Grassmann and Weyl embeddings of orthogonal grassmannians

Abstract

Given a non-singular quadratic form q of maximal Witt index on $V := V(2n+1,\mathbb{F})$, let Δ be the building of type B _n formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δ _k be the k-grassmannian of Δ. Let ε _k be the embedding of Δ _k into PG(⋀^k V) mapping every point 〈v ₁,v ₂,…,v _k 〉 of Δ _k to the point 〈v ₁∧v ₂∧⋯∧v _k 〉 of PG(⋀^k V). It is known that if $\mathrm{char}(\mathbb{F})\neq2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}$. In this paper we give a new very easy proof of this fact. We also prove that if $\mathrm{char}(\mathbb{F}) = 2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}$. As a consequence, when 1<k<n and $\mathrm{char}(\mathbb{F}) = 2$ the embedding ε _k is not universal. Finally, we prove that if $\mathbb{F}$ is a perfect field of characteristic p>2 or a number field, n>k and k=2 or 3, then ε _k is universal.

1 Introduction

1.1 Definitions and notation

Let $V := V(2n+1,\mathbb{F})$ for a field $\mathbb{F}$ and let q be a non-singular quadratic form of V of Witt index n. Let Δ be the building of type B _n where the elements of type k=1,2,…,n are the k-dimensional subspaces of V totally singular for q.

Let $\mathcal{G}_{k}$ and Δ _k be the k-grassmannians of PG(V) and Δ respectively. We recall that $\mathcal{G}_{k}$ is a point-line geometry where the points are the k-dimensional subspaces of V and the lines are the sets

$$l_{X,Y} := \bigl\{Z \mid X \subset Z\subset Y ,\ \mathrm{dim}(Z) = k \bigr\} $$

for subspaces X and Y of V with dim(X)=k−1, dim(Y)=k+1 and X⊂Y. The grassmannian Δ _k is a subgeometry of $\mathcal{G}_{k}$. The points of Δ _k are the k-subspaces of V that are totally singular for q. When k<n the lines of Δ _k are the lines l _X,Y of $\mathcal{G}_{k}$ with Y totally singular. When k=n the lines of Δ _n are the sets

$$l_{X}:=\bigl\{Z\mid X \subset Z\subset X^{\perp} ,\ \mathrm{dim}(Z) = n,\ Z \mbox{ totally singular}\bigr\} $$

where X is a totally singular (n−1)-subspace of V and X ^⊥ is the subspace orthogonal to X with respect to q. Note that the points of l _X form a conic in the projective plane PG(X ^⊥/X). The geometry Δ _n is often called the dual of Δ ₁. The latter is the polar space associated to the building Δ.

Let W _k :=⋀^k V. The natural projective embedding $e_{k}:\mathcal{G}_{k}\rightarrow\mathrm{PG}(W_{k})$ of $\mathcal{G}_{k}$ maps every k-subspace 〈v ₁,v ₂,…,v _k 〉 of V to the point 〈v ₁∧v ₂∧⋯∧v _k 〉 of PG(W _k ). Let $\varepsilon_{k}:= {e_{k}}|_{\varDelta _{k}}$ be the restriction of e _k to Δ _k . When k<n the mapping ε _k is a projective embedding of Δ _k into the subspace 〈ε _k (Δ _k )〉 of PG(W _k ) spanned by ε _k (Δ _k ). We call ε _k the Grassmann embedding of Δ _k .

If k=n then ε _n maps lines of Δ _n onto non-singular conics of PG(W _n ). So, ε _n is not a projective embedding. Indeed a projective embedding of a point-line geometry Γ into the projective space PG(W) of a vector space W is an injective mapping ε from the point-set of Γ to the set of points of PG(W) such that ε maps every line of Γ surjectively onto a line of PG(W) and ε(Γ) spans PG(W) (see [19], for instance). The dimension of W is taken as the (vector) dimension dim(ε) of ε. Borrowing a word from [22], we say that an injective mapping ε from the point set of Γ to the set of points of PG(W) is a veronesean embedding if it maps every line of Γ onto a non-singular conic of PG(W) and ε(Γ) spans PG(W). (Of course, the underlying division ring of W is assumed to be commutative.) We put dim(ε)=dim(W), as for projective embeddings. With this terminology, ε _n is a veronesean embedding of Δ _n . We call it the Grassmann veronesean embedding of Δ _n , also the Grassmann embedding of Δ _n , for short. According to the previous conventions, dim(ε _n ):=dim(〈ε _n (Δ _n )〉).

We recall that Δ _n admits a projective embedding, namely the spin embedding. We shall denote it by the symbol ε _spin. The embedding ε _spin is hosted by the so-called spin module, namely the Weyl module V(ω _n ) (see below). Note that dim(V(ω _n ))=2ⁿ. Hence dim(ε _spin)=2ⁿ.

Henceforth $G := \mathrm{SO}(2n+1,\mathbb{F})$ is the stabilizer of the form q in $\mathrm{SL}(V) = \mathrm{SL}(2n+1,\mathbb{F})$. The group G also acts on W _k , according to the following rule:

$$g(v_1\wedge\cdots\wedge v_k) = g(v_1) \wedge\cdots\wedge g(v_k) \quad \mbox{for $g\in G$ and $v_{1}, \ldots, v_{k}\in V$}. $$

Note that $\mathrm{SO}(2n+1,\mathbb{F}) = \mathrm{PSO}(2n+1,\mathbb {F})$, namely G is the adjoint Chevalley group of type B _n defined over $\mathbb{F}$. The universal Chevalley group of type B _n is the spin group $\widetilde {G} = \mathrm{Spin}(2n+1,\mathbb{F})$. It acts faithfully on V(ω _n ). If $\mathrm{char}(\mathbb{F}) = 2$ then $\widetilde{G} = G$. On the other hand, if $\mathrm{char}(\mathbb{F}) \neq2$ then $\widetilde{G} = 2^{\cdot}G$, a non-split central extension of G by a group of order two.

Finally, we fix some notation for Weyl modules. Let ω ₁,ω ₂,…,ω _n be the fundamental dominant weights for the root system of type B _n , numbered in the usual way (see the picture at the beginning of this introduction). For a positive integral combination λ of ω ₁,…,ω _n , we denote by V(λ) the Weyl module over $\mathbb{F}$ with λ as the highest weight. The group $\widetilde{G}$ acts on V(λ). If its action is unfaithful then $\widetilde{G}$ induces G on V(λ), namely V(λ) is a G-module. On the other hand, if $\widetilde{G}$ acts faithfully on V(λ) and $\widetilde{G}\neq G$ then V(λ) is a $\widetilde{G}$-module but not a G-module. For instance, if $\mathrm {char}(\mathbb{F})\neq2$ then V(ω _n ) is a $\widetilde {G}$-module but not a G-module. On the other hand, V(ω ₁),V(ω ₂),…,V(ω _n−1) and V(2ω _n ) are G-modules.

Throughout this paper λ _k :=ω _k for k=1,2,…,n−1 and λ _n =2ω _n . Note that $\mathrm{dim}(V(\lambda_{k})) = {{2n+1}\choose k}$, as one can see by applying the Weyl dimension formula (see [18, 24.3], for instance).

1.2 Dimensions

The G-module 〈ε _k (Δ _k )〉 is a homomorphic image of V(λ _k ) (see [12]; also Blok [3, Sect. 9]). We say that V(λ _k ) hosts ε _k (also that ε _k lives in V(λ _k )) if 〈ε _k (Δ _k )〉≅V(λ _k ) (isomorphism of G-modules). Equivalently, $\mathrm{dim}(\varepsilon_{k}) = {{2n+1}\choose k}$, namely ε _k (Δ _k ) spans W _k .

When $\mathrm{char}(\mathbb{F}) \neq2$ the G-module V(λ _k ) is irreducible, hence 〈ε _k (Δ _k )〉=V(λ _k ) (see [12]; also Blok [3, Sect. 9]). We state this fact as a theorem.

Theorem 1.1

Let $\mathrm{char}(\mathbb{F})\neq2$. Then $\mathrm{dim}(\varepsilon_{k}) = {{2n+1}\choose k}$. In other words, V(λ _k ) hosts ε _k .

In Sect. 2 we shall give a different and very easy proof of Theorem 1.1, relying only on elementary properties of quadratic forms in odd characteristic, without asking the irreducibility of V(λ _k ) for help.

By contrast, when $\mathrm{char}(\mathbb{F}) = 2$ the following holds:

Theorem 1.2

Let $\mathrm{char}(\mathbb{F}) = 2$. Then $\mathrm{dim}(\varepsilon_{k}) = {{2n+1}\choose k} - {{2n+1}\choose{k-2}}$, where, if k=1, we adopt the convention that ${{2n+1}\choose{-1}} = 0$.

We shall prove this theorem in Sect. 3.3, obtaining it as a consequence of a more detailed statement valid when $\mathbb{F}$ is perfect (see below, Theorem 1.3). Before stating the latter result we must recall a few facts on symplectic grassmannians and their natural embeddings.

Put $V' := V(2n,\mathbb{F})$, let Δ ^sp be the building of type C _n associated to the symplectic group $\mathrm{Sp}(2n,\mathbb{F})$ in its natural action on V′ and, for k=1,2,…,n, let $\varDelta ^{sp}_{k}$ be the k-grassmannian of Δ ^sp. Then $\varDelta ^{sp}_{k}$ is a subgeometry of the k-grassmannian ${\mathcal{G}}'_{k}$ of PG(V′). Put $W'_{k}:=\bigwedge^{k} V'$ and let $e'_{k}:{\mathcal{G}}_{k}'\rightarrow \mathrm{PG}(W'_{k})$ be the natural embedding of ${\mathcal{G}}'_{k}$, sending every totally isotropic subspace 〈v ₁,…,v _k 〉 of V′ to the point 〈v ₁∧⋯∧v _k 〉 of $\mathrm{PG}(W'_{k})$. Let $\varepsilon^{sp}_{k} := e'_{k}|_{\varDelta ^{sp}_{k}}$ be the restriction of $e'_{k}$ to $\varDelta ^{sp}_{k}$. Then $\varepsilon^{sp}_{k}$ is a projective embedding of $\varDelta ^{sp}_{k}$, called the natural or Grassmann embedding of $\varDelta ^{sp}_{k}$. It is well known that $\mathrm{dim}(\varepsilon^{sp}_{k}) = {{2n}\choose k}-{{2n}\choose{k-2}}$.

Let now $\mathrm{char}(\mathbb{F}) = 2$. If moreover $\mathbb{F}$ is perfect then Δ≅Δ ^sp. Indeed, denoted by N ₀ the nucleus of the quadratic form q, namely the radical of the bilinear form associated to q, the projection from V to V/N ₀≅V′ induces an isomorphism from Δ ₁ to $\varDelta ^{sp}_{1}$, which can be regarded as an isomorphism from Δ to Δ ^sp and immediately induces an isomorphism from Δ _k to $\varDelta ^{sp}_{k}$ for every k>1. Thus, both embeddings ε _k and $\varepsilon^{sp}_{k}$ can be considered for Δ _k .

Let k>1. Given an element X of Δ of type k−1, let St(X) be its upper residue, formed by the elements of Δ of type k,k+1,…,n that contain X. We call St(X) the star of X. Clearly, St(X) is (the building of) an orthogonal polar space of rank n−k+1 defined in X ^⊥/X. Still assuming that $\mathrm{char}(\mathbb{F}) = 2$, let n _X be the nucleus of a quadratic form associated to the polar space St(X). Then n _X =N _X /X where N _X =〈X,N ₀〉. Clearly, N _X is a point of $\mathcal{G}_{k}$ and, since n _X belongs to X ^⊥/X, which is spanned by the 1-dimensional subspaces Y/X for Y ranging in the set of points of St(X), the point e _k (N _X ) of PG(W _k ) belongs to 〈ε _k (Δ _k )〉. Put $\mathcal {N}_{k}:=\langle e_{k}(N_{X})\rangle_{X\in \varDelta _{k-1}}$. Clearly, $\mathcal{N}_{k}$ is stabilized by G.

In Sect. 3 we shall prove that the mapping $\iota_{k-1}:\varDelta _{k-1}\rightarrow\mathrm{PG}(\mathcal{N}_{k})$ sending every point X of Δ _k−1 to e _k (N _X ) is a projective embedding and that $\mathcal{N}_{k}$ defines a quotient $\varepsilon_{k}/\mathcal{N}_{k}$ of ε _k . More precisely, when k<n or k=n but $\mathbb{F}$ is perfect then $\varepsilon_{k}/\mathcal{N}_{k}$ is a projective embedding in the usual sense, mapping lines of Δ _k onto lines of $\mathrm{PG}(\langle \varepsilon_{k}(\varDelta _{k})\rangle/\mathcal{N}_{k})$. On the other hand, let k=n and let $\mathbb{F}$ be non-perfect. Then $\varepsilon_{n}/\mathcal {N}_{n}$ is a lax embedding as defined in [24], namely the image of a line of Δ _n under $\varepsilon_{n}/\mathcal{N}_{n}$ is properly contained in a line of $\mathrm{PG}(\langle\varepsilon_{n}(\varDelta _{n})\rangle/\mathcal{N}_{n})$. In Sect. 3.3 we will prove the following:

Theorem 1.3

Let $\mathbb{F}$ be a perfect field of characteristic 2 and let k>1.

• (1) $\iota_{k-1}\cong\varepsilon^{sp}_{k-1}$. Consequently, $\mathrm{dim}(\mathcal{N}_{k}) = {{2n}\choose{k-1}}-{{2n}\choose{k-3}}$.

• (2) $\varepsilon_{k}/\mathcal{N}_{k}\cong\varepsilon^{sp}_{k}$, whence $\mathrm{dim}(\langle\varepsilon_{k}(\varDelta _{k})\rangle /\mathcal{N}_{k}) = {{2n}\choose{k}}-{{2n}\choose{k-2}}$.

Conjecture 1

The equalities $\mathrm{dim}(\mathcal{N}_{k})={{2n}\choose {k-1}} - {{2n}\choose{k-3}}$ and $\mathrm{dim}(\varepsilon_{k}/\mathcal{N}_{k})={{2n}\choose k} - {{2n}\choose{k-2}}$ also hold if $\mathbb{F}$ is non-perfect.

Both claims of Conjecture 1 hold true when n≤4 and $\mathbb{F}$ is any field of characteristic 2, as one can check by crude computations.

