ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 151–170
https://doi.org/10.26493/1855-3974.2201.b65
(Also available at http://amc-journal.eu)
Geometry of the parallelism in polar spine
spaces and their line reducts
Krzysztof Petelczyc , Krzysztof Prażmowski , Mariusz Żynel
Faculty of Mathematics, University of Białystok,
Ciołkowskiego 1 M, 15-245 Białystok, Poland
Received 18 December 2019, accepted 22 November 2020, published online 20 October 2021
Abstract
The concept of the spine geometry over a polar Grassmann space belongs to a wide
family of partial affine line spaces. It is known that the geometry of a spine space over
a projective Grassmann space can be developed in terms of points, so called affine lines,
and their parallelism (in this case the parallelism is not intrinsically definable as it is not
Veblenian). This paper aims to prove an analogous result for the polar spine spaces. As a
by-product we obtain several other results on primitive notions for the geometry of polar
spine spaces.
Keywords: Grassmann space, projective space, polar space, spine space, coplanarity, pencil of lines.
Math. Subj. Class. (2020): 51A15, 51A45
Introduction
Some properties of the polar spine spaces were already established in [8], where the class
of such spaces was originally introduced. Its definition resembles the definition of a spine
space defined within a (projective) Grassmann space (= the space Pk(V) of pencils of
k-subspaces in a fixed vector space V), cf. [12, 13]. In every case, a spine space is a
fragment of a (projective) Grassmannian whose points are subspaces which intersect a fixed
subspace W in a fixed dimension m. In case of polar spine spaces we consider a two-
step construction, in fact: we consider the subspaces of V that are totally isotropic (self
conjugate, singular) under a fixed nondegenerate reflexive bilinear form ξ on V, and then
we restrict this class to the subspaces which touch W in dimension m.
It is a picture which is seen from the view of V. Clearly, W can be extended to a
subspace M of V with codimension 1 and then M yields a hyperplane M of the polar space
E-mail addresses: kryzpet@math.uwb.edu.pl (Krzysztof Petelczyc), krzypraz@math.uwb.edu.pl (Krzysztof
Prażmowski), mariusz@math.uwb.edu.pl (Mariusz Żynel)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
152 Ars Math. Contemp. 20 (2021) 151–170
Q0 determined by ξ in V. In other words, the projective points on W that are points of Q0
yield a subspace W of Q0 extendable to a hyperplane. Recall that situation of this sort was
already investigated in [10]. The isotropic k-subspaces of V are the (k − 1)-dimensional
linear subspaces of Q0, and first-step restriction yields the so called polar Grassmann space
Qk−1 = Pk−1(Q). The points of Qk−1 which touch W in dimension m are – from view
of Q – the elements of Qk−1 which touch W in dimension m− 1. So, a polar spine space
is also the fragment of a polar Grassmannian which consists of subspaces which touch a
fixed subspace extendable to a hyperplane in a fixed dimension. The analogy seems full.
In particular, when W is a hyperplane of V i.e. W is a hyperplane of Q0 then a k-
subspace of Q either is contained in W or it touches it in dimension k − 1. It is seen that
in this case the only reasonable value of m is m = k − 1 and the obtained structure is
the Grassmannian of subspaces of the affine polar space obtained from Q0 by deleting W
(cf. [3, 11]). So, the class of polar spine spaces contain Grassmannians of k-subspaces of
arbitrary polar slit space: of a polar space with a subspace (extendable to a hyperplane)
removed, see [10]. An interesting case appears, in particular, when we assume that W is
isotropic.
1 Generalities
This section is quoted after [8] with slight modifications.
1.1 Point-line spaces and their fragments
A point-line structure B = ⟨S,L⟩, where the elements of S are called points, the elements
of L are called lines, and where L ⊂ 2S , is said to be a partial linear space, or a point-line
space, if two distinct lines share at most one point and every line is of size (cardinality) at
least 2 (cf. [2]).
A subspace of B is any set X ⊆ S with the property that every line which shares with
X two or more points is entirely contained in X . We say that a subspace X of B is strong
if any two points in X are collinear. If S is strong, then B is said to be a linear space.
Let us fix a nonempty subset H ⊂ S and consider the set
L|H :=
{
k ∩H : k ∈ L and |k ∩H| ≥ 2
}
. (1.1)
The structure
M := B ↾ H = ⟨H,L|H⟩
is a fragment of B induced by H and itself it is a partial linear space. The incidence relation
in M is again ∈, inherited from B, but limited to the new point set and line set. Following a
standard convention we call the points of M proper, and the points in S \H improper. The
set S \ H will be called the horizon of M. To every line L ∈ L|H we can assign uniquely
the line L ∈ L, the closure of L, such that L ⊆ L. For a subspace X ⊆ H the closure of X
is the minimal subspace X of B containing X . A line L ∈ L|H is said to be a projective
line if L = L, and it is said to be an affine line if |L \ L| = 1. With every affine line L one
can correlate the point L∞ ∈ S \H by the condition L∞ ∈ L\L. We write A for the class
of affine lines. In what follows we consider sets H which satisfy the following
|L \ H| ≤ 1 or |L ∩H| ≤ 1 for all L ∈ L.
Note that the above holds when H or S \ H is a subspace of B, but the above does not
force H or S \ H to be a subspace of B. In any case, under this assumption every line
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 153
is either projective or affine. In case L|H contains projective or affine lines only, then
M is a semiaffine geometry (for details on terminology and axiom systems see [18]). In
this approach an affine space is a particular case of a semiaffine space. For affine lines
L1, L2 ∈ L|H we can define a parallelism in a natural way:
L1, L2 are parallel (L1 ∥ L2) iff L1 ∩ L2 ∩ (S \ H) ̸= ∅.
