum Math. 11 (1999), 349–367. [32] J.A. Thas and H. Van Maldeghem, Lax embeddings of generalized quadrangles in finite projective spaces. Proc. London Math. Soc. (3) 82 (2001), 402–440. [33] J.A. Thas and H. Van Maldeghem, Classifcation of embeddings of the flag geometries of projective planes in finite projective spaces. Parts 1, 2 and 3, J. Combin. Th. Ser. A 90 (2000), 159–172, 241–256, 173–196. [34] J.A. Thas and H. Van Maldeghem, Full embeddings of the finite dual split Cayley hexagons. To appear. [35] J. Tits, Buildings of Spherical Type ans Finite BN-pairs. Lecture Notes in Math. 386 Springer 1974. [36] H. Van Maldeghem, Generalized Polygons. Birkha˝ user, Basel 1998.

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[37] A.L. Wells jr., Universal projective embeddings of the Grassmann, half spinor and dual orthogonal geometries. Quart. J. Math. Oxford Ser (2) 34 (1983), 375–386. Eva Ferrara Dentice Seconda Universit` a di Napoli Dipartimento di Matematica via Vivaldi, 43 81100 - Caserta - Italy e-mail: [email protected] Giuseppe Marino Universit` a di Napoli “Federico II” Dipartimento di Matematica “R. Caccioppoli” via Cintia, 80126 - Napoli - Italy e-mail: [email protected] Antonio Pasini Universit` a di Siena Dipartimento di Matematica Pian dei Matellini, 22 53100 - Siena - Italy e-mail: [email protected] Received: March, 2004

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Lax Projective Embeddings of Polar Spaces - Springer Link

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