Conjecture 2

Let $\mathrm{char}(\mathbb{F}) = 2$ and k>2. Then the kernel of the projection of V(λ _k ) onto 〈ε _k (Δ _k )〉 is isomorphic to V(λ _k−2).

1.3 Results and conjectures on universality

Following Kasikova and Shult [19], we say that a projective embedding of a point-line geometry Γ is relatively universal when it is not a proper quotient of any larger embedding of Γ. A projective embedding ε of Γ is absolutely universal if all embeddings of Γ defined over the same division ring as ε are quotients of ε. If all projective embeddings of Γ are defined over the same division ring (as is the case for Δ _k ) then the absolutely universal embedding of Γ, if it exists, is uniquely determined up to isomorphisms. Clearly, every absolutely universal projective embedding is relatively universal. If Γ admits the absolutely universal embedding then the converse also holds true: all relatively universal embeddings of Γ are absolutely universal. In this case we may simply speak of universal embeddings, dropping the words ‘absolutely’ or ‘relatively’. We can do so when dealing with Δ _k . Indeed Δ _k admits the absolutely universal projective embedding (Kasikova and Shult [19]).

As remarked earlier, the G-module 〈ε _k (Δ _k )〉 is a quotient of V(λ _k ). Indeed an embedding $\widetilde{\varepsilon}_{k}$ of Δ _k can be created in PG(V(λ _k )). More explicitly, if v ₀ is a highest weight vector of V(λ _k ), then the G-orbit of 〈v ₀〉 corresponds to the set of points of Δ _k and, if P _k is the minimal fundamental parabolic subgroup of G of type k and L ₀ is the P _k -orbit of 〈v ₀〉, then the G-orbit of L ₀ corresponds to the set of lines of Δ _k . The embedding $\widetilde {\varepsilon}_{k}$ is projective when k<n and veronesean when k=n. The projection of V(λ _k ) onto 〈ε _k (Δ _k )〉 is a morphism from $\widetilde{\varepsilon}_{k}$ to the Grassmann embedding ε _k . Following Blok [3], when k<n we call $\widetilde{\varepsilon}_{k}$ the Weyl embedding of Δ _k . We call $\widetilde{\varepsilon}_{n}$ the Weyl veronesean embedding of Δ _n .

It is well known that ε ₁ is universal (Tits [23, Chap. 8]), no matter what $\mathrm{char}(\mathbb{F})$ is. Hence $\widetilde {\varepsilon}_{1} = \varepsilon_{1}$, for any field $\mathbb{F}$. By Theorem 1.1, if $\mathrm{char}(\mathbb{F})\neq2$ then $\widetilde {\varepsilon}_{k} = \varepsilon_{k}$.

Let $\mathrm{char}(\mathbb{F}) = 2$ and k>1. Then $\mathrm {dim}(\varepsilon_{k}) < \mathrm{dim}(\widetilde{\varepsilon}_{k})$ by Theorem 1.2. In this case ε _k is a proper quotient of $\widetilde{\varepsilon}_{k}$. We state this fact as a corollary, but keeping aside the case k=n for the moment, since ε _n is not a projective embedding. We will turn back to ε _n in a few lines.

Corollary 1.4

Let $\mathrm{char}(\mathbb{F}) = 2$ and 1<k<n. Then ε _k is not universal.

On the other hand, the following is quite plausible.

Conjecture 3

The Weyl embedding $\widetilde{\varepsilon}_{k}$ is universal for any k=2,3,…,n−1 and any field $\mathbb{F}$.

The following theorem, to be proved in Sect. 4, is one of the reasons that make us believe that the previous conjecture holds true.

Theorem 1.5

Let $\mathbb{F}$ be a perfect field of positive characteristic or a number field.

• (1) If n>2 then the Weyl embedding $\widetilde{\varepsilon}_{2}$ is universal.

• (2) Let n>3 and $\mathbb{F}\neq\mathbb{F}_{2}$. Then the Weyl embedding $\widetilde{\varepsilon}_{3}$ is universal.

The same conclusion as in (1) of Theorem 1.5 has been obtained by Cooperstein [14], but under the stronger assumption that $\mathbb{F}$ is a finite field of prime order. In fact Cooperstein [14] proves that when $|\mathbb{F}|$ is a prime integer, Δ ₂ can be generated by ${{2n+1}\choose2}$ points. The universality of $\widetilde{\varepsilon}_{2}$ follows from this fact.

We now turn to the veronesean embeddings ε _n and $\widetilde {\varepsilon}_{n}$. Relative universality can be defined for veronesean embeddings just in the same way as for projective embeddings. Let ε be a veronesean embedding of a point-line geometry Γ. The linear hull of ε can be defined in the same way as for projective embeddings (see [21], for instance) and it is characterized as an initial object in the full sub-category of the category of veronesean embeddings of Γ formed by those embeddings e′ for which Hom(e′,e)≠∅ (see [21] for details). We say that e is relatively universal if it is its own linear hull. Thus, it makes sense to ask whether ε _n or $\widetilde{\varepsilon}_{n}$ are relatively universal or not. By Theorem 1.2 we immediately obtain the following:

Corollary 1.6

If $\mathrm{char}(\mathbb{F}) = 2$ then ε _n is not relatively universal.

Actually, when $\mathrm{char}(\mathbb{F}) = 2$ the Weyl veronesean embedding $\widetilde{\varepsilon}_{n}$ is not relatively universal either (see [11]), but perhaps $\widetilde{\varepsilon}_{n}$ is relatively universal when $\mathrm{char}(\mathbb{F})\neq2$.

We warn that now we are not allowed to jump from relative universality to absolute universality as we can do when dealing with projective embeddings of Δ _k . Indeed we do not know if Δ _n admits an absolutely universal veronesean embedding when $\mathbb{F}\neq\mathbb {F}_{2}$. (If $\mathbb{F}= \mathbb{F}_{2}$ then Δ _n admits an absolutely universal veronesean embedding, obtained by taking the point-set of Δ _n as a basis of an $\mathbb{F}_{2}$-vector space.)

Another important difference exists between veronesean and projective embeddings: the dimension of a projective embedding of a point-line geometry Γ cannot be larger than the minimal number of points needed to generate Γ while the dimension of a veronesean embedding of Γ can be far larger than that number. For instance, if $\mathrm{char}(\mathbb{F}) \neq2$ then Δ _n can be generated by 2ⁿ points (Blok and Brouwer [4], Cooperstein and Shult [15]), whence every projective embedding of Δ _n is at most 2ⁿ-dimensional. Actually dim(ε _spin)=2ⁿ. Therefore ε _spin is universal when $\mathrm{char}(\mathbb{F})\neq2$. By contrast, $\mathrm {dim}(\widetilde {\varepsilon}_{n}) = {{2n+1}\choose n} > 2^{n}$. In fact, the usual notion of generation is unfit for veronesean embeddings. We will say more on this point in Sect. 4.4.

As recalled above, ε _spin is universal when $\mathrm{char}(\mathbb{F}) \neq2$. On the other hand, let $\mathbb {F}$ be a perfect field of characteristic 2. Then ε _spin is a quotient of $\varepsilon^{sp}_{n}$ (Blok, Cardinali and De Bruyn [5]; see also Cardinali and Lunardon [10]). In this case V(λ _n )=V(2ω _n ) admits a chain of submodules V(λ _n )⊃A⊃B⊃C⊃0 where $\mathrm{dim}(A) = {{2n+1}\choose n}-2^{n}$, $\mathrm{dim}(B) = {{2n+1}\choose{n-1}}$ and $\mathrm{dim}(C) = {{2n+1}\choose{n-2}}$, C is the kernel of the projection of $\widetilde{\varepsilon}_{n}$ onto ε _n , V(λ _n )/B hosts $\varepsilon^{sp}_{n}$ (by Lemma 1.3) and V(λ _n )/A≅V(ω _n ) hosts ε _spin. Moreover, $B/C = \mathcal{N}_{n}$ hosts $\varepsilon^{sp}_{n-1}$ by Lemma 1.3 and A/B hosts a projective embedding of $\varDelta ^{sp}_{n-2}$ (see [5], also [10]).

1.4 Non-universality of $\varepsilon^{sp}_{k}$ when $\mathrm{char}(\mathbb{F}) = 2$ and k<n

It is known that $\varDelta ^{sp}_{k}$ admits the absolutely universal projective embedding, for every k=1,2,…,n (Kasikova and Shult [19]). When $\mathrm{char}(\mathbb{F}) \neq2$, the absolutely universal projective embedding of $\varDelta ^{sp}_{k}$ is just $\varepsilon^{sp}_{k}$ [1, 2, 6, 13]. On the other hand, it is well known that $\varepsilon^{sp}_{1}$ is not universal when $\mathrm{char}(\mathbb{F}) = 2$ (Tits [23, Chap. 8]; see also De Bruyn and Pasini [17] for the non-perfect case). If $\mathbb{F}$ is a perfect field of characteristic 2 and 1<k<n then $\varepsilon^{sp}_{k} \cong\varepsilon_{k}/\mathcal{N}_{k}$ by (1) of Lemma 1.3. Therefore:

Corollary 1.7

Let $\mathbb{F}$ be a perfect field of characteristic 2 and let k<n. Then the embedding $\varepsilon^{sp}_{k}$ is not universal.

In this corollary, the restriction k<n is essential. Indeed the isomorphism $\varepsilon^{sp}_{n} \cong\varepsilon_{n}/\mathcal{N}_{n}$ gives no information on the linear hull of $\varepsilon^{sp}_{n}$, since ε _n is not a projective embedding. In fact, if $|\mathbb {F}| > 2$ then $\varepsilon^{sp}_{n}$ is universal (Cooperstein [13] in the finite case, De Bruyn and Pasini [16] for the infinite case). On the other hand, if $\mathbb{F}= \mathbb{F}_{2}$ then $\varepsilon^{sp}_{n}$ is not universal (Li [20], Blokhuis and Brouwer [9]).

2 An elementary proof of Theorem 1.1

Throughout this section $\mathrm{char}(\mathbb{F}) \neq2$. We also assume that k>1, since when k=1 there is nothing to prove. Indeed ε ₁ is the natural embedding of the polar space Δ ₁ into W ₁=V. Obviously, 〈ε ₁(Δ ₁)〉=W ₁.

From now on we will often take the liberty of using the symbol Δ _k to denote both the point-line geometry Δ _k and its point-set. However these little abuses will be harmless. The context will always help to avoid any confusion.

For h=0,1,…,k let $\mathcal{G}_{k}^{(h)}$ be the set of k-subspaces X of V such that cod _X (X∩X ^⊥)≤h, where ⊥ is the orthogonality relation defined by the bilinear form f _q associated to q. So,

$$\varDelta _k = \mathcal{G}_k^{(0)} \subset \mathcal{G}^{(1)}_k \subset \cdots \subset \mathcal{G}_k^{(k-1)} \subset\mathcal{G}_k^{(k)} = \mathcal{G}_k. $$

Lemma 2.1

For every h=1,…,k, if $X\in\mathcal{G}_{k}^{(h)}$ then there exists a line l of $\mathcal{G}_{k}$ through X such that $|l\cap\mathcal {G}_{k}^{(h-1)}| \geq2$.

Proof

Assume firstly that h=k and let $X\in\mathcal{G}^{(k)}_{k}\setminus \mathcal{G}^{(k-1)}_{k}$, namely X is a k-subspace of V such that X∩X ^⊥=0. Then q induces a non-singular quadratic form on X. Hence X contains at least one (k−1)-subspace Z such that q induces a non-singular form on Z. Consequently Z∩Z ^⊥=0, because $\mathrm{char}(\mathbb{F}) \neq2$. Therefore V=Z⊕Z ^⊥ and q induces a non-singular quadratic form q′ on Z ^⊥. Clearly, dim(Z ^⊥)=(2n+1)−(k−1)=2n+2−k. By this fact and the well known Grassmann formula for dimensions of sums and intersections of subspaces one easily sees that every n-subspace of V meets Z ^⊥ non-trivially. In particular, every maximal totally singular subspace of V has non-trivial intersection with Z ^⊥. It follows that Z ^⊥ contains at least one singular point of PG(V). On the other hand, q′ is non-singular. It is also trace-valued, because $\mathrm{char}(\mathbb{F})\neq2$. Hence Z ^⊥ is spanned by the singular points contained in it (compare Tits [23, Lemma 8.1.6]).

Clearly, dim(Z ^⊥∩X)=1. Let x be a non-zero vector in Z ^⊥∩X. Suppose that every singular point of Z ^⊥ is orthogonal to x. Then Z ^⊥⊆X ^⊥ because Z ^⊥ is spanned by its singular points and X=〈x,Z〉. This forces X⊆Z, contrary to the choice of Z. It follows that $x\not\perp x_{1}$ for at least one singular point 〈x ₁〉 of PG(Z ^⊥). The non-degenerate projective line 〈x,x ₁〉 of PG(Z ^⊥) contains one more singular point 〈x ₂〉 of PG(Z ^⊥). Let X _i :=〈Z,x _i 〉, i=1,2. Then $X_{i}\cap X_{i}^{\perp}= \langle x_{i}\rangle$. Therefore $X_{i}\in \mathcal{G}_{k}^{(k-1)}$. Moreover, X,X ₁ and X ₂ contain Z and are contained in the (k+1)-space Y:=〈X,x ₁〉=〈X,x ₂〉. The line l _Z,Y of $\mathcal{G}_{k}$ has the required properties: it contains X and two points of $\mathcal{G}_{k}^{(k-1)}$, namely X ₁ and X ₂.

Let now h<k. Put R=X∩X ^⊥, X _R :=X/R and V _R :=R ^⊥/R. Then dim(V _R )=2n+1−2(k−h) and q induces a non-singular quadratic form q _R on V _R , with maximal Witt index n−k+h. (We warn that the hypothesis that $\mathrm{char}(\mathbb{F}) \neq2$ is implicitly used in this reduction.) We can now argue as in the previous case, replacing X with X _R , V with V _R and q with q _R . We leave the details for the reader. □

We recall that a set S of points of a point-line geometry Γ is a subspace of Γ if S contains every line l of Γ such that |l∩S|≥2. Intersections of subspaces are still subspaces. So, given a set of points S of Γ we can consider the span 〈S〉 _Γ of S in Γ, namely the smallest subspace of Γ containing S, defined as the intersection of all subspaces containing S. We say that a set S of points of Γ generates Γ if 〈S〉 _Γ =Γ.