In what follows we assume that the notion of ‘a plane’ (= 2-dimensional strong sub-
space) is meaningful in B: e.g. B is an exchange space, or a dimension function is defined
on its strong subspaces. In the article in most parts we consider B such that its planes are,
up to an isomorphism, projective planes. We say that E is a plane in M if E is a plane in B.
Observe that there are two types of planes in M: projective and semiaffine. A semiaffine
plane E arises from E by removing a point or a line. In result we get a punctured plane
or an affine plane respectively. For lines L1, L2 ∈ L|H we say that they are coplanar and
write
L1 π L2 iff there is a plane E such that L1, L2 ⊂ E. (1.2)
Let E be a plane in M and U ∈ E. The set
p(U,E) :=
{
L ∈ L|H : U ∈ L ⊆ E
}
(1.3)
will be called a pencil of lines if U is a proper point, or a parallel pencil otherwise. The
point U is said to be the vertex and the plane E is said to be the base plane of that pencil.
We write
L1 ρ L2 iff there is a pencil p such that L1, L2 ∈ p. (1.4)
1.2 Cliques
Let ϱ be a binary symmetric relation defined on a set X. A subset of X is said to be a
ϱ-clique iff every two elements of this set are ϱ-related.
For any x1, x2, . . . , xs in X we introduce
∆sϱ(x1, x2, . . . , xs) iff ̸= (x1, x2, . . . , xs) and xi ϱ xj for all i, j = 1, . . . , s
and for all y1, y2 ∈ X if y1, y2 ϱ x1, x2, . . . , xs then y1 ϱ y2, (1.5)
cf. analogous definition of ∆ϱs in [9]. For short we will frequently write ∆ϱ instead of ∆
s
ϱ.
Next, we define
[|x1, x2, . . . , xs|]ϱ :=
{
y ∈ X : y ϱ x1, x2, . . . xs
}
. (1.6)
It is evident that if ∆ϱ(x1, . . . , xs) holds (and ϱ is reflexive) then [|x1, . . . , xs|]ϱ is the
(unique) maximal ϱ-clique which contains {x1, . . . , xs}. Finally, for an arbitrary integer
s ≥ 3 we put
Ksϱ =
{
[|x1, x2, . . . , xs|]ϱ : x1, x2, . . . , xs ∈ X and ∆ϱ(x1, x2, . . . , xs)
}
. (1.7)
Then we write
Kϱ :=
∞⋃
s=3
Ksϱ.
In most of the interesting situations there is an integer smax such that Kϱ =
⋃smax
s=3 K
s
ϱ =
K∗(ϱ), where
K∗(ϱ) is the set of maximal ϱ-cliques.
154 Ars Math. Contemp. 20 (2021) 151–170
1.3 Grassmann spaces and spine spaces
We start with some constructions of a general character. Let X be a nonempty set and
let P be a family of subsets of X . Assume that there is a dimension function dim: P →
{0, . . . , n} such that B = ⟨P,⊂,dim⟩ is an incidence geometry, cf. e.g. [1]. Write Pk for
the set of all U ∈ P with dim(U) = k.
Given H ∈ Pk−1 and B ∈ Pk+1 with H ⊂ B, a k-pencil over B is a set of the form
p(H,B) = {U ∈ Pk : H ⊂ U ⊂ B}.
The idea behind this concept is the same as in (1.3), though this definition is more general.
The family of all such k-pencils over B will be denoted by Pk. Then, the structure
Pk(B) = ⟨Pk,Pk⟩
will be called a Grassmann space over B (cf. [5, Section 2.1.3]). It is a partial linear space
for 0 < k < n.
Let us fix W ∈ P and an integer m. We will write
Fk,m(B,W ) := {U ∈ Pk : dim(U ∩W ) = m}.
The fragment
Ak,m(B,W ) := Pk(B) ↾ Fk,m(B,W )
will be called a spine space over B determined by W . It will be convenient to have an
additional symbol for the line set of a spine space, which is
Gk,m(B,W ) := Pk|Fk,m(B,W ).
What follows are more specific examples of the above constructions that we actu-
ally investigate in our paper. Let V be a vector space and let Sub(V) be the set of all
vector subspaces of V. Then Pk(V) is a partial linear space called a projective Grass-
mann space. In particular P1(V) is the projective space over V. It is well known that
Pk(V) ∼= Pk−1(P1(V)).
Let W ∈ Sub(V). The spine space Ak,m(V,W ) was introduced in [12] and developed
in [13, 14, 15, 16]. Note that Ak,m(V,W ) ∼= Ak−1,m−1(P1(V),Sub1(W )). The concept
of a spine space makes a little sense without the assumption that
0, k − n+ w ≤ m ≤ k,w, (1.8)
where w = dim(W ). It is a partial linear space when (1.8) is satisfied.
For possibly maximal values of m we get Ak,k(V,W ) = Pk(W ), where the points are
basically vector subspaces of W , and Ak,w(V,W ) ∼= Pk−w(V/W ), where the points are
those vector subspaces of V which contain W . Therefore, we assume that
m < k,w. (1.9)
Now, let ξ be a nondegenerate reflexive bilinear form of index r on V. For U,W ∈
Sub(V) we write U ⊥ W iff ξ(U,W ) = 0, meaning that ξ(u,w) = 0 for all u ∈ U ,
w ∈ W . Then the set of all totally isotropic subspaces of V w.r.t. ξ is
Q := {U ∈ Sub(V) : U ⊥ U},
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 155
and Qk := Q ∩ Subk(V). The set Qk is nonempty iff
k ≤ r. (1.10)
Provided that 2 ≤ r the structure Q = P1(Q) is a classical polar space embeddable into
the projective space P1(V). It is clear that Q ∼= ⟨Q1,Q2,⊂⟩ and usually polar space is
defined that way.