Proposition 2.2

The point-set of Δ _k generates $\mathcal{G}_{k}$.

Proof

By Lemma 2.1, every point of $\mathcal{G}_{k}^{(h)}$ belongs to at least one line meeting $\mathcal{G}_{k}^{(h-1)}$ in two distinct points. Hence $\mathcal{G}_{k}^{(h)}\subseteq\langle\mathcal{G}_{k}^{(h-1)}\rangle_{\mathcal {G}_{k}}$. So, $\mathcal{G}_{k} = \langle\mathcal{G}^{(0)}_{k}\rangle_{\mathcal{G}_{k}}$, namely Δ _k spans $\mathcal{G}_{k}$. □

By Proposition 2.2, 〈ε _k (Δ _k )〉=W _k . Equivalently, $\mathrm{dim}(\varepsilon_{k}) = \mathrm {dim}(W_{k}) = {{2n+1}\choose k}$. This forces V(λ _k )=W _k , as claimed in Theorem 1.1.

3 A quotient of ε _k when $\mathrm{char}(\mathbb{F}) = 2$

Throughout this section, $\mathrm{char}(\mathbb{F}) = 2$ and k>1. Up to rescaling the form q when $\mathbb{F}$ is non-perfect, we can assume to have chosen an ordered basis B=(e ₁,e ₂,…,e _2n+1) of V with respect to which

$$q(x_1,\ldots, x_{2n+1}) = \sum_{i=1}^nx_ix_{n+i} + x_{2n+1}^2. $$

We set I:={1,2,…,2n+1} and $B_{\wedge}:= (e_{J})_{J\in{I\choose k}}$, where $I\choose k$ stands for the set of subsets of I of size k and $e_{J} = e_{j_{1}}\wedge e_{j_{2}}\wedge \cdots\wedge e_{j_{k}}$ for every k-subset J={j ₁,…,j _k } of I, with the convention that j ₁<j ₂<⋯<j _k .

The radical of the bilinear form associated to q is a 1-dimensional subspace N ₀ of V. It is called the nucleus of the quadric $Q(2n,\mathbb{F} )$ of PG(V) represented by the equation q(x ₁,…,x _2n+1)=0, also the nucleus of q, for short. With B as above, N ₀ is spanned by the vector n ₀=(0,0,…,0,1).

As in Sect. 2, in the sequel we will freely use symbols as Δ _k and Δ _k−1 to denote both point-line geometries and their point-sets. In order to avoid duplication of notation, we will also often use the same symbols for vector subspaces and the corresponding projective subspaces. Every time the context will make it clear if we are considering vector or projective spaces.

3.1 The subspace $\mathcal{N}_{k}$ and the embedding ι _k−1

Given a point X of Δ _k−1 let St(X) be its star, defined as in the introduction of this paper. As noticed there, St(X) is isomorphic to an orthogonal polar space of rank n−k+1, naturally embedded in X ^⊥/X. Let q _X be a quadratic form of X ^⊥/X associated to that polar space and let n _X be its nucleus. Then n _X =N _X /X for a uniquely determined k-subspace N _X of V containing X and contained in X ^⊥. On the other hand, N ₀⊂X ^⊥ and N ₀∩X=0. Hence 〈X,N ₀〉 is a k-subspace of X ^⊥ containing X. Moreover, 〈X,N ₀〉⊆X ^⊥⊥. Therefore n _X =〈X,N ₀〉/X, namely

$$N_X = \langle X,N_0\rangle. $$

We warn that N _X is totally isotropic but it is not totally singular. Hence N _X is a point of $\mathcal{G}_{k}$ (actually it belongs to $\mathcal{G}_{k}^{(1)}$) but it is not a point of Δ _k . Put $\mathcal{N}_{k} := \langle e_{k}(N_{X})\rangle_{X\in \varDelta _{k-1}}$. We call $\mathcal{N}_{k}$ the global nucleus of ε _k .

Lemma 3.1

$\mathcal{N}_{k}\subseteq\langle\varepsilon_{k}(\varDelta _{k})\rangle$.

Proof

The mapping sending every Y∈Δ _k ∩St(X) to Y/X is isomorphic to the natural embedding of the polar space St(X). Hence the vector space X ^⊥/X is spanned by the 1-dimensional subspaces Y/X for Y∈Δ _k ∩St(X). Consequently, $e_{k}(N_{X})\in\langle\varepsilon_{k}(Y)\rangle_{Y\in \varDelta _{k}\cap\mathrm{St}(X)}\subseteq\langle\varepsilon_{k}(\varDelta _{k})\rangle$. Therefore $\mathcal{N}_{k}\subseteq\langle\varepsilon_{k}(\varDelta _{k})\rangle$. □

For every X∈Δ _k−1, put ι _k−1(X):=e _k (N _X ).

Lemma 3.2

The mapping ι _k−1 is a projective embedding of Δ _k−1 into $\mathrm{PG}(\mathcal{N}_{k})$.

Proof

Let $\nu_{k-1}:\varDelta _{k-1}\rightarrow\mathcal{G}_{k}$ be the mapping sending every point X of Δ _k−1 to the point N _X =〈X,N ₀〉 of $\mathcal{G}_{k}$. Then ι _k−1=e _k ∘ν _k−1. It is easily seen that the mapping ν _k−1 is an embedding of Δ _k−1 into a subgeometry of $\mathcal{G}_{k}$, namely it is injective and it maps lines of Δ _k−1 onto lines of $\mathcal{G}_{k}$. On the other hand e _k , being a projective embedding, is injective and maps lines of $\mathcal{G}_{k}$ onto lines of PG(W _k ). Therefore ι _k−1 is injective and maps lines of Δ _k−1 onto lines of PG(W _k ) (contained in $\mathrm{PG} (\mathcal{N}_{k})$). □

We shall now give an explicit description of ι _k−1. In the sequel we regard a vector of V as the same thing as its sequence of coordinates with respect to the basis B. Coordinates in W _k are given with respect to the standard basis B _∧ of W _k , defined at the beginning of Sect. 3.

For X∈Δ _k−1, let {x ₁,…,x _k−1} be a basis of the (k−1)-subspace X. Let M _X =(x ₁,…,x _k−1) be the [(k−1)×(2n+1)]-matrix with x ₁,…,x _k−1 as the rows and let $M'_{X} = (x_{1},\ldots, x_{k-1}, n_{0})$ be the [k×(2n+1)]-matrix obtained by adding n ₀ to M _X as a further row. Let $(X_{J})_{J\in {I\choose k}}$ be the sequence of coordinates of a representative vector v _X of ι _k−1(X)=e _k (N _X ). Since N _X =〈X,N ₀〉=〈x ₁,…,x _k−1,n ₀〉, we can assume to have chosen v _X in such a way that X _J is the determinant of the (k×k)-submatrix of $M'_{X}$ formed by the columns indexed by the elements of J. Recall that n ₀=(0,0,…,1). Hence X _J =0 whenever 2n+1∉J while if 2n+1∈J then X _J is the determinant of the [(k−1)×(k−1)]-submatrix of M _X formed by the columns indexed by elements of J∖{2n+1}. So, regarding N _X /N ₀=〈X,N ₀〉/N ₀ as a point of $\mathcal {G}_{k-1}$ and $(X_{J})_{2n+1\in J\in{I\choose k}}$ as a vector of W _k−1, we have $(X_{J})_{2n+1\in J\in{I\choose k}} = e_{k-1}(N_{X}/N_{0})$. Suppose to have ordered the set $I\choose k$ in such a way that the k-subsets containing 2n+1 come as last. Then we can rephrase the above as follows:

Lemma 3.3

The last ${2n}\choose{k-1}$ coordinates of ι _k−1(X) are the same as the coordinates of e _k−1(N _X /N ₀). The remaining coordinates of ι _k−1(X) are null.

Proposition 3.4

Let $\mathbb{F}$ be perfect. Then $\iota_{k-1}\cong\varepsilon^{sp}_{k-1}$.

Proof

When $\mathbb{F}$ is perfect the mapping sending every totally singular subspace X of V to 〈X,N ₀〉/N ₀ is an isomorphism from Δ to Δ ^sp. The isomorphism $\iota_{k-1}\cong \varepsilon^{sp}_{k-1}$ immediately follows from this remark and Lemma 3.3. □

3.2 The quotient $\varepsilon_{k}/\mathcal{N}_{k}$

By Lemma 3.1 we know that $\mathcal{N}_{k}\subseteq\langle \varepsilon_{k}(\varDelta _{k})\rangle$. In this subsection we shall prove that $\mathcal{N}_{k}$ satisfies both the following:

• (Q1) $\mathrm{PG}(\mathcal{N}_{k})\cap\varepsilon_{k}(\varDelta _{k}) = \emptyset$;

• (Q2) $\langle\varepsilon_{k}(X_{1}), \varepsilon_{k}(X_{2})\rangle \cap \mathcal{N}_{k} = 0$ for any two distinct points X ₁ and X ₂ of Δ _k .

Properties (Q1) and (Q2) allow us to define the quotient $\varepsilon_{k}/\mathcal{N}_{k}$ as the composition of ε _k with the canonical projection of 〈ε _k (Δ _k )〉 onto $\langle \varepsilon_{k}(\varDelta _{k})\rangle/\mathcal{N}_{k}$. In view of (Q1), this composition is a mapping from the point-set of Δ _k to the set of points of the projective space $\mathrm{PG}(\langle\varepsilon_{k}(\varDelta _{k})\rangle/\mathcal{N}_{k})$ of $\langle\varepsilon_{k}(\varDelta _{k})\rangle /\mathcal{N}_{k}$, sending X∈Δ _k to the point $\langle \varepsilon_{k}(X), \mathcal{N}_{k}\rangle/\mathcal{N}_{k}$. By (Q2), this mapping is injective.

When k<n the mapping $\varepsilon_{k}/\mathcal{N}_{k}$ maps every line of Δ _k bijectively onto a line of $\mathrm{PG}(\langle \varepsilon_{k}(\varDelta _{k})\rangle/\mathcal{N}_{k})$. Hence it is a projective embedding. As we shall see at the end of this subsection, when $\mathbb{F}$ is perfect $\varepsilon_{n}/\mathcal{N}_{n}$ is a projective embedding in the usual sense, mapping every line of Δ _n bijectively onto a line of $\mathrm{PG} (\langle\varepsilon_{n}(\varDelta _{n})\rangle/\mathcal{N}_{n})$. If $\mathbb{F}$ is non-perfect then $\varepsilon_{n}/\mathcal{N}_{n}$ maps every line l of Δ _n into a line $\bar{l}$ of $\mathrm{PG}(\langle\varepsilon_{n}(\varDelta _{n})\rangle/\mathcal{N}_{n})$, but not all points of $\bar{l}$ are images of points of l by $\varepsilon_{n}/\mathcal{N}_{n}$.

In view of the above, it is convenient to slightly revise our terminology. From now on we say that a projective embedding ε:Γ→PG(W) as defined in Sect. 1.1 is a full projective embedding. On the other hand, following [24], if a mapping ε maps the lines of Γ injectively but possibly non-surjectively into lines of PG(W) then we say that ε is a lax projective embedding. So, we can rephrase as follows what we have said above: if k<n then $\varepsilon_{k}/\mathcal{N}_{k}$ is full while $\varepsilon_{n}/\mathcal {N}_{n}$ is full when $\mathbb{F}$ is perfect and it is lax but not full when $\mathbb{F}$ is non-perfect.

Lemma 3.5

Condition (Q1) holds.

Proof

By way of contradiction, let $\varepsilon_{k}(X)\in\mathrm {PG}({\mathcal {N}}_{k})$ for some X∈Δ _k . The group $G = \mathrm {SO}(2n+1,\mathbb{F} )$ acts transitively on Δ _k and stabilizes ${\mathcal{N}}_{k}$. Hence $\varepsilon_{k}(\varDelta _{k}) \subseteq\mathrm{PG}({\mathcal {N}}_{k})$. This is a contradiction because every vector $(X_{J})_{J\in{I\choose k }}\in {\mathcal{N}}_{k}$ has X _J =0 whenever 2n+1∉J, but only some of the vectors of ε _k (Δ _k ) have this property. □

Lemma 3.6

Let k<n. Then (Q2) holds.

Proof

By way of contradiction, let $\langle\varepsilon_{k}(X_{1}), \varepsilon_{k}(X_{2})\rangle\cap{\mathcal{N}}_{k}\neq0$ for two distinct points X ₁,X ₂∈Δ _k . As (Q1) holds, $\langle\varepsilon_{k}(X_{1}), \varepsilon_{k}(X_{2})\rangle\cap\mathcal{N}_{k}$ does not contain any point of ε _k (Δ _k ). Hence $\langle\varepsilon_{k}(X_{1}), \varepsilon_{k}(X_{2})\rangle\cap\mathcal{N}_{k}$ is a point $n_{X_{1},X_{2}}$ of PG(W _k )∖ε _k (Δ _k ). Since ε _k is full, the points X ₁ and X ₂ cannot be collinear in Δ _k . Let d=d(X ₁,X ₂) be the distance between X ₁ and X ₂ in the collinearity graph of Δ _k . We have d>1, since X ₁ and X ₂ are non-collinear.

The group G acts transitively on the pairs of points of Δ _k at distance d and stabilizes $\mathcal{N}_{k}$. Hence 〈ε _k (X),ε _k (Y)〉 meets $\mathcal{N}_{k}$ in a point n _X,Y ∈PG(W _k )∖ε _k (Δ _k ), for every pair of points X,Y∈Δ _k at distance d.

For any two collinear points Y ₁,Y ₂ of Δ _k we can pick a point X at distance d from both Y ₁ and Y ₂. Clearly, the point ε _k (X) does not belong to the projective line 〈ε _k (Y ₁),ε _k (Y ₂)〉. Consequently, $n_{X,Y_{1}}\neq n_{X,Y_{2}}$ and the points ε _k (X),ε _k (Y ₁) and ε _k (Y ₂) span a projective plane which contains both of the lines 〈ε _k (Y ₁),ε _k (Y ₂)〉 and $\langle n_{X,Y_{1}}, n_{X,Y_{2}}\rangle$. These two lines, being coplanar, meet in a point, say z. On the one hand z∈ε _k (Δ _k ), as 〈ε _k (Y ₁),ε _k (Y ₂)〉⊂ε _k (Δ _k ) (recall that Y ₁ and Y ₂ are collinear in Δ _k ). On the other hand, $z\in\mathrm{PG}(\mathcal{N}_{k})$, since $\langle n_{X,Y_{1}}, n_{X,Y_{2}}\rangle \subseteq\mathcal{N}_{k}$. Hence $z\in\varepsilon_{k}(\varDelta _{k})\cap \mathrm{PG}(\mathcal{N}_{k})$, contrary to (Q1). We have reached a final contradiction. □

We now turn to the case k=n. Recall that the lines of Δ _n are the stars of the elements of Δ of type n−1. For every (n−1)-element X of Δ, the image ε _n (St(X)) of St(X) by ε _n is a conic C _X of PG(W _k ), spanning a plane π _X of PG(W _n ). Moreover, π _X ∩ε _n (Δ _n )=C _X . The nucleus of C _X is the point ν _X :=e _n (N _X ). (Recall that N _X =〈X,N ₀〉.)

Lemma 3.7

We have $\pi_{X}\cap\mathcal{N}_{n} = \nu_{X}$, for every (n−1)-element X of Δ.

Proof

Clearly, $\nu_{X}\in\pi_{X}\cap\mathcal{N}_{n}$. By way of contradiction, suppose that $\pi_{X}\cap\mathcal{N}_{n}$ is larger than ν _X . Since $C_{X}\cap\mathrm{PG}(\mathcal{N}_{n}) = \emptyset$ by (Q1), $\pi_{X}\cap\mathcal{N}_{n}$ is a projective line through ν _X . If $\mathbb{F}$ is perfect then the line $\pi_{X}\cap\mathcal{N}_{n}$ is tangent to C _X , namely it meets C _X in one point. This is impossible, since $C_{X}\cap\mathrm{PG}(\mathcal {N}_{n}) = \emptyset$.

Therefore $\mathbb{F}$ is non-perfect. Let $\widehat{\mathbb{F}}$ be the quadratic closure of $\mathbb{F}$. Put $\widehat{V} := \widehat{\mathbb{F}}\otimes_{\mathbb{F}}V$ (where vectors are linear combinations of the vectors of B with coefficients taken from $\widehat{\mathbb{F}}$) and $\widehat{W}_{n} := \bigwedge^{n}\widehat {V}$ ($= \widehat{\mathbb{F}}\otimes_{\mathbb{F}}W_{n}$). The form q naturally extends to a non-singular quadratic form $\hat{q}$ of $\widehat{V}$, admitting the same expression as q with respect to B. Denoted by $\widehat {\varDelta }$ the building of type B _n associated to $\hat{q}$, every element X of Δ is the intersection $X = V\cap\widehat{X}$ of V with a uniquely determined element $\widehat{X}$ of $\widehat{\varDelta }$, of the same type as X (in fact $\widehat{X} = \widehat{\mathbb {F}}\otimes_{\mathbb{F}} X$). Accordingly, $\mathcal{G}_{n}$ and Δ _n can be regarded as $\mathbb{F}$-subgeometries of the n-grassmannians $\widehat{\mathcal{G}}_{n}$ and $\widehat {\varDelta }_{n}$ of $\mathrm{PG}(\widehat{V})$ and $\widehat{\varDelta }$ respectively and, if $\hat{e}_{n}$ and $\hat{\varepsilon}_{n}$ are the natural embeddings of $\widehat{\mathcal{G}}_{n}$ and the Grassmann embedding of $\widehat {\varDelta }_{n}$, then e _n and ε _n are induced by $\hat{e}_{n}$ and $\hat{\varepsilon}_{n}$. Clearly, the global nucleus $\widehat {\mathcal{N}}_{n}$ of $\hat{\varepsilon}_{n}$ contains the $\widehat{\mathbb{F} }$-tensorization $\widehat{\mathbb{F}}\otimes_{\mathbb{F}}\mathcal {N}_{n}$ of the global nucleus $\mathcal{N}_{n}$ of ε _n .

Turning to the plane π _X and the conic C _X , we have $\pi_{X} = \mathrm{PG} (W_{k})\cap\hat{\pi}_{X}$ for a uniquely determined plane $\hat{\pi }_{X}$ of $\mathrm{PG}(\widehat{W}_{k})$ (in fact $\hat{\pi}_{X} = \widehat {\mathbb{F}}\otimes_{\mathbb{F}}\pi_{X}$) and $C_{X} = \pi_{X}\cap\widehat{C}_{X}$ for a uniquely determined conic $\widehat{C}_{X}$ of $\hat{\pi}_{X}$. The nucleus $\hat{\nu}_{X}$ of $\widehat{C}_{X}$ coincides with (the 1-subspace of $\widehat{W}_{n}$ spanned by) the nucleus ν _X of C _X . By assumption, $\pi_{X}\cap \mathcal{N}_{n}$ is a line of π _X through ν _X . Hence $\hat {\pi}_{X}\cap \widehat{N}_{n}$ contains a line of $\hat{\pi}_{X}$ through $\hat{\nu}_{X}$. However this is impossible, in view of the first paragraph of this proof. Indeed $\widehat{\mathbb{F}}$ is perfect. □

Lemma 3.8

Let k=n. Then (Q2) holds.

Proof

By way of contradiction, let $\langle\varepsilon_{n}(X_{1}), \varepsilon_{n}(X_{2})\rangle\cap{\mathcal{N}}_{n}\neq0$ for two distinct points X ₁,X ₂∈Δ _n . By (Q1), $n_{X_{1},X_{2}} := \langle\varepsilon_{n}(X_{1}), \varepsilon_{n}(X_{2})\rangle\cap{\mathcal{N}}_{n}$ is a point of PG(W _n )∖ε _n (Δ _n ).

Let d=d(X ₁,X ₂) be the distance between X ₁ and X ₂ in the collinearity graph of Δ _n . Suppose firstly that d=1, namely X ₁ and X ₂ are collinear. Then X ₁∩X ₂∈Δ _n−1 and St(X ₁∩X ₂) is the line of Δ _n through X ₁ and X ₂. The image of St(X ₁∩X ₂) by ε _n is a conic $C_{X_{1}\cap X_{2}}$ spanning a plane $\pi_{X_{1}\cap X_{2}}$ of PG(W _n ). The projective line 〈ε _n (X ₁),ε _n (X ₂)〉 is contained in $\pi_{X_{1}\cap X_{2}}$. Hence $n_{X_{1}, X_{2}}\in\pi_{X_{1}\cap X_{2}}$. However, $\pi_{X_{1}\cap X_{2}}\cap\mathcal{N}_{n}$ is the nucleus $\nu_{X_{1}\cap X_{2}}$ of the conic $C_{X_{1}\cap X_{2}}$, by Lemma 3.7. Hence $n_{X_{1},X_{2}} = \nu_{X_{1}\cap X_{2}}$, namely 〈ε _n (X ₁),ε _n (X ₂)〉 is a line of $\pi_{X_{1}\cap X_{2}}$ through the nucleus of $C_{X_{1}\cap X_{2}}$. On the other hand, 〈ε _n (X ₁),ε _n (X ₂)〉 contains two distinct points of the conic $C_{X_{1}\cap X_{2}}$, namely ε _n (X ₁) and ε _n (X ₂). Thus, we have got a secant line of a conic passing through the nucleus of that conic. This is impossible. Therefore d>1.

As in the proof of Lemma 3.6, the distance-transitivity of G on the collinearity graph of Δ _n implies that $\langle \varepsilon_{n}(X),\varepsilon_{n}(Y)\rangle\cap\mathcal{N}_{n}\neq0$ for any two points X,Y∈Δ _n at mutual distance d. We now choose two collinear points Y ₁,Y ₂ of Δ _n and a point X at distance d from both of them. Then X has distance d−1 from a unique point Y ₀ of the line St(Y ₁∩Y ₂) of Δ _n through Y ₁ and Y ₂. The point ε _n (X) does not belong to the plane $\pi_{Y_{1}\cap Y_{2}}$, since $\varepsilon_{n}(\mathrm {St}(Y_{1}\cap Y_{2})) = \pi_{Y_{1}\cap Y_{2}}\cap\varepsilon_{n}(\varDelta _{n})$, and X∉St(Y ₁∩Y ₂). Hence ε _n (X),ε _n (Y ₁) and ε _n (Y ₂) span a 3-dimensional subspace S of PG(W _n ). We have d(X,Y)=d for every Y∈St(Y ₁∩Y ₂)∖{Y ₀}. Hence $\langle\varepsilon_{n}(X), \varepsilon_{n}(Y)\rangle\cap\mathcal{N}_{n} \neq0$ for every such Y. Let σ be the subspace of S spanned by the points n _X,Y for Y∈St(Y ₁∩Y ₂)∖{Y ₀}. Clearly, $\sigma \subseteq\mathcal{N}_{n}$.

Suppose firstly that σ contains (or is) a plane. Then $\sigma \cap\pi_{Y_{1}\cap Y_{2}}$ has projective dimension at least 1. Accordingly, $\pi_{Y_{1}\cap Y_{2}}\cap\mathcal{N}_{n}$ contains at least a line. This contradicts Lemma 3.7. Hence σ must be a line. Suppose $|\mathbb{F}| > 2$. If Y ₃∈St(Y ₁∩Y ₂)∖{Y ₀,Y ₁,Y ₂} (≠∅ because $|\mathbb{F}| > 2$) then ε _n (Y ₁), ε _n (Y ₂) and ε _n (Y ₃) are non-collinear in the projective plane $\pi_{Y_{1}\cap Y_{2}}$. Hence the points $n_{X,Y_{1}}, n_{X,Y_{2}}$ and $n_{X,Y_{3}}$ are non-collinear as well, contrary to the fact that σ is a line. We are forced to conclude that $\mathbb{F}= \mathbb {F}_{2}$. The line σ meets $\pi_{Y_{1}\cap Y_{2}}$ in a point. On the other hand, $\sigma\subseteq\mathcal{N}_{n}$. Hence $\sigma\cap\pi_{Y_{1}\cap Y_{2}}$ is the nucleus $\nu_{Y_{1}\cap Y_{2}}$ of the conic $C_{Y_{1}\cap Y_{2}}$, by Lemma 3.7. Let π be the plane spanned by $n_{X, Y_{1}}$, $n_{X,Y_{2}}$ and ε _n (X). Then $\pi\cap\pi_{Y_{1}\cap Y_{2}}$ is a line, say l. The line l belongs to $\pi_{Y_{1}\cap Y_{2}}$ and contains the nucleus $\nu_{Y_{1}\cap Y_{2}}$ of the conic $C_{Y_{1}\cap Y_{2}}$ as well as two points of it, namely ε _n (Y ₁) and ε _n (Y ₂). This is obviously impossible. We have reached a final contradiction. □

So, the mapping $\varepsilon_{k}/\mathcal{N}_{k}$ is well-defined and injective. As remarked at the beginning of this subsection, if k<n then $\varepsilon_{k}/\mathcal{N}_{k}$ is a full projective embedding.

Let k=n. For every (n−1)-element X of Δ let $\bar{\lambda }_{X}$ be the set of lines of π _X through ν _X and λ _X the set of lines of π _X tangent to C _X . Clearly, $\lambda_{X}\subseteq\bar{\lambda}_{X}$. Moreover, by Lemma 3.7, the mapping θ _X sending every line $l\in\bar{\lambda}_{X}$ to $\langle l, \mathcal{N}_{n}\rangle$ is a bijection from $\bar{\lambda}_{X}$ to a line $\overline{L}_{X}$ of $\mathrm{PG}(\varepsilon_{n}(\varDelta _{n})/\mathcal {N}_{n})$. The set L _X :=θ _X (λ _X ) is contained in $\overline{L}_{X}$. Moreover, if ζ _X is the bijection from St(X) to λ _X sending every Y∈St(X) to the line 〈ε _n (Y),ν _X 〉 of π _X , then the composite η _X :=θ _X ∘ζ _X is a bijection from St(X) to L _X . Clearly, η _X is the mapping induced by $\varepsilon_{n}/\mathcal{N}_{n}$ on the line St(X) of Δ _n .

If $\mathbb{F}$ is perfect then $\lambda_{X} = \bar{\lambda}_{X}$. In this case $L_{X} = \overline{L}_{X}$. Hence $\varepsilon_{n}/\mathcal{N}_{n}$ maps the line St(X) of Δ _n onto a line of $\mathrm {PG}(\varepsilon_{n}(\varDelta _{n})/\mathcal{N}_{n})$. On the other hand, if $\mathbb{F}$ is non-perfect then λ _X is a proper subset of $\bar{\lambda}_{X}$. Accordingly, $L_{X}\subset\overline{L}_{X}$. In this case $\varepsilon_{n}/\mathcal{N}_{n}$ maps the line St(X) onto a proper subset of a line of $\mathrm{PG} (\varepsilon_{n}(\varDelta _{n})/\mathcal{N}_{n})$. Summarizing:

Lemma 3.9

Let k=n. If $\mathbb{F}$ is perfect then $\varepsilon_{n}/\mathcal {N}_{n}$ is a full projective embedding. If $\mathbb{F}$ is non-perfect then $\varepsilon_{n}/\mathcal{N}_{n}$ is a non-full lax embedding.

Proposition 3.10

Let $\mathbb{F}$ be perfect. Then $\varepsilon_{k}/\mathcal{N}_{k}\cong \varepsilon^{sp}_{k}$, for k=1,2,…,n.