A polar Grassmann space is the structure Pk(Q). It is a partial linear space whenever
k < r. (1.11)
Note that Pk(Q) ∼= Pk−1(P1(Q)).
Finally,
M := Ak,m(Q,W ),
a polar spine space, the main subject of our paper, arises. Note that we have M ∼=
Ak−1,m−1(P1(Q),Sub(W ) ∩Q1).
Let rW = ind(ξ ↾ W ) be the index of the form ξ restricted to W . If rW < m, then there
is no totally isotropic subspace of V, which meets W in some m-dimensional subspace.
Every U ∈ Q can be extended to an Y ∈ Qr. Assume that dim(Y ∩ W ) > r − k + m
for all Y ∈ Qr. This means that all totally isotropic subspaces of V, which meet W in
some m-dimensional subspace, are at most (k − 1)-dimensional. On the other hand, this
assumption implies rW > r − k +m. Thus
m ≤ rW ≤ r − k +m (1.12)
is a sufficient condition for Fk,m(Q,W ) ̸= ∅.
Warning. The condition (1.12) is – in the context above – only sufficient. As we shall see
there are sets W such that r − k +m < rW but Fk,m(Q,W ) ̸= ∅. Clearly, the condition
m ≤ rW is necessary.
Under (1.12) no point of M is isolated and M is a partial linear space. Now, let us
have a look at the structure of strong subspaces of polar spine spaces. Following [13] they
are called: α-stars, ω-stars, α-tops and ω-tops. For details see Table 2. Actually, this is
an ‘adaptation’ of the classification of strong subspaces of Ak,m(V,W ) (consult [13]) to
the case when we restrict Pk(V) to Pk(Q). With a slight abuse of language all sets of
the type T α and T ω we call tops, and sets of the form Sα and Sω stars. But note that
due to some specific values of r, k,m and dim(Y ∩ W ) with Y ∈ Qr families of some
of these types may be empty. Moreover, stars and tops consist of strong subspaces of M,
but stars or tops of some kind may be not maximal among strong. In general, Sω and T α
consist of projective spaces, while the other consist of proper slit spaces (cf. [4, 18]), but if
Fk−1,m(Q,W ) ∋ H ⊂ Y ∈ Fr,m(Q,W ) then [H,Y ]k ∩ Fk,m(Q,W ) = [H,Y ]k ∈ Sα
is a projective space as well.
Generally, H ∈ Pk−1 determines a star and B ∈ Pk+1 determines a top as follows
S(H) = {U ∈ Pk : H ⊂ U}, T(B) = {U ∈ Pk : U ⊂ B}.
Here, we occasionally make use of this convention in the context of polar spine spaces,
where Pk = Fk,m(Q,W ).
156 Ars Math. Contemp. 20 (2021) 151–170
2 Lines classification and existence problems
In analogy to [13, 17] the lines of M can be of three sorts: affine (in A), α-projective
(in Lα), and ω-projective (in Lω). To be more concrete, comp. Table 1, these are pencils
L = p(H,B) ∩ Fk,m(Q,W ) such that (we consider parameters k,m,Q,W as fixed)
A: H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W ); in this case L∞ = H+(W ∩B) =
(H +W ) ∩ B. Note that L∞ ⊂ B ∈ Q and therefore L∞ ∈ Fk,m+1(Q,W ). In
other words, L∞ is a point of Ak,m+1(Q,W ).
Lα: H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m(Q,W ).
Lω: H ∈ Fk−1,m−1(Q,W ), B ∈ Fk+1,m+1(Q,W )
(cf. Table 1).
Note that if rW < m + 1 (in view of the global assumption rW ≥ m this means
rW = m) then A ∪ Lω = ∅. Looking at [8, Lemma 1.6] we see that in this case M is
disconnected as well or m = w. In the latter case also A ∪ Lω = ∅. Besides, this also
contradicts (1.9). Consequently, for rW < m+1 the horizon Ak,m+1(Q,W ) of M looses
its sense.
The problem whether one of the three above classes of lines is nonempty reduces, in
fact, to the problem whether the corresponding class of ‘possible tops’ of these lines is
nonempty. More precisely, we have the following criterion.
Lemma 2.1.
(i) Let B ∈ Fk+1,m(Q,W ); then T(B) ̸= ∅.
(ii) Let B ∈ Fk+1,m(Q,W ) and U ∈ T(B); then there is an L = [H,B]k ∈ Lα such
that U ∈ L.
So, if Fk+1,m(Q,W ) ̸= ∅ then Lα ̸= ∅.
(iii) Let B ∈ Fk+1,m+1(Q,W ); then T(B) ̸= ∅.
(iv) Let B ∈ Fk+1,m+1(Q,W ) and U ∈ T(B). Then there are:
• an L′ = [H ′, B]k ∈ Lω (provided that m > 0) such that U ∈ L′ and
• an L′′ = [H ′′, B]k ∈ A such that U ∈ L′′.
Consequently, if Fk+1,m+1(Q,W ) ̸= ∅ then A ≠ ∅, and Lω ̸= ∅ when m > 0.
Proof. To justify (i) present B in the form B = (B ∩ W ) ⊕ D, where D ∩ W = Θ
and dim(D) = k + 1 − m. Let D′ be a (k − m)-dimensional subspace of D and put
H := (B ∩W ) +D′. To justify (ii) we simply use (i) with B replaced by U to obtain the
subspace H .
To justify (iii) we present B in the form B = (B ∩W )⊕D (now, dim(D) = k −m)
and proceed analogously to (i): U = D + Z, where Z is an m-dimensional subspace of
B ∩ W . To justify (iv) to get H ′ we apply (iii) with B replaced by U , and to get H ′′ we
apply (i) with B replaced by U .
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 157
Note that, sufficient conditions for the existence of the corresponding subspaces B in
Lemma 2.1, i.e. for Fk+1,m(Q,W ),Fk+1,m+1(Q,W ) ̸= ∅ are
m ≤ rW ≤ r − (k + 1) +m and m+ 1 ≤ rW ≤ r − k +m, respectively.
We say that an U ∈ Fk,m(Q,W ) is an α-point iff each top containing U is of type α,
i.e. each line through U is of type α. Similarly, an U ∈ Fk,m(Q,W ) is an ω-point iff each
top containing U is of type ω, i.e. each line through U is either affine or of type ω.
Lemma 2.2. Let U ∈ Fk,m(Q,W ).
(i) There is B such that U ⊂ B ∈ Fk+1,m(Q,W ) ∪ Fk+1,m+1(Q,W ).
(ii) U is an α-point iff U⊥ ∩W ⊂ U . In this case
w ≤ k +m. (2.1)
Otherwise, if U⊥ ∩W ̸⊂ U then there is a B ∈ Fk+1,m+1(Q,W ) such that U ⊂ B.
(iii) U is an ω-point when U⊥ ⊂ U +W . In this case
w ≥ n+m− 2k. (2.2)
Otherwise, if U⊥ ̸⊂ U +W then there is a B ∈ Fk+1,m(Q,W ) such that U ⊂ B.
Proof. Clearly, U is not maximal isotropic, so there is a B such that U ⊂ B ∈ Qk+1. As
in [12] we obtain m ≤ dim(B ∩W ) ≤ m+ 1. This justifies (i).
To justify (ii) note that every B ∈ Qk+1 containing U belongs to Fk+1,m(Q,W ), and
then U ≺ B ⊂ U⊥ and U ∩W ⊂ B ∩W ⊂ U⊥ ∩W . If we have dim(B ∩W ) = m for
all B, then dim(U⊥ ∩W ) = m and U ∩W = U⊥ ∩W . As U ⊂ U⊥ by definition of U ,
the obtained condition is equivalent to U⊥ ∩W ⊂ U .
In this case we have W = (U ∩W )⊕D, where D is contained in a linear complement
of U⊥. D is at most codim(U⊥) = k-dimensional, so dim(W ) ≤ m+ k.
To justify (iii) note, first, that if U ⊂ B ∈ Qk+1 then B ∈ Fk+1,m+1(Q,W ). So, if
U ≺ B ∈ Q, then B = U ⊕ ⟨y⟩ with y ∈ U⊥ \ U . If U⊥ ⊂ U +W then y = u + w for
some u ∈ U and w ∈ W \ U and then B = U ⊕ ⟨w⟩. So, B ∩W = (U ∩W ) ⊕ ⟨w⟩. If
there is y ∈ U⊥ \ (U +W ), then U + ⟨y⟩ intersects W in U ∩W .
If U is as required above then n− k = dim(U⊥) ≤ dim(U +W ) = w+ k−m. This
gives w ≥ m+ n− 2k.
From Lemmas 2.1 and 2.2(ii), 2.2(iii) we infer the following geometrical fact.
Corollary 2.3.
(i) If w > k + m then through each point of M there passes an ω-line and an affine
line.
(ii) If w < n+m− 2k then through every point of M there passes an α-line.
Combining Lemmas 2.2(i) with 2.1(ii) and 2.1(iv) we obtain the following Corollary, a
weakening of Corollary 2.3 but with more general assumptions.
Corollary 2.4. If U ∈ Fk,m(Q,W ) then there is a line in Gk,m(Q,W ) through U . Con-
sequently, if Fk,m(Q,W ) ̸= ∅, then Gk,m(Q,W ) ̸= ∅.
158 Ars Math. Contemp. 20 (2021) 151–170
Comments to Lemma 2.2.
ad (ii) Condition (2.1) is a necessary condition for the existence of an α-point.
By (1.12) and (2.1) we get rW ≤ r− k+m ≤ r−w (this implies r− rW ≥ w).
This condition is not inconsistent. So, it may happen that M contains both α-
points and ω-tops.
One can note (it is, practically, proved in the proof of Lemma 2.2(ii) that if (2.1)
is satisfied and U ∈ Qk then there is a subspace W such that U is an α-point in
Ak,m(Q,W ) and dim(W ) = w.
ad (iii) Analogously, condition (2.2) is a necessary condition for the existence of an ω-
point.
It is seen that (under suitable assumption, obtained by (1.12) and (2.2): r− rW ≥
k −m ≥ n− k − w) the space M may contain both ω-points and α-tops.
And there do exist W for which associated spine spaces contain an ω-point.
As an immediate consequence of Lemma 2.2(iii) we obtain the following.
Corollary 2.5. Assume that w < n +m + 1 − 2k. Then, for every U ∈ Fk,m+1(Q,W )
there is L ∈ Ak,m(Q,W ) such that U = L∞.
3 Examples, particular cases
Let us examine in some detail polar spine spaces of some, particularly natural classes.
3.1 Grassmannians of affine polar spaces
Assume that W is a hyperplane of P; in turn this is equivalent to say that Sub1(W ) is a
hyperplane in Q. In this case we have
m = k − 1 and (3.1)
dim(W ∩ Y ) =
{
r when Y ⊂ W
r − 1 when Y ̸⊂ W
for every Y ∈ Qr. (3.2)
It is clear that in this case
Fk,m(Q,W ) ̸= ∅; in view of Corollary 2.4, Ak,m(Q,W ) is nontrivial
simply, because it is impossible to have Qk ⊂ Subk(W ). However, this case raises several
degenerations concerning the structure of strong subspaces of M.