Proof

We recall that, since $\mathbb{F}$ is assumed to be perfect, the mapping sending every element X of Δ to 〈X,N ₀〉/N ₀ is an isomorphism from Δ to a model of Δ ^sp realized inside V/N ₀. For X∈Δ _k , let $(X_{J})_{J\in{I\choose k}}$ be the family of coordinates of ε _k (X) with respect to the basis B _∧ of W _k . If we take only those coordinates X _J with 2n+1∉J then we get a family of coordinates for the image of X by $\varepsilon_{k}/{{\mathcal {N}}_{k-1}}$. It is not difficult to see that these coordinates are just the same as those that we obtain if we apply $\varepsilon^{sp}_{k}$ to 〈X,N ₀〉/N ₀. □

3.3 Proof of Theorems 1.2 and 1.3

Propositions 3.4 and 3.10 yield Theorem 1.3. Turning to the proof of Theorem 1.2, suppose firstly that $\mathbb{F}$ is perfect. We also assume k>1, since the statement of Theorem 1.2 is trivial when k=1. Under these hypotheses, Theorem 1.3 and the equality $\bigl[{{2n}\choose{k-1}}-{{2n}\choose{k-3}} \bigr] + \bigl[{{2n}\choose {k}}-{{2n}\choose{k-2}} \bigr] = {{2n+1}\choose{k}}-{{2n+1}\choose {k-2}}$ imply that

$$ \mathrm{dim}(\varepsilon_k) = {{2n+1}\choose k} - {{2n+1}\choose{k-2}}. $$

(1)

Suppose now that $\mathbb{F}$ is non-perfect. Let $\widehat{\mathbb {F}}$ be a perfect extension of $\mathbb{F}$ (e.g. the quadratic closure of $\mathbb {F}$). Let $\widehat {V} = \widehat{\mathbb{F}}\otimes_{\mathbb{F}}V$ and define $\widehat {\varDelta }$, $\widehat {\varDelta }_{k}$ and $\hat{\varepsilon}_{k}$ accordingly (see the proof of Lemma 3.7). Then (1) holds for $\hat{\varepsilon}_{k}$. Dimensions cannot decrease when tensorizing with field extensions. Therefore:

$$ \mathrm{dim}(\varepsilon_k) \leq{{2n+1} \choose k} - {{2n+1}\choose{k-2}}. $$

(2)

On the other hand, $\mathbb{F}$ contains $\mathbb{F}_{2}$. Let $\varDelta _{k}^{0}$ be the subgeometry of Δ _k formed by the subspaces spanned by $\mathbb{F}_{2}$-linear combinations of the vectors of B and $\varepsilon_{k}^{0}$ the embedding induced by ε _k on $\varDelta _{k}^{0}$. All vectors of $\varepsilon_{k}^{0}(\varDelta _{k}^{0})$ are $\mathbb{F}_{2}$-linear combinations of vectors of B _∧. Thus (1) holds for $\varepsilon_{k}^{0}$, since $\mathbb{F}_{2}$ is perfect. It follows that 〈ε _k (Δ _k )〉 contains an independent set of ${{2n+1}\choose k} - {{2n+1}\choose{k-2}}$ vectors. Consequently,

$$ \mathrm{dim}(\varepsilon_k) \geq{{2n+1} \choose k} - {{2n+1}\choose{k-2}}. $$

(3)

Equation (1) follows from (2) and (3).

4 Proof of Theorem 1.5

In our proof of Theorem 1.5 we will go back and forth between B _n -buildings and D _n -buildings. So, we must firstly spend a few words on buildings of type D _n , their grassmannians and embeddings.

4.1 Grassmannians of D _n -buildings and their embeddings

Henceforth Δ ⁺ stands for the building of type D _n defined over $\mathbb{F}$. It can be constructed as follows. Given a non-singular quadratic form q ⁺ of Witt index n in $V' = V(2n,\mathbb{F})$, the non-trivial subspaces of V′ totally singular for q ⁺, with their dimensions taken as types, form a non-thick building Δ′ of Coxeter type C _n . The building Δ ⁺ is obtained from Δ′ as follows: drop the elements of type n−1 and partition the set of n-elements in two families such that two n-elements X and Y are in the same family precisely if codim _X (X∩Y) is even. Two elements X and Y not in the same family are declared to be incident precisely when dim(X∩Y)=n−1.

It is customary to choose the integers n−1 and n as types for these two families, but the following different convention better suits our needs in this section: we take the pairs (n,0) and (n,1) as types for them.

We allow n=3. Recall that the diagram D ₃ is the same as A ₃, but with the usual types 1, 2, 3 replaced with (3,0),1 and (3,1), respectively. If n=3 then Δ ⁺ is isomorphic to $\mathrm{PG}(3,\mathbb {F})$ via the Klein correspondence, the elements of Δ ⁺ of type 1, (3,0) and (3,1) being respectively the lines, the points and the planes of $\mathrm{PG} (3,\mathbb{F})$ (or lines, planes and points, if we prefer so).

For k<n the k-grassmannian of Δ′ is defined just in the same way as the k-grassmannian Δ _k of Δ. We call it the k-grassmannian of Δ ⁺ and we denote it by the symbol $\varDelta ^{+}_{k}$, although when k=n−1 this convention is not so consistent with the terminology commonly used in the literature. (If we followed the custom, we should rather call $\varDelta ^{+}_{n-1}$ the {(n,0),(n,1)}-grassmannian of Δ ⁺.) The (n,0)- and (n,1)-grassmannian can also be defined, called half spin geometries in the literature, but we are not interested in them here.

The 1-grassmannian $\varDelta ^{+}_{1}$ of Δ ⁺ is the polar space defined by q ⁺ on V′. Identify V′ with a hyperplane of $V = (2n+1,\mathbb{F})$ suitably chosen so that $\varDelta ^{+}_{1}$ is the polar space induced by Δ ₁ on V′. Similarly, $\varDelta ^{+}_{k}$ is the subgeometry induced by Δ _k on the set of k-subspaces of V′. Note that the points of $\varDelta _{n-1}^{+}$ are the {(n,0),(n,1)}-flags of Δ ⁺ while the lines of $\varDelta ^{+}_{n-1}$ correspond to flags of Δ ⁺ type {n−2,(n,0)} or {n−2,(n,1)}. In particular, if n=3 then $\varDelta ^{+}_{2}$ is the so-called root-subgroup geometry of $\mathrm{SL}(4,\mathbb{F})$, with the point-plane flags of $\mathrm{PG}(3,\mathbb{F} )$ as points and the line-plane and point-line flags of $\mathrm {PG}(3,\mathbb{F})$ as lines.

For k=1,2,…,n−1 we can define a projective embedding $\varepsilon^{+}_{k}$ of $\varDelta _{k}^{+}$ into a subspace $\langle \varepsilon_{k}(\varDelta _{k}^{+})\rangle$ of $W'_{k} = \bigwedge^{k} V'$ sending every point 〈v ₁,…,v _k 〉 of $\varDelta _{k}^{+}$ to the point 〈v ₁∧⋯∧v _k 〉 of $\mathrm{PG}(W'_{k})$. We call $\varepsilon^{+}_{k}$ the Grassmann embedding of $\varDelta ^{+}_{k}$. Moreover, let μ ₁,…,μ _n−2,μ _n,0 and μ _n,1 be the fundamental dominant weights of the root system of type D _n , corresponding to the nodes 1,2,…,n−2,(n,0) and (n,1) of the D _n -diagram in the obvious way. Put μ _n−1:=μ _n,0+μ _n,1. Then for k=1,2,…,n−1 the Weyl module V(μ _k ) hosts a projective embedding $\widetilde{\varepsilon}^{+}_{k}$ of $\varDelta ^{+}_{k}$ and $\varepsilon^{+}_{k}$ is a quotient of $\widetilde{\varepsilon}^{+}_{k}$. We call $\widetilde {\varepsilon}^{+}_{k}$ the Weyl embedding of $\varDelta ^{+}_{k}$.

By the Weyl dimension formula, $\mathrm{dim}(\widetilde{\varepsilon }^{+}_{k}) = {{2n}\choose k}$. Clearly, $\widetilde{\varepsilon}^{+}_{1} \cong \varepsilon^{+}_{1}$.

Proposition 4.1

Let k>1.

• (1) If $\mathrm{char}(\mathbb{F})\neq2$ then $\varepsilon^{+}_{k} \cong \widetilde {\varepsilon}^{+}_{k}$, namely $\mathrm{dim}({\varepsilon}^{+}_{k}) = {{2n}\choose k}$.

• (2) Let $\mathrm{char}(\mathbb{F}) = 2$. Then $\mathrm {dim}({\varepsilon}^{+}_{k}) = {{2n}\choose k}-{{2n}\choose{k-2}}$. In this case $\varepsilon^{+}_{k}$ is a proper quotient of the Weyl embedding $\widetilde{\varepsilon}_{k}^{+}$.

Proof

Claim (1) can be proved by just the same argument used for Theorem 1.1. Indeed Lemma 2.1 still holds if we replace the (2n+1)-dimensional space V with the 2n-dimensional space V′, the quadratic form q with q ⁺ and $\mathcal{G}_{k}$ with the k-grassmannian $\mathcal{G}'_{k}$ of PG(V′). The proof of Lemma 2.1 remains valid for this setting word for word. By that lemma, when $\mathrm{char}(\mathbb{F})\neq2$ the point-set of $\varDelta ^{+}_{k}$ spans the k-grassmannian $\mathcal{G}'_{k}$ of $\mathrm {PG}(2n-1,\mathbb{F})$ (compare Proposition 2.2). Hence $\langle\varepsilon^{+}_{k}(\varDelta _{k}^{+})\rangle= W'_{k}$, namely $\mathrm{dim}({\varepsilon}^{+}_{k}) = {{2n}\choose k}$.

Let $\mathrm{char}(\mathbb{F}) = 2$. Let f be the bilinearization of q ⁺ and $\varDelta ^{sp}_{1}$ the polar space defined by f on V′. Then $\varDelta ^{+}_{1}$ is a subgeometry (but not a subspace) of $\varDelta ^{sp}_{1}$. Accordingly, $\varDelta ^{+}_{k}$ is a subgeometry of $\varDelta ^{sp}_{k}$ and the natural embedding $\varepsilon^{sp}_{k}$ of $\varDelta ^{sp}_{k}$ (see Introduction) induces $\varepsilon^{+}_{k}$ on $\varDelta ^{+}_{k}$. We have $\mathrm{dim}(\varepsilon^{sp}_{k}) = {{2n}\choose k}-{{2n}\choose {k-2}}$. Hence $\mathrm{dim}(\varepsilon^{+}_{k}) \leq{{2n}\choose k}-{{2n}\choose{k-2}}$. Moreover, claim (2) holds when $\mathbb{F}$ is perfect. Explicitly:

(2′):

If $\mathbb{F}$ is perfect then $\mathrm {dim}(\varepsilon^{+}_{k}) = {{2n}\choose k}-{{2n}\choose{k-2}}$.

Indeed, let $\mathbb{F}$ be perfect. As $\varDelta ^{+}_{k}$ is a subgeometry of $\varDelta ^{sp}_{k}$ which in its turn is a subgeometry of $\mathcal{G}'_{k}$, in order to prove (2′) it suffices to prove that $\varDelta ^{sp}_{k}$ is contained in the subgeometry $\langle \varDelta ^{+}_{k}\rangle_{\mathcal{G}'_{k}}$ spanned by $\varDelta ^{+}_{k}$. Let X be a k-dimensional subspace of V′ totally isotropic for f. We must prove that $X\in\langle \varDelta ^{+}_{k}\rangle_{\mathcal{G}'_{k}}$. Since $\mathbb{F}$ is perfect $\varDelta _{1}^{sp}\cong \varDelta _{1}$. However $\varDelta ^{+}_{1}$ is a geometric hyperplane of Δ ₁. Hence $\varDelta ^{+}_{1}$ is also a geometric hyperplane of $\varDelta ^{sp}_{k}$ too. Consequently either $X\in \varDelta ^{+}_{k}$ or X ₀:={x∈X∣q ⁺(x)=0} is a hyperplane of X and $X\subseteq X_{0}^{\perp}$. In the first case there is nothing to prove. Assume the latter. Then $X_{0}\in \varDelta ^{+}_{k-1}$ and $\mathrm{dim}(X_{0}^{\perp}/X_{0}) = 2n-2k-2$. Given a flag F of Δ ⁺ of type {1,…,k−1} containing X ₀, no matter which, $\mathrm{Res}_{\varDelta ^{+}}(F)$ is isomorphic to the building of type D _n−k−1 associated to a non-degenerate quadratic form $q_{X_{0}}^{+}$ of $X_{0}^{\perp}/X_{0}$ of maximal Witt index n−k−1. The k-subspaces of V′ containing X ₀ and contained in $X_{0}^{\perp}$ form a subspace $\mathcal{G}'_{k}(X_{0})$ of $\mathcal{G}'_{k}$ and the function mapping $Y\in\mathcal{G}'_{k}(X_{0})$ onto Y/X ₀ is an isomorphism $\pi_{X_{0}}$ from the geometry induced by $\mathcal{G}'_{k}$ on $\mathcal{G}'_{k}(X_{0})$ to $\mathrm{PG} (X_{0}^{\perp}/X_{0})$. Clearly $\pi_{X_{0}}$ maps the set of k-elements of Δ ⁺ containing X ₀ onto the set of points of the quadric $Q_{X_{0}}^{+}$ associated to $q_{X_{0}}^{+}$. The point $\pi_{X_{0}}(X)$ of $\mathrm{PG} (X_{0}^{\perp}/X_{0})$ does not belong to $Q_{X_{0}}^{+}$. On the other hand, every point of $\mathrm{PG}(X_{0}^{\perp}/X_{0})$ belongs to a secant line of $Q_{X_{0}}^{+}$. Hence there is a line L of $\mathcal{G}'_{k}$ contained in $\mathcal{G}'_{k}(X_{0})$ and containing X and two distinct points of $\varDelta ^{+}_{k}$. Therefore $X\in\langle \varDelta ^{+}_{k}\rangle_{\mathcal{G}'_{k}}$, as we wished to prove. Claim (2′) is proved.

Having proved (2′) in the perfect case, claim (2) in the non-perfect case follows by descent to $\mathbb{F}_{2}$, as in the proof of Theorem 1.2 (see Sect. 3.3). □

The geometry $\varDelta ^{+}_{k}$ admits the absolutely universal embedding (Kasikova and Shult [19] for k<n−1 and Blok and Pasini [8] for k=n−1). Therefore, if $\widetilde{\varepsilon}^{+}_{k}$ is relatively universal then it is also absolutely universal. It follows from Tits [23, 8.4.3] that $\varepsilon^{+}_{1}$ ($= \widetilde {\varepsilon}^{+}_{1}$) is universal for any field $\mathbb{F}$. On the other hand, by Proposition 4.1, when $\mathrm{char}(\mathbb{F}) = 2$ and k>1 the Grassmann embedding $\varepsilon^{+}_{k}$ is not universal. In the sequel, as a by-product of our proof of Theorem 1.5, we shall show that, under the hypotheses assumed on $\mathbb{F}$ in Theorem 1.5, the Weyl embedding $\widetilde{\varepsilon}^{+}_{k}$ is universal for k=2 or 3.