Lemma 3.1.
(i) Let B ∈ Subk+1(V). Then either dim(B∩W ) = k+1 = m+2 (and then B ⊂ W )
or dim(B ∩ W ) = k = m + 1. Therefore, there is no strong subspace in T α.
Moreover, by the same reasons, Lα = ∅.
(ii) If B ∈ Fk+1,m+1(Q,W ) then T(B)∩Fk,m(Q,W ) ∈ T ω is a k-dimensional punc-
tured projective space.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 159
(iii) Let X = [H,Y ]k ∩ Fk,m(Q,W ), H ∈ Subk−1(V), Y ∈ Qr. Assume that
dim(H ∩ W ) = m = k − 1 i.e. H ⊂ W . If Y ⊂ W then, clearly, X = ∅. If
Y ̸⊂ W then X ∈ Sα is a (r − k)-dimensional affine space.
(iv) Let X = [H,Y ]k ∩ Fk,m(Q,W ), H ∈ Subk−1(V), Y ∈ Qr. Assume that
dim(H ∩ W ) = m − 1 = k − 2 i.e. H ̸⊂ W . Then Y ̸⊂ W and, consequently,
dim(Y ∩ W ) = r − 1. In this case X ∈ Sω is a (r − k)-dimensional projective
space.
Corollary 3.2. If 4 ≤ k + 2 ≤ r then every line of M has at least two extensions to a
maximal at least 2-dimensional strong subspace: one to a top, and one to a star.
3.2 Spine spaces with isotropic ‘holes’
Next, let us assume that W ∈ Q i.e. W is isotropic. In this case we have
rW = w. (3.3)
So, let m < w, k; let us take arbitrary D ∈ Subm(W ) and Y ∈ Qr with W ⊂ Y . Then
there is Y0 ∈ Qr such that Y ∩ Y0 = D. Consider any U such that dim(U) = k and
D ⊂ U ⊂ Y0; then U ∈ Fk,m(Q,W ). Thus we have proved that
Fk,m(Q,W ) ̸= ∅; in view of Corollary 2.4, Ak,m(Q,W ) is nontrivial.
Note that if we assume (1.10) then k+w−m ≤ r+ r−m ≤ n−m ≤ n follows, so (1.8)
holds as well.
Next, let us pay attention to the problem of extending lines. Namely, let L = p(H,B) ∈
Lω . So, dim(B ∩ W ) = m + 1. Suppose that r = m + 1; then we obtain contradictory
m < k < r = m+ 1. As above, we extend W to a maximal isotropic Y and find maximal
isotropic Y ′ with Y ∩ Y ′ = B. This proves
Lemma 3.3. If k < r−1, then every line in Lω can be extended to an at least 2-dimensional
star.
4 Binary collinearity
Let us start with a Chow’like result concerning binary collinearity λ of points in a polar
spine space M = Ak,m(Q,W ) defined for some integers k,m and a fixed subspace W of
a vector space V equipped with a suitable form ξ. To this aim standard reasoning similar
to this of [6, 7, 17] can be used:
a line through two distinct points is the intersection of all the maximal
λ-cliques which contain these points.
In the sequel we intensively analyse Table 2. Let U1 λ U2, U1 ̸= U2. Put L = U1, U2.
Evidently, every line L = p(H,B) can be extended to a top T = T(B) ∩ Fk,m(Q,W ),
which is a (k − m)-dimensional (T ∈ T α) or a k-dimensional (T ∈ T ω) slit space. We
have assumed that k > 1. So, when m < k − 1 then T is greater than L. For any triangle
U1, U2, U3 ∈ T we have ∆λ(U1, U2, U3) and T = [|U1, U2, U3|]λ.
If L is an α-projective line or an affine line then it has at least one extension to a star S
in Sα, which are (r − k)-dimensional slit spaces. Consequently, L = T ∩ S. Assume that
k < r − 1, so L ⊊ S.
160 Ars Math. Contemp. 20 (2021) 151–170
In this point we can choose one of the following two ways. Firstly, we notice that there
is a finite system U1, U2, . . . , Ut ∈ S such that ∆λ(U1, U2, . . . , Ut), so S′ := S =
[|U1, U2, . . . , Ut|]λ. Secondly, we can extend U1, U2 to any triangle U1, U2, U3 ∈ S and
note that S′ := [|U1, U2, U3|]λ is the union of all the extensions of the plane spanned by
U1, U2, U3 to a maximal λ-clique. In both cases L = S′ ∩ T and thus L can be defined in
terms of λ.
A problem may arise when L ∈ Lω . In this case each extension of L to a star S
is contained in a segment [H,Y ]k with a maximal totally isotropic extension Y of B ⊃
H and it has dimension dim(W ∩ Y ) − m. So, it may degenerate to the line L when
dim(W ∩ Y ) = m + 1. Is it possible that every such an extension Y intersects W in
dimension m + 1? Recall that the condition L ∈ Lω yields dim(B ∩ W ) = m + 1 and
therefore we obtain W ∩ Y = W ∩ B, for every Qr ∋ Y ⊃ B. So, our problematic case
reduces to the question: for which B ∈ Fk+1,m+1(Q,W ) there is no reasonable extension
Y and: when each such a B has a required extension. Note that to find Y it suffices to find
D such that B ≺ D ∈ Q and dim(D ∩ W ) = m + 2; then Y is an extension of D to a
maximal totally isotropic subspace. On the other hand, the existence of D in question can
be assured by a suitable substitution in Lemma 2.2(ii), which yields a sufficient condition
for the existence of our Y :
w > k +m+ 2. (4.1)
As a consequence we can formulate the following result.