4.2 Back and forth between B _n and D _n

In this subsection k is 2 or 3, n>k and

$$\eta_k:\varDelta _k\rightarrow\mathrm{PG}(U_k), \qquad \eta^+_k:\varDelta _k^+\rightarrow\mathrm{PG} \bigl(U'_k\bigr) $$

are given projective embeddings of Δ _k and $\varDelta ^{+}_{k}$, for some $\mathbb{F}$-vector spaces U _k and $U'_{k}$. As in the previous sections, we will freely use the symbols Δ _k and $\varDelta ^{+}_{k}$ to denote the point-line geometries Δ _k and $\varDelta ^{+}_{k}$ as well as their point-sets. We will do the same with other symbols like Δ _k,a , Δ _k,H etc. (see below). We denote spans in Δ _k , $\varDelta ^{+}_{k}$, Δ ₁ and $\varDelta ^{+}_{1}$ by the symbols $\langle.\rangle_{\varDelta _{k}}$, $\langle.\rangle_{\varDelta ^{+}_{k}}$, $\langle. \rangle_{\varDelta _{1}}$ and $\langle. \rangle_{\varDelta ^{+}_{1}}$ respectively, keeping the symbol 〈.〉 for spans in U _k or $U'_{k}$.

Let H be a non-singular hyperplane of the polar space Δ ₁ such that the polar space Δ _1,H induced by Δ ₁ on H is isomorphic to $\varDelta ^{+}_{1}$. Let Δ _k,H be the subgeometry of Δ _k induced on the set of totally singular k-subspaces of V contained in H. Then $\varDelta _{k,H}\cong\nobreak \varDelta ^{+}_{k}$. The embedding η _k induces on Δ _k,H a projective embedding η _k,H :Δ _k,H →PG(U _k,H ) where U _k,H :=〈η _k (Δ _k,H )〉.

Let a be a point of Δ ₁ exterior to H and Δ _k,a the subgeometry of Δ _k induced on the set of totally singular k-subspaces of V containing a. Then Δ _k,a is isomorphic to the (k−1)-grassmannian $\overline{\varDelta }_{k-1}$ of a building $\overline{\varDelta }$ of type B _n−1 defined over $\mathbb{F}$. The embedding η _k,a :Δ _k,a →PG(U _k,a ) induced by η _k on Δ _k,a , where U _k,a :=〈η _k (Δ _k,a )〉, can be regarded as a projective embedding of $\overline{\varDelta }_{k-1}$.

When k=2 let l ₀ be a line of Δ ₁ not contained in H∪a ^⊥ and such that a ^⊥∩l ₀≠H∩l ₀. Put $S_{2} := \langle\{l_{0}\}\cup \varDelta _{2,a}\cup \varDelta _{2,H}\rangle_{\varDelta _{2}}$.

When k=3 (hence n>3) the subgeometry Δ _1,a,H of Δ ₁ induced on a ^⊥∩H is isomorphic to the polar space associated with a non-singular quadratic form of $V(2n-1,\mathbb{F})$ of Witt index n−1. It is well known that the latter admits a generating set of 2n−1 points. Hence the same holds for Δ _1,a,H . Let {p ₁,…,p _2n−1} be a spanning set of 2n−1 points of Δ _1,a,H . For every i=1,…,2n−1 let α _i be a plane of Δ ₁ through p _i such that α _i ∩H∩a ^⊥={p _i }. Put $S_{3} := \langle\{\alpha_{i}\}_{i=1}^{2n-1}\cup \varDelta _{3,a}\cup \varDelta _{3,H}\rangle_{\varDelta _{3}}$.

Lemma 4.2

S ₂=Δ ₂.

Proof

We firstly prove the following:

• (1) All lines of Δ ₁ coplanar with a in Δ ₁ belong to S ₂.

Let α be a plane of Δ ₁ through a and l a line of α. If either a∈l or l⊂H there is nothing to prove. Let a∉l⊈H and let p=l∩H. Then the lines of α through p are the points of a line L of Δ ₂. The line L contains l, α∩H and the line $\langle a, p\rangle_{\varDelta _{1}}$ of Δ ₁ through a and p. Clearly, α∩H∈Δ _2,H and $\langle a, p\rangle_{\varDelta _{1}}\in \varDelta _{2,a}$. Hence l∈S ₂. Claim (1) is proved.

• (2) All lines of Δ ₁ coplanar with l ₀ belong to S ₂.

Let α be a plane of Δ ₁ through l ₀. Then α contains three lines of S ₂, namely l ₀, α∩H and α∩a ^⊥ (the latter belongs to S ₂ by (1)). These three lines form a triangle, as l ₀∩a ^⊥∉H by assumption. It is now easy to see that all lines of α belong to S ₂.

• (3) If l is a line of Δ ₁ meeting l ₀ non-trivially, then l belongs to S ₂.

Let p:=l∩l ₀. Suppose firstly that p∈H. We can choose a plane α of Δ ₁ through l such that $l_{0}^{\perp}\cap \alpha \neq\alpha\cap H$. We have α∩H∈Δ _2,H and $l_{0}^{\perp}\cap\alpha\in S_{2}$ by (2). Moreover, both α∩H and $\alpha\cap l_{0}^{\perp}$ pass through p∈l. Therefore l∈S ₂.

Let p∉H. By (2), we can assume that l and l ₀ are non-coplanar. We consider two cases. Suppose firstly that we can choose a plane α of Δ ₁ through l ₀ such that a ^⊥∩α∩H≠l ^⊥∩α∩H. Let β be the plane of Δ ₁ spanned by l and α∩l ^⊥. Then α∩l ^⊥∈S ₂ by (2), a ^⊥∩β∈S ₂ by (1) and β∩H∈Δ _2,H . Since a ^⊥∩α∩H≠l ^⊥∩α∩H, these three lines form a triangle. Hence all lines of β belong to S ₂. In particular, l∈S ₂.

Assume now that a ^⊥∩α∩H=l ^⊥∩α∩H for every plane α on l ₀. Clearly, there is at least one plane β through l containing a line m through p, non-coplanar with l ₀ and such that a ^⊥∩α∩H≠m ^⊥∩α∩H. Then m∈S ₂ by the previous paragraph. As $l_{0}^{\perp}\cap \beta\in S_{2}$ by (2), all lines of β through p belong to S ₂.

• (4) If l is a line of Δ ₁ meeting $l_{0}^{\perp}$ non-trivially, then l∈S ₂.

By (3) we can assume that l∩l ₀=∅. Suppose firstly that $l\subseteq l_{0}^{\perp}$. Pick a point p∈l ₀, $p\not\perp a$. By (3), all lines of the plane α:=〈p,l〉 through p are in S ₂. On the other hand, the line α∩a ^⊥ belongs to S ₂ by (1). It does not pass through p, as p∉a ^⊥ by assumption. Therefore all lines of α belong to S ₂. In particular, l∈S ₂.

Suppose that $l\cap l_{0}^{\perp}$ is a point, say $p_{1} := l\cap l_{0}^{\perp}$. We may also assume that p ₁∉l ₀, by (3). Then p ₂:=l ₀∩l ^⊥ is a point. Let $\alpha= \langle p_{2}, l\rangle_{\varDelta _{1}}$. All lines of the plane α passing through p ₂ belong to S ₂, by (3). If p ₂∉H then α∩H is a line of α in S ₂ not through p ₂. It follows that all lines of α belong to S ₂. In particular, l∈S ₂.

Let now p ₂∈H. The line a ^⊥∩α belongs to S ₂ by (1). It does not pass through p ₂, as a ^⊥∩l ₀∉H by assumption. As above, all lines of α belong to S ₂. Whence l∈S ₂.

We can now finish the proof of the lemma. Let l be any line of Δ ₁. We may assume that a∉l⊈H. In view of (1)–(4) we can also assume that l∩a ^⊥ is a point and $l\cap l_{0}^{\perp}= \emptyset$. Suppose that we can choose a plane α on l such that the point $p := l_{0}^{\perp}\cap\alpha$ does not belong to H∩a ^⊥. By (4), all lines of α through p belong to S ₂. Moreover, a ^⊥∩α∈S ₂ by (1) and α∩H∈Δ _2,H ⊆S ₂. Thus, α contains at least three lines of S ₂ forming a triangle. Therefore all lines of α belong to S ₂. In particular, l∈S ₂.

Finally, suppose that $p = l_{0}^{\perp}\cap\alpha\in H\cap a^{\perp}$ for every plane α through l. Pick a point p ₁∈l not in H and consider a line l ₁ of α through p ₁, different from either of the lines l and $l_{2} := \langle p_{1}, p\rangle_{\varDelta _{1}}$. Clearly, we can choose a plane α ₁ on l ₁ such that $l_{0}^{\perp}\cap\alpha_{1}\notin H\cap a^{\perp}$. Hence l ₁∈S ₂ by the previous paragraph. Moreover, l ₂∈S ₂ by (4). Hence l∈S ₂. □

Corollary 4.3

Suppose that $\mathrm{dim}(U_{2,H})\leq{{2n}\choose2}$. Then $\mathrm {dim}(U_{2}) \leq{{2n+1}\choose2}$.

Proof

This follows from Lemma 4.2, recalling that Δ _2,a is isomorphic to a polar space of type B _n−1, that every projective embedding of such a polar space has dimension equal to 2n−1 or possibly 2n−2 (the latter only when $\mathrm{char}(\mathbb {F}) =2$) and noticing that $1 + (2n-1) + {{2n}\choose2} = {{2n+1}\choose 2}$. □

Lemma 4.4

S ₃=Δ ₃.

Proof

It will be useful to have fixed some terminology. In the sequel, a totally singular 4-subspace of V will be called a space of Δ ₁, for short. We say that a point p∈a ^⊥∩H is S ₃-full if all planes of Δ ₁ on p belong to S ₃.

• (1) All planes of Δ ₁ contained in a ^⊥ belong to S ₃.

Let α be a plane of Δ ₁ contained in a ^⊥. If a∈α then α∈Δ _3,a ⊆S ₃. Suppose that a∉α and let X be the space of Δ ₁ spanned by a and α. Then all planes of X through a belong to S ₃. Moreover, X∩H∈Δ _3,H ⊆S ₃. It follows that all planes of X belongs to S ₃. In particular, α∈S ₃.

• (2) Let p∈a ^⊥∩H and let α ₀ be a plane of Δ ₁ on p such that a ^⊥∩H∩α ₀={p} and α ₀∈S ₃. Then p is S ₃-full.

Let Δ _3,p be the subgeometry of Δ ₃ induced on the set of planes of Δ ₁ through p. Let $\overline{\varDelta }_{3,p}$ be the induced subgeometry of Δ _3,p formed by those planes α such that α∩a ^⊥∩H={p}. So, $\alpha_{0}\in \overline{\varDelta }_{3,p}$. It is not difficult to see that $\overline {\varDelta }_{3,p}$ is connected.

Let now X be a space of Δ ₁ through p containing α ₀. Then X contains three planes of S ₃ through p, namely α ₀, X∩H and X∩a ^⊥ (which belongs to S ₃ by (1)). The intersection of these three planes is the point p. Hence all planes of X through p belong to S ₃. Therefore, every plane through p contained in a common space with α ₀ belongs to S ₃. Let now α be any plane in $\overline{\varDelta }_{3,p}$ contained in a common space with α ₀. We can repeat the above argument with α ₀ replaced by α, thus obtaining that all planes through p contained in a common space with α belong to S ₃. In this way, exploiting the connectedness of $\overline{\varDelta }_{3,p}$ we obtain the result that every plane through p contained in a common space with a plane $\alpha\in\overline{\varDelta }_{3,p}$ belongs to S ₃. However, every plane β∈Δ _3,p is contained in the same space as a plane $\alpha\in\overline{\varDelta }_{3,p}$. Therefore Δ _3,p ⊆S ₃, namely p is S ₃-full.

• (3) Let x and y be two points of a ^⊥∩H collinear in Δ _1,a,H and z another point on the line l of Δ _1,a,H spanned by x and y. If both x and y are S ₃-full then z is S ₃-full.

It is easy to see that there exists at least one space X of Δ ₁ containing l and such that a ^⊥∩X=H∩X=l. Let X be such a space. As both x and y are S ₃-full, all planes of Δ ₁ contained in X and containing either x or y belong to S ₃. It follows that all planes contained in X belong to S ₃. In particular, all planes through z contained in X belong to S ₃. On the other hand, at least one of these planes meets a ^⊥ and H in distinct lines. Therefore z satisfies the hypotheses of (2). Hence z is S ₃-full.

We can now finish the proof of the lemma. By (2), the points p ₁,…,p _2n−1 considered in the definition of S ₃ are S ₃-full. Moreover they span Δ _1,a,H . Hence all points of a ^⊥∩H are S ₃-full, by (3). On the other hand, every plane of Δ ₁ meets a ^⊥∩H non-trivially. Hence every plane of Δ ₁ belongs to S ₃. □

Corollary 4.5

Suppose that $\mathrm{dim}(U_{3,H})\leq{{2n}\choose3}$ and $\mathrm {dim}(U_{3,a})\leq{{2n-1}\choose2}$. Then $\mathrm{dim}(U_{3}) \leq {{2n+1}\choose3}$.

Proof

Note that $(2n-1) + {{2n-1}\choose2} + {{2n}\choose3} = {{2n+1}\choose3}$. The conclusion follows from this equality and Lemma 4.4. □

We now turn to $\varDelta ^{+}_{k}$ and its embedding $\eta_{k}^{+}:\varDelta _{k}^{+}\rightarrow\mathrm{PG}(U'_{k})$. Let H be a non-singular hyperplane of the polar space $\varDelta ^{+}_{1}$. The polar space $\varDelta ^{+}_{1,H}$ induced by $\varDelta ^{+}_{1}$ on H is isomorphic to the polar space $\overline{\varDelta }_{1}$, for a building $\overline{\varDelta }$ of type B _n−1 defined over $\mathbb{F}$. Let $\varDelta ^{+}_{k,H}$ be the subgeometry of $\varDelta ^{+}_{k}$ induced on H. Then $\varDelta ^{+}_{k,H}$ is isomorphic to the k-grassmannian $\overline{\varDelta }_{k}$ of $\overline{\varDelta }$. The embedding $\eta^{+}_{k}:\varDelta ^{+}_{k}\rightarrow\mathrm{PG}(U'_{k})$ induces on $\varDelta ^{+}_{k,H}$ a projective embedding $\eta^{+}_{k,H}:\varDelta ^{+}_{k,H}\rightarrow \mathrm{PG}(U'_{k,H})$ where $U'_{k,H} := \langle\eta^{+}_{k}(\varDelta ^{+}_{k,H})\rangle$.

Let a be a point of $\varDelta ^{+}_{1}$ exterior to H and $\varDelta ^{+}_{k,a}$ the subgeometry of $\varDelta ^{+}_{k}$ induced on the set of points $X\in \varDelta ^{+}_{k}$ such that a∈X. Then $\varDelta ^{+}_{k,a}$ is isomorphic to the (k−1)-grassmannian $\overline{\varDelta }^{+}_{k-1}$ of a building $\overline{\varDelta }^{+}$ of type D _n−1 defined over $\mathbb{F}$. The embedding $\eta^{+}_{k,a}:\varDelta ^{+}_{k,a}\rightarrow\mathrm {PG}(U'_{k,a})$ induced by $\eta^{+}_{k}$ on $\varDelta ^{+}_{k,a}$, where $U'_{k,a} := \langle\eta^{+}_{k}(\varDelta ^{+}_{k,a})\rangle$ can be regarded as a projective embedding of $\overline{\varDelta }^{+}_{k-1}$.

When k=2 let l ₀ be a line of $\varDelta ^{+}_{1}$ not contained in H∪a ^⊥ and such that a ^⊥∩l ₀≠H∩l ₀. Put $S^{+}_{2} := \langle\{l_{0}\}\cup \varDelta ^{+}_{2,a}\cup \varDelta ^{+}_{2,H}\rangle_{\varDelta ^{+}_{2}}$.

When k=3 the subgeometry $\varDelta ^{+}_{1,a,H}$ of $\varDelta ^{+}_{1}$ induced on a ^⊥∩H is isomorphic to the polar space associated with a non-singular quadratic form of $V(2n-2,\mathbb{F})$ of Witt index n−1. It is well known that the latter can be spanned by 2n−2 points. Hence the same holds for $\varDelta ^{+}_{1,a,H}$. Given a spanning set {p ₁,…,p _2n−2} of $\varDelta ^{+}_{1,a,H}$, for every i=1,…,2n−2 we pick a plane α _i of $\varDelta _{1}^{+}$ through p _i such that α _i ∩H∩a ^⊥={p _i }. We put $S^{+}_{3} := \langle\{\alpha_{i}\}_{i=1}^{2n-2}\cup \varDelta ^{+}_{3,a}\cup \varDelta ^{+}_{3,H}\rangle_{\varDelta ^{+}_{3}}$.

The next lemma can be proved by arguments very similar to those exploited in the proofs of Lemmas 4.2 and 4.3. We leave the details for the reader.

Lemma 4.6

We have $S^{+}_{2} = \varDelta _{2}^{+}$ and $S^{+}_{3} = \varDelta ^{+}_{3}$.

Lemma 4.6 immediately implies the following:

Corollary 4.7

• (1) If $\mathrm{dim}(U'_{2,H})\leq{{2n-1}\choose2}$ then $\mathrm {dim}(U'_{2}) \leq{{2n}\choose2}$.

• (2) Let $\mathrm{dim}(U'_{3,H})\leq{{2n-1}\choose3}$ and $\mathrm {dim}(U'_{3,a})\leq{{2n-2}\choose2}$. Then $\mathrm{dim}(U'_{3}) \leq {{2n}\choose3}$.

4.3 Proof of Theorem 1.5. The case k=2

Let k=2. Assume firstly that n=3. Then, as we have recalled before, the points of $\varDelta ^{+}_{2}$ can be regarded as point-plane flags of $\mathrm{PG}(3,\mathbb{F})$. The Weyl embedding $\widetilde {\varepsilon}^{+}_{2}:\varDelta ^{+}_{2}\rightarrow\mathrm{PG}(W'_{2})$ can be described as follows: $W'_{2}$ is the vector space of null-traced (4×4)-matrices and, for every non-zero vector v of $V(4,\mathbb{F})$ and every non-trivial linear functional f of $V(4,\mathbb{F})$ such that f(v)=0, the flag {〈v〉,Ker(f)} of $\mathrm{PG}(3,\mathbb{F})$ is mapped by $\widetilde {\varepsilon}^{+}_{2}$ onto the linear subspace of $W'_{2}$ spanned by the matrix f⊗v. Note that $\mathrm{dim}(W'_{2}) = 4^{2}-1 = 15 = {{2\cdot3}\choose2}$. The next lemma is contained in the main result of Völklein [25] as a special case.

Lemma 4.8

Let $\mathbb{F}$ be a perfect field of positive characteristic or a number field and let n=3. Then the Weyl embedding $\widetilde{\varepsilon }^{+}_{2}$ is universal.

The next theorem can be proved by induction on n, using Lemma 4.8 to start and Corollary 4.3 combined with part (1) of Corollary 4.7 to go on.

Theorem 4.9

Let $\mathbb{F}$ be a perfect field of positive characteristic or a number field and let n≥3. Then every projective embedding of Δ ₂ has dimension at most ${{2n+1}\choose2}$ and every projective embedding of $\varDelta _{2}^{+}$ has dimension at most ${{2n}\choose2}$.

Since $\mathrm{dim}(\widetilde{\varepsilon}_{2}) = {{2n+1}\choose2}$ and $\mathrm{dim}(\widetilde{\varepsilon}^{+}_{2}) = {{2n}\choose2}$, Theorem 4.9 immediately implies the following corollary, which contains part (1) of Theorem 1.5.

Corollary 4.10

Let $\mathbb{F}$ be a perfect field of positive characteristic or a number field and let n≥3. Then both $\widetilde{\varepsilon}_{2}$ and $\widetilde{\varepsilon}^{+}_{2}$ are universal.

4.4 Quasi-veronesean embeddings of projective spaces

We can deal with the case k=3 by induction just as we have done for k=2 in the previous subsection, but in order to start the induction we need an analog of Lemma 4.8 for n=4. We will obtain such a lemma in the next subsection. In the present subsection we prove a preliminary result to be exploited in the proof of that lemma. We firstly state a few definitions.

The following class of embeddings includes both projective and veronesean embeddings. Given a point-line geometry Γ and a vector space U defined over a commutative division ring, a quasi-veronesean embedding of Γ in PG(U) is an injective mapping ν from the point-set of Γ to the set of points of PG(U) such that ν(Γ) spans PG(U) and every line of Γ is mapped by ν onto either a non-singular conic or a line of PG(U). We set dim(ν):=dim(U), as usual.

Given a quasi-veronesean embedding ν:Γ→PG(U), the span 〈ν(l)〉 in PG(U) of the image ν(l) of a line l of Γ is uniquely determined by any three of its points. This suggests to consider the following notions. A set of points X of Γ is a 3-subspace of Γ if every line meeting X in at least three points is contained in X. Intersections of 3-subspaces are 3-subspaces. So, for every set X of points, the intersection $\langle X\rangle^{3}_{\varGamma}$ of all 3-subspaces of Γ containing X is the smallest 3-subspace of Γ containing X. We call it the 3-span of X in Γ. Note that every subspace of Γ as defined in Sect. 2 is also a 3-subspace, but the converse is false in general. Hence $\langle X\rangle^{3}_{\varGamma}\subseteq\langle X\rangle_{\varGamma}$, possibly with strict inclusion. We say that X 3-generates Γ if $\langle X\rangle^{3}_{\varGamma}= \varGamma$.

Lemma 4.11

Let $\mathbb{F}\neq\mathbb{F}_{2}$. Then, for every positive integer d, every quasi-veronesean embedding of $\mathrm{PG}(d,\mathbb{F})$ is at most ${{d+2}\choose 2}$-dimensional.

Proof

It suffices to prove that $\mathrm{PG}(d,\mathbb{F})$ can be 3-generated by a set of ${{d+2}\choose2}$ points. Explicitly,

• (1) $\mathrm{PG}(d,\mathbb{F})$ admits a 3-generating set of size ${{d+2}\choose2}$.

It is convenient to combine (1) with the following:

• (2) If $X\subset\mathrm{PG}(d,\mathbb{F})$ is such that $|X| = {{d+1}\choose2}$ and $\langle X\rangle^{3}_{\mathrm{PG}(d,\mathbb{F})}$ is a hyperplane of $\mathrm{PG}(d,\mathbb{F})$, then we can find a set of points $Y\subset\mathrm{PG}(d,\mathbb{F})$ such that |Y|=d+1, Y∩X=∅ and X∪Y 3-generates $\mathrm {PG}(d,\mathbb{F})$.

We shall prove the conjunction of (1) and (2) by induction on d. If d=1 there is nothing to prove. Let d>1. Assume that we have already proved that (2) holds for that d. Every hyperplane H of $\mathrm{PG} (d,\mathbb{F})$ admits a 3-generating set X of size ${{d+1}\choose 2}$, by the inductive hypothesis on (1). By (2), we can enlarge X to a 3-generating set X∪Y of $\mathrm{PG}(d,\mathbb{F})$ of size $d+1 + {{d+1}\choose 2} = {{d+2}\choose2}$. So, (1) holds for d.

It remains to prove (2). Given X as in (2), let $H_{1} = \langle X\rangle^{3}_{\mathrm{PG}(d,\mathbb{F})}$ and let H ₂ be another hyperplane of $\mathrm{PG}(d,\mathbb{F} )$, different from H ₁. By the inductive hypothesis on (1), H ₁∩H ₂ admits a 3-generating set X′ of size ${d\choose2}$. By the inductive hypothesis on (2), we can find Y′⊂H ₂ such that |Y′|=d, Y′∩X′=∅ and $\langle X'\cup Y'\rangle^{3}_{\mathrm{PG}(d,\mathbb{F})} = H_{2}$. Thus, $\langle X\cup Y'\rangle^{3}_{\mathrm{PG}(3,\mathbb{F})}\supseteq H_{1}\cup H_{2}$. Pick now a point $p\in \mathrm{PG}(d,\mathbb{F})\setminus(H_{1}\cup H_{2})$ and let H ₃ be the unique hyperplane of $\mathrm{PG}(d,\mathbb{F})$ containing {p}∪H ₁∩H ₂. Every point x of $\mathrm{PG}(d,\mathbb{F})$ exterior to H ₁∪H ₂∪H ₃ belongs to a line through p meeting H ₁∪H ₂ in two distinct points. Hence $\langle X\cup Y'\cup\{p\}\rangle^{3}_{\mathrm{PG}(d,\mathbb{F})}$ contains all points of $\mathrm{PG}(d,\mathbb{F})$ except possibly those of H ₃ different from p and exterior to H ₁∩H ₂. As $\mathbb{F}\neq\mathbb{F}_{2}$, this set of points is enough to 3-generate $\mathrm{PG}(d,\mathbb{F})$. So, Y:=Y′∪{p} has the properties required in (2). □

Remarks

• 1. The hypothesis $\mathbb{F}\neq\mathbb{F}_{2}$ cannot be dropped from Lemma 4.11. Indeed let Γ be a point-line geometry where every line has just 3 points and let P be the point-set of Γ. Then every subset of P is a 3-subspace of Γ and Γ admits a (universal) veronesean embedding $\nu:\varGamma \rightarrow\mathrm{PG}(\mathbb{F}_{2}^{P})$ where $\mathbb{F}_{2}^{P}$ is the $\mathbb{F}_{2}$-vector space of all functions $f:P\rightarrow\mathbb{F}_{2}$ and ν sends p∈P to the characteristic function of {p}. Clearly, dim(ν)=|P|.

• 2. Note that ${{d+2}\choose2}$ is indeed the dimension of the usual veronesean embedding of $\mathrm{PG}(d,\mathbb{F})$, sending a vector $(x_{i})_{i=0}^{d}\in V(d+1,\mathbb{F})$ to the vector $(x_{i}x_{j})_{i\leq j}\in V\bigl({{d+2}\choose2},\mathbb{F}\bigr)$. So, by Lemma 4.11, that embedding is relatively universal when $\mathbb{F}\neq\mathbb{F}_{2}$.

4.5 An analog of Lemma 4.8 for n=4

In this subsection $\mathbb{F}$ is either a perfect field of positive characteristic different from $\mathbb{F}_{2}$ or a number field, n=4 and $\eta_{3}^{+}:\varDelta _{3}^{+}\rightarrow\mathrm{PG}(U'_{3})$ is a projective embedding of $\varDelta ^{+}_{3}$.

For a point p of $\varDelta _{1}^{+}$, we denote by $\varDelta ^{+}_{3,p}$ the subgeometry of $\varDelta ^{+}_{3}$ induced on the set of planes of $\varDelta ^{+}_{1}$ containing p and $\eta^{+}_{3,p}$ is the restriction of $\eta^{+}_{3}$ to $\varDelta ^{+}_{3,p}$. Clearly, $\varDelta _{3,p}^{+}$ is isomorphic to the 2-grassmannian $\overline{\varDelta }^{+}_{2}$ of a building $\overline {\varDelta }^{+}$ of type D ₃ defined over $\mathbb{F}$ and $\eta^{+}_{3,p}$ can be regarded as a projective embedding of $\overline{\varDelta }^{+}_{2}$ into $U'_{3,p} := \langle\eta^{+}_{3}(\varDelta _{3,p}^{+})\rangle$. By Corollary 4.10, $\eta^{+}_{3,p}$ is a quotient of the Weyl embedding of $\overline {\varDelta }^{+}_{2}$. Hence $\mathrm{dim}(U'_{3,p})\leq\nobreak 15$.

Given two non-collinear points a and b of $\varDelta ^{+}_{1}$, let $\varDelta ^{+}_{3,a,b}$ be the set of planes of $\varDelta ^{+}_{1}$ contained in a ^⊥∩b ^⊥ and $U'_{3,a,b} := \langle\eta^{+}_{3}(\varDelta ^{+}_{3,a,b})\rangle$.

Lemma 4.12

$\mathrm{dim}(U'_{3,a,b})\leq20$.