Theorem 4.1 (The Chow Theorem for M). If m < k − 1, k < r − 1, and each line in
Lω can be extended to at least 2-dimensional star (which is assured, e.g. by (4.1)) then the
structures M and ⟨Fk,m(Q,W ),λ⟩ are definitionally equivalent.
In particular, in view of Corollary 3.2 and Lemma 3.3, the Chow theorem holds in M
when W is an isotropic subspace and k < r−1, and it holds in M when W is a hyperplane
and 4 ≤ k + 2 ≤ r.
One can continue these investigations in the fashion of [17] considering graphs of
collinearity with some sorts of lines distinguished (λα, λω , λα∨ω etc.). Observing cri-
teria in Lemma 2.2 and Corollary 2.5 we see that it may be a hard work: α-points and
ω-points may appear, ‘deep’ improper points may appear as well.
5 Maximal cliques of λσ
Let σ be a one of the symbols
α, ω, α ∨ ω, α+, ω+.
The classes Lσ with σ ∈ {α, ω} are already defined (usually, the arguments like k,m,
V, Q, W will be omitted, if unnecessary or fixed). Next, Lσ+ := Lσ ∪ A, and, finally
Lα∨ω = Lα ∪ Lω . It is evident that
Mσ := ⟨Fk,m(Q,W ),Lσk,m(Q,W )⟩ (5.1)
is a partial linear space for every admissible symbol σ as above, but it may be trivial for
particular values of k,m, r, w etc.: it may have a void line set. Let us write λσ for the
binary collinearity of points of Mσ . Let λaf be the binary collinearity in
A = ⟨Fk,m(Q,W ),Ak,m, ∥⟩.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 161
In the first part of this section we shall determine (maximal) cliques of λσ for particular
values of σ as above. Clearly, each such a clique is a λ-clique. So, it is contained in an
appropriate strong subspace of M.
We begin with some results which state, generally, that the affine lines in many cases
can be ‘eliminated’: they are definable in terms of other projective lines.
Proposition 5.1. Assume that m > 0 or w < r−k. Then for arbitrary triple U1, U2, U3 ∈
Fk,m(Q,W ) we have
there is a line L0 ∈ A s.t. U1, U2, U3 ∈ L0 ⇐⇒
there is a triangle L1, L2, L3 ∈ Lα∨ωk,m s.t. Ui ∈ Li for i = 1, 2, 3
& there is no L ∈ Lα∨ω s.t. Ui, Uj ∈ L for some 1 ≤ i < j ≤ 3. (5.2)
Proof. Let U1, U2, U3 ∈ L0 ∈ A; then we can write L0 = p(H,B) ∩ Fk,m(Q,W )
for suitable H,B. As in the proof of Lemma 6.6 we examine extensions of L0 to maximal
strong subspaces of M. First, let us have a look at T(B)∩Fk,m(Q,W ). It is an affine space
only when m = 0; otherwise it contains a nonaffine semiaffine plane A which contains L0.
The lines on A are all in Lα∨ω except the direction of L0. It suffices to find adequate
triangle on A to justify (⇒:) of (5.2).
Next, assume that m = 0 and take a look at extensions of L0 of the form [H,Y ]k ∩
Fk,m(Q,W ), then B ⊂ Y ∈ Qr. This extension is an affine space when dim(W ∩ Y ) =
r − k. If there is no such Y , which is assured by the condition assumed, our extension
contains a plane A as above and (⇒:) of (5.2) is justified.
To prove (⇐:) it suffices to note that a triangle spans a plane A in M. Since this plane
contains projective lines it is not affine, and since there are non projectively joinable points
on A it contains just one direction of affine lines. The rest is evident.
Thus we have proved the following result.
Proposition 5.2. Under assumptions made in Proposition 5.1 the class Ak,m(Q,W ) is
definable in Mα∨ω . That means: M is definable in Mα∨ω .
Remark 5.3. Analysing the proof of Proposition 5.1 one can note an even more detailed
result:
(i) If m > 0 then A is definable in Mω and therefore then Mω+ is definable in Mω .
(ii) If every affine line L = p(H,B) can be extended to a non-affine star (dim(W∩Y ) ≥
r − k +m − 3 for some maximal isotropic Y containing B) then A is definable in
Mα. So, Mα
+
is definable in Mα.
For an arbitrary set X of points we write
L(X) = {L ∈ Gk,m(Q,W ) : L ⊂ X}.
Let us remind well known and fundamental classification of lines in strong subspaces of
M.
Fact 5.4. Let X be a strong subspace of M and X = L(X).
If X ∈ T α then X ⊂ Lα, if X ∈ Sα then X ⊂ Lα
+
,
if X ∈ T ω then X ⊂ Lω
+
, if X ∈ Sω then X ⊂ Lω.
162 Ars Math. Contemp. 20 (2021) 151–170
Let us note an elementary
Fact 5.5. Let S be a n0-dimensional slit space with a w0-dimensional hole i.e. let S result
from a n0-dimensional projective space by deleting a w0-dimensional subspace D. Let L0
be the class of projective lines of S and λ0 be the binary collinearity determined by L0.
Then
(i) The maximal affine subspaces of S (i.e. maximal strong subspace w.r.t. to the family
of affine lines of S) are w0+1 dimensional affine spaces. Two such subspaces either
coincide or are disjoint.
(ii) The maximal projective subspaces of S are (n0 − w0 − 1)-dimensional projective
spaces. These are linear complements of D and the elements of K∗(λ0).
(iii) Let X be a maximal projective subspace of S; then X ∈ Kn0−w0λ0 .
If w0 ≤ n0 − 3 (i.e. every projective line of S has two distinct extensions to maximal
projective subspaces) then the Chow Theorem holds:
The class L0 is definable in terms of λ0.
Observing Table 2 and Fact 5.5 we conclude with the following.
Corollary 5.6.
(i) The maximal λα-cliques are (k −m)-dimensional projective tops: elements of T α,
and (r +m− k − dim(W ∩ Y ))-dimensional projective spaces of the form
[H,E]k, where H ⊂ E ⊂ Y , E ∩ ((W ∩ Y ) +H) = H
contained in a suitable element [H,Y ]k ∩ Fk,m(Q,W ) of Sα.
(ii) The maximal λω-cliques are (dim(W ∩ Y )−m)-dimensional projective stars: ele-
ments of Sω , and m-dimensional projective spaces of the form
[G,B]k, where G ⊂ B, G ∩ (B ∩W ) = Θ
contained in a suitable element T(B) ∩ Fk,m(Q,W ) of T ω .
(iii) The maximal λα
+
-cliques are elements of T α ∪ Sα, and the maximal λω
+
-cliques
are elements of T ω ∪ Sω.
(iv) K∗(λα∨ω) = K∗(λα) ∪ K∗(λω), so the maximal λα∨ω-cliques are of the form (i)
and of the form (ii) above.
Corollary 5.7. The following variants of the Chow Theorem hold in projective reducts
of M.
(i) If m > 1 then Mω is definable in ⟨Fk,m(Q,W ),λω⟩.
(ii) If every projective line L = p(H,B) ∈ Lα can be extended to a non-affine star
(dim(W ∩ Y ) ≤ r − k +m − 2 for some maximal isotropic Y containing B) then
Mα is definable in ⟨Fk,m(Q,W ),λα⟩.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 163
6 Parallelism, horizon, projective completion(s)
Let us summarize the following
(i) {L∞ : L ∈ Ak,m} ⊂ Fk,m+1(Q,W ).
(ii) by Lemma 2.2(ii) {L∞ : L ∈ Ak,m} ⊃ {U ∈ Fk,m+1(Q,W ) : U⊥ ∩W ̸⊂ U},
(iii) {L∞ : L ∈ Ak,m} ⊃ Fk,m+1(Q,W ), when w < n+m+1− 2k by Corollary 2.5.
Note 6.1. The set {L∞ : L ∈ Ak,m} will be frequently referred to as the horizon of M.
We warn that, generally it does not coincide with the horizon Qk \Fk,m(Q,W ) as defined
in Section 1.
Note that the inequality in (iii) above is only sufficient. One can compute e.g.
Lemma 6.2. Let W ∈ Q. Then the claim of Corollary 2.5 holds i.e. for every U ∈
Fk,m+1(Q,W ) there is an L ∈ Ak,m(Q,W ) such that U = L∞. Consequently,
{L∞ : L ∈ Ak,m} = Fk,m+1(Q,W ).
Proof. By assumption, dim(U ∩ W ) = m + 1. There are extensions Y1, Y2 ∈ Qr such
that U ⊂ Y1, W ⊂ Y2, and Y1 ∩ Y2 = U ∩ W . Take B ∈ [U, Y1]k+1; then B ∈
Fk+1,m+1(Q,W ) and we are through.
For a subset X of Fk,m(Q,W ) we write
X∞ := {N∞ : Ak,m ∋ N ⊂ X}.
Lemma 6.3. Let L = p(H,B) ∈ Lωk,m+1 ∪ Lαk,m+1.
(i) If L ∈ Lα then there is in M a plane A = [G,B]k ∩ Fk,m(Q,W ) with G ∈
Fk−2,m(Q,W ) such that A∞ = L.
(ii) Assume that w < n + m − 2k. If L ∈ Lω then A = [H,E]k ∩ Fk,m(Q,W ) with
some E ∈ Fk+2,m+2(Q,W ) is a plane in M such that A∞ = L.
Proof. Ad (i): By assumption, B ∈ Fk+1,m+1(Q,W ) and H ∈ Fk−1,m+1(Q,W ). There
is a point U ∈ L, so U ∈ Fk,m+1(Q,W ). By Lemma 2.1(iii) there is an H0 such that
U ≻ H0 ∈ Fk−1,m(Q,W ). Set G = H0 ∩H; clearly, dim(G) = k − 2, so [G,B]k is a
plane in Pk(Q). Taking into account the fact that H,H0 ≻ G we obtain dim(G ∩W ) ∈
{m+1,m} and dim(G∩W ) ∈ {m,m−1}. Thus dim(G∩W ) = m. As L ⊂ [G,B]k and
[G,B]k ⊃ [H0, B]k while [H0, B]k ∩ Fk,m(Q,W ) ∈ Ak,m we get that A ∩ Fk,m(Q,W )
is a plane in M with A∞ = L.
Ad (ii): By assumption, B ∈ Fk+1,m+2(Q,W ) and H ∈ Fk−1,m(Q,W ). As above,
we take any U ∈ L, so U ∈ Fk,m+1(Q,W ). By assumption of (ii) (they yield w < n +
(m+2)−2(k+1)) and Lemma 2.2(iii) there is an E such that B ≺ E ∈ Fk+2,m+2(Q,W ).