Proof

The polar space $\varDelta ^{+}_{1}$ induces on a ^⊥∩b ^⊥ the line grassmannian of a projective geometry $\varPi_{a,b}\cong\mathrm {PG}(3,\mathbb{F})$. So, the set $\varDelta ^{+}_{3,a,b}$ is partitioned in two sets P ₀ and P ₁, corresponding to the points and the planes of Π _a,b .

Let p∈a ^⊥∩b ^⊥. The residue res(p) of p in Δ ⁺ is a D ₃-building, $\varDelta ^{+}_{3,p}$ is the 2-grassmannian of the building res(p) and the subgeometry $\varDelta ^{+}_{2,p}$ induced by $\varDelta ^{+}_{2}$ on the set of lines of $\varDelta ^{+}_{1}$ through p is the polar space associated to res(p). The lines l _a and l _b of $\varDelta ^{+}_{1}$ joining p with a or b respectively, are points of the polar space $\varDelta ^{+}_{2,p}$ and $S_{p} := \varDelta ^{+}_{3,p}\cap \varDelta ^{+}_{3,a,b}$ is the set of lines of $\varDelta ^{+}_{2,p}$ contained in $l_{a}^{\perp}\cap l_{b}^{\perp}$. The set S _p is partitioned in two families P _p,0=S _p ∩P ₀ and P _p,1=S _p ∩P ₁. In the polar space $\varDelta ^{+}_{2,p}$, the sets P _p,0 and P _p,1 form the two families of lines of a grid. On the other hand, p is a line of the projective geometry Π _a,b . Accordingly, P _p,0 is the set of points of the line p of Π _a,b and P _p,1 is the set of planes of Π _a,b through p.

The polar space $\varDelta ^{+}_{2,p}$ can also be regarded as the line-grassmannian of a projective geometry $\varPi_{p} \cong\mathrm {PG}(3,\mathbb{F})$. The lines l _a and l _b appear as two skew lines in Π _p while the elements of P _p,0 and P _p,1 are point-plane flags, formed by a point in l _a and a plane through l _b or a point of l _b and a plane on l _a . We can assume that the elements of P _p,0 are flags {x,X} with x a point of l _a and X a plane through l _b while those of P _p,1 are flags {x,X} with x∈l _b and X⊃l _a .

The embedding $\eta^{+}_{3,p}:\varDelta ^{+}_{3,p}\rightarrow\mathrm {PG}(U'_{3,p})$ is a quotient of the Weyl embedding $\widetilde{\varepsilon }^{+}_{2,p}:\varDelta ^{+}_{3,p}\rightarrow\mathrm{PG}(\widetilde{U})$, where $\widetilde {U}$ is the vector space of null-traced 4×4-matrices with entries in $\mathbb{F}$. We have given an explicit description of this embedding in Sect. 4.3. Comparing that description with the above characterization of P _p,0 and P _p,1 as point-plane flags of Π _p , one can see that for i=0,1 the embedding $\widetilde {\varepsilon}^{+}_{2,p}$ maps P _p,i onto a non-singular conic C _p,i of $\mathrm{PG}(\widetilde{U})$. Explicitly, if {u ₁,u ₂,u ₃,u ₄} is a basis of $V(4,\mathbb{F})$ such that l _a and l _b correspond to the lines 〈u ₁,u ₂〉 and 〈u ₃,u ₄〉 of $\mathrm{PG}(3,\mathbb{F}) \cong\varPi_{p}$, then $C_{p,0} = \{\langle M_{0}(s,t)\rangle\}_{s,t\in\mathbb{F}}$ and $C_{p,1} = \{ \langle M_{1}(s,t)\rangle\}_{t,s\in\mathbb{F}}$ where

$$M_0(t,s) = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -ts & s^2 & 0 & 0 \\ -t^2 & ts & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right ], \qquad M_1(t,s) = \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -ts & s^2\\ 0 & 0 & -t^2 & ts \end{array} \right ]. $$

Let φ be the projection of $\widetilde{U}$ over $U'_{3,p}$. Since $\eta^{+}_{3,p}$ is injective, 〈C _p,i 〉∩ker(φ) is either the null subspace or the nucleus of the conic C _p,i , the latter case possibly occurring only if $\mathrm {char}(\mathbb{F} ) = 2$. In the first case φ maps C _p,i onto a conic of $\mathrm{PG} (U'_{3,p})$. In the second case, since $\mathbb{F}$ is perfect by assumption, every line of 〈C _p,i 〉 through the nucleus of C _p,i meets C _p,i in exactly one point. Hence φ maps C _p,i onto a line of $\mathrm{PG}(U'_{3,p})$. Thus, $\eta^{+}_{3,p}$ maps P _p,i onto either a conic or a line.

Let now $\eta^{+}_{3,a,b}$ be the restriction of $\eta^{+}_{3}$ to $\varDelta ^{+}_{3,a,b}$. Clearly, $\eta^{+}_{3,a,b}$ and $\eta^{+}_{3,p}$ induce the same mapping on $\varDelta ^{+}_{3,a,b}\cap \varDelta ^{+}_{3,p}$. As remarked above, every point p∈a ^⊥∩b ^⊥ is a line of Π _a,b and P _p,0 is the set of points of that line. Moreover P ₀ is the set of points of Π _a,b . As $\eta^{+}_{3,p}$ maps P _p,0 onto either a conic or a line, $\eta^{+}_{3,a,b}$ is a quasi-veronesean embedding of Π _a,b . By Lemma 4.11, $\mathrm {dim}(\langle\eta^{+}_{3,a,b}(P_{0})\rangle)\leq10$. By a dual argument, $\mathrm{dim}(\langle\eta^{+}_{3,a,b}(P_{1})\rangle)\leq10$. Hence $\mathrm{dim}(\langle\eta^{+}_{3,a,b}(P_{0}\cup P_{1})\rangle)\leq20$. □

The polar space $\varDelta ^{+}_{1,a,b}$ induced by $\varDelta ^{+}_{1}$ on a ^⊥∩b ^⊥ can be generated by six points, say p ₁,p ₁,…,p ₆. For every i=1,2,…,6, let α _i be a plane of $\varDelta ^{+}_{1}$ on p _i such that a ^⊥∩b ^⊥∩α _i ={p _i }. Put $S := \langle\{\alpha_{i}\}_{i=1}^{6}\cup \varDelta ^{+}_{3,a}\cup \varDelta ^{+}_{3,b}\cup \varDelta ^{+}_{3,a,b}\rangle_{\varDelta ^{+}_{3}}$.

Lemma 4.13

$S = \varDelta ^{+}_{3}$.

Proof

Throughout the proof of this lemma the words ‘point’, ‘line’, ‘plane’ and ‘space’ refer to a point, a line, a plane or a 3-space, respectively, of the polar space $\varDelta _{1}^{+}$. We say that a point p∈a ^⊥∩b ^⊥ is S-full if all planes on p belong to S. We chop our proof in a series of steps.

• (1) Every plane contained in a common space with either a or b belongs to S.

Let X be a space on a. Then b ^⊥∩X is a plane. It belongs to $\varDelta _{3,a,b}^{+}$. Hence it belongs to S. On the other hand, a∉b ^⊥∩X since $a\not\perp b$ by assumption. Moreover, all planes of X through a belong to $\varDelta ^{+}_{3,a}$, whence they belong to S. It follows that all planes of X are in S. Claim (1) follows.

• (2) Given a point p∈a ^⊥∩b ^⊥, if there is a plane α ₀ on p such that α ₀∈S and α ₀∩a ^⊥∩b ^⊥={p}, then p is S-full.

Let X be a space on α ₀. By (1), both planes a ^⊥∩X and b ^⊥∩X belong to S. These two planes meet α ₀ in distinct lines passing through p. Therefore, and since α ₀∈S, all planes of X through p belong to S. Let H be the set of planes through p that meet a ^⊥∩b ^⊥ in at least a line. The complement $\varDelta ^{+}_{3,p}\setminus H$ of H in $\varDelta ^{+}_{3,p}$ is a connected subgeometry of $\varDelta ^{+}_{3,p}$. It contains α ₀, which belongs to S. Hence, by the above, $\varDelta ^{+}_{3,p}\setminus H\subseteq S$. Moreover, still by the above, every plane through p contained in a common subspace with a plane of $\varDelta ^{+}_{3,p}\setminus H$ belongs to S. On the other hand, a plane through p is not contained in a common space with any of the planes of $\varDelta ^{+}_{3,p}\setminus H$ only if it belongs to $\varDelta ^{+}_{3,a,b}$. If so, it belongs to S. Therefore S contains all planes through p.

• (3) Let x,y,z be three points of a line l⊂a ^⊥∩b ^⊥ and suppose that both x and y are S-full. Then z is S-full too.

There exists at least one space X containing l and such that a ^⊥∩b ^⊥∩X=l. As x and y are S-full, all planes of X through either x or y belong to S. Hence all planes of X belong to S. On the other hand a ^⊥∩b ^⊥∩X=l. Therefore there exists at least one plane α of X containing z and such that α∩a ^⊥∩b ^⊥={z}. This plane belongs to S, as all planes of X belong to S. So, z satisfies the hypotheses of (2). By (2), z is S-full.

We can now finish the proof of the lemma. For every i=1,2,…,6, we have chosen the plane α _i on p _i in such a way that the hypotheses of (2) hold for p _i and α _i . Hence p _i is S-full. On the other hand, p ₁,…,p ₆ span $\varDelta ^{+}_{1,a,b}$, by assumption. Therefore every point of a ^⊥∩b ^⊥ is S-full, by (3). Finally, every plane meets a ^⊥∩b ^⊥ in at least a point. Hence every plane belongs to S. □

Theorem 4.14

Let $\mathbb{F}$ be either a perfect field of positive characteristic, different from $\mathbb{F}_{2}$, or a number field. Let n=4. Then the Weyl embedding $\widetilde{\varepsilon}^{+}_{3}$ is universal.

Proof

By Lemma 4.13, for every projective embedding $\eta^{+}_{3}$ of $\varDelta ^{+}_{3}$ we have

$$\mathrm{dim}\bigl(\eta^+_3\bigr)~ \leq~ 6 + \mathrm{dim}\bigl( \eta^+_{3,a}\bigr) + \mathrm {dim}\bigl(\eta^+_{3,b}\bigr) + \mathrm{dim}\bigl(U'_{3,a,b}\bigr). $$

On the other hand, $\mathrm{dim}(\eta^{+}_{3,a})$ and $\mathrm {dim}(\eta^{+}_{3,b})$ are less or equal to 15 by Lemma 4.8 and $\mathrm{dim}(U'_{3,a,b})\leq20$ by Lemma 4.12. Hence $\mathrm{dim}(\eta^{+}_{3})\leq56$. However $\mathrm{dim}(\widetilde {\varepsilon}^{+}_{3}) = 56$ and $\varDelta ^{+}_{3}$ admits the absolutely universal embedding [8]. Hence $\widetilde{\varepsilon}^{+}_{3}$ is universal. □

4.6 Proof of Theorem 1.5. The case k=3

By an inductive argument on n, using Theorem 4.14 to start and Corollary 4.5 combined with part (2) of Corollary 4.7 to go on, we obtain the following:

Theorem 4.15

Let $\mathbb{F}$ be a perfect field of positive characteristic, different from $\mathbb{F}_{2}$ or a number field and let n>3. Then every projective embedding of Δ ₃ has dimension at most ${{2n+1}\choose3}$ and every projective embedding of $\varDelta _{3}^{+}$ has dimension at most ${{2n}\choose3}$.

Since $\mathrm{dim}(\widetilde{\varepsilon}_{3}) = {{2n+1}\choose3}$ and $\widetilde{\varepsilon}^{+}_{3} = {{2n}\choose3}$, Theorem 4.15 immediately implies the following corollary, which contains part (2) of Theorem 1.5.

Corollary 4.16

Let $\mathbb{F}$ be a perfect field of positive characteristic, different from $\mathbb{F}_{2}$, or a number field and let n>3. Then both $\widetilde {\varepsilon}_{3}$ and $\widetilde{\varepsilon}^{+}_{3}$ are universal.

4.7 Remarks

• 1. The assumptions n>2 when k=2 and n>3 when k=3 cannot be removed from Theorem 1.5. Indeed when n=k=2 or 3 the Weyl embedding $\widetilde{\varepsilon}_{k}$ is veronesean. Regretfully, we do not know so much on veronesean embeddings. We guess that $\widetilde{\varepsilon}_{n}$ is relatively universal when $\mathrm {char}(\mathbb{F})\neq2$, but so far we have not found a way to prove this conjecture, even in the case of n=2. On the other hand, if $\mathbb {F}$ is a perfect field of characteristic 2 then $\widetilde{\varepsilon}_{n}$ is not universal, for any n (see [11]).

• 2. One might believe that the ideas exploited in Sect. 4.3 can be re-used to obtain results similar to Corollaries 4.3, 4.5 and 4.7 for any k<n, but things are not so easy as they look at first glance. For instance, let k=4. Instead of choosing a generating set p ₁,p ₂,…,p _2n−1 for the polar space induced by Δ ₁ on a ^⊥∩H and suitable planes α ₁,α ₂,…,α _2n−1 on p ₁,p ₂,…,p _2n−1 as we have done for Δ ₃, we could consider a generating set {l ₁,…,l _m } of the 2-grassmannian $\overline {\varDelta }_{2}$ of that polar space and a suitable 4-space on each of l ₁,…,l _m . However, for this move to be effective we need $m = {{2n-1}\choose2}$. Thus, we should know that $\overline{\varDelta }_{2}$ admits a generating set of size ${{2n-1}\choose2}$. However we do not know if this is true in general. It is true when $\mathbb{F}$ is a finite prime field (Cooperstein [14]), but perhaps it is false for other fields (compare Blok and Pasini [7]). Anyway, Theorem 1.5 is of no help here. That theorem only tells us that every projective embedding of $\overline{\varDelta }_{2}$ is at most ${{2n-1}\choose2}$-dimensional. It says nothing on generating sets.

  We face similar difficulties if, in the attempt to generalize Theorem 4.14 to $\varDelta ^{+}_{n-1}$ with n>4, we try to rephrase the proof of Lemma 4.13. Besides this, in order to generalize Theorem 4.14 we must preliminarily prove an analog of Lemma 4.11 for quasi-veronesean embeddings of half spin geometries, obtaining an upper bound for the dimension of such an embedding, but this does not look so easy to do. Perhaps, it is equivalent to determine an upper bound for the dimension of a quasi-veronesean embedding of Δ _n .