Next, there is B0 ∈ Fk+1,m+1(Q,W ) with U ⊂ E: B = U + ⟨b⟩ with a b ∈ W and
E = B + ⟨e⟩ with an e /∈ W ; we take B0 = U + ⟨e⟩. Clearly, E = B + B0 and
[H,B0]k ∩ Fk,m(Q,W ) ∈ Ak,m. As above we argue that A = [H,E]k ∩ Fk,m(Q,W ) is
a plane in M, and L = A∞.
164 Ars Math. Contemp. 20 (2021) 151–170
Roughly speaking, Lemma 6.3 gives sufficient condition under which a (projective) line
L of Ak,m+1(Q,W ) can be considered as a ‘horizon’ – the set of improper points of a plane
in Ak,m(Q,W ). On the other hand, considering classification of planes in Ak,m(V,W )
presented in some details in [14] we easily conclude with the following
Lemma 6.4. Let X ⊂ Subk(V) and A = X ∩ Fk,m(Q,W ) be a plane of M such that
A∞ is a line of Ak,m+1(Q,W ). Then one of the following holds:
(i) X = [G,B]k for some G ∈ Fk−2,m(Q,W ), B ∈ Fk+1,m+1(Q,W ).
(ii) X = [H,E]k for some H ∈ Fk−1,m(Q,W ) and E ∈ Fk+2,m+2(Q,W ).
Conversely, if X is defined by (i) then X ∩ Fk,m+1(Q,W ) =
(
X ∩ Fk,m(Q,W )
)∞ ∈
Lαk,m+1, and if (ii) holds, then X ∩ Fk,m+1(Q,W ) ∈ Lωk,m+1.
So, Lemma 6.4 states that the ‘horizon’ of any (affine) plane of Fk,m(Q,W ) is a (pro-
jective) line of Fk,m+1(Q,W ). As usually, the conditions of Lemma 6.3 are only suffi-
cient. Dealing with concrete cases one should look for suitable extendability more or less
‘by hand’. Let us quote an example:
Lemma 6.5. Let W ∈ Q. If L = p(H,B) ∈ Lωk,m+1 then A = [H,E]k ∩ Fk,m(Q,W )
with some E ∈ Fk+2,m+2(Q,W ) is a plane in M such that A∞ = L.
Hint. With the reasoning as in the proof of Lemma 6.3(ii) we look for an E such that
B ≺ E ∈ Fk+2,m+2(Q,W ). It suffices to find an E such that E ∩ W = B ∩ W just
considering suitable maximal isotropic extensions of B and W .
To accomplish this part of investigations on the parallelism let us check if directions are
‘isolated’: when for an affine line L of M there are other lines parallel to L and coplanar
with L; with the plane in question being affine in M.
Lemma 6.6. Let L = p(H,B) ∈ Ak,m and U = L∞.
(i) Assume that k > m + 1. There is an L0 = p(H0, B) ∈ Lαk,m+1 such that U ∈ L0
and A = [H0∩H,B]k∩Fk,m(Q,W ) is a plane in M such that A∞ = L0. We have
dim((H0 ∩H) ∩W ) = m− 1.
(ii) If B has an extension to a Y ∈ Qr such that dim(W ∩ Y ) ≥ m + 2 (this yields,
necessarily, m+ 2 ≤ rW ) then there exists an L1 = p(H,B1) ∈ Lωk,m+1 such that
U ∈ L1 and A = [H,B + B1]k is a plane in M such that A∞ = L1. We have
dim((B1 +B) ∩W ) = m+ 2.
Proof. Let us begin with a reminder: H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W ). Have
a look at the extension of L to a top T = T(B) ∩ Fk,m(Q,W ) (an ω-top in this case).
Since k > m + 1, this is a semiaffine space, and its hole is at least 1-dimensional. Let L0
be any line of Pk(V) contained in this hole and A be the plane spanned by L ∪ L0. That
way we justify (i).
Next, let us look for appropriate extension of L to an α-star S = [H,Y ]k∩Fk,m(Q,W ).
In general, it is a (r−k)-dimensional semiaffine space. Since Qk+1 ̸= ∅ we have k+1 ≤ r.
So, S is at least a line. To assure that the hole of S contains at least a line of Pk(V) we
must assume that dim(W ∩ Y ) ≥ m+ 2. That way we justify (ii).
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 165
Let us remind that for distinct affine lines L1, L2 contained in a strong subspace of
Ak,m(V,W ) their parallelism ∥ can be characterized by the following formula (so called
Veblenian parallelism).
L1 ∥v L2 ⇐⇒ there are lines L′1, L′2 s.t. |L′1 ∩ L′2| = 1,
and L′1 ∩ L′2 ∩ Li = ∅, |L′i ∩ Lj | = 1 for i = 1, 2, (6.1)
and then L1 ∥ L2 iff L1 ∥v L2. It is easy to note that the same formula (6.1) characterizes
parallelism of affine lines contained in a common strong subspace of M.
Let us begin with a special form of connectedness of the space of lines over M:
Lemma 6.7. Let U ∈ Fk,m+1(Q,W ) and L∞1 = U = L∞2 for L1, L2 ∈ Ak,m. Moreover,
assume that k ≤ r − 2. Then there are lines M1, . . . ,Mt ∈ Ak,m (t ≤ r + 1) such that
L1 = M1, L2 = Mt, and M∞i = U , Mi,Mi+1 are in a strong (semiaffine) subspace of
M or Mi = Mi+1, for i = 1, . . . , t− 1.
Proof. Write M1 := L1. We have H1, H2 ⊂ U ⊂ B1, B2, U ∈ Fk,m+1(Q,W ) and Bi ∈
Fk+1,m+1(Q,W ), Hi ∈ Fk−1,m(Q,W ) for i = 1, 2. Put N1 := [H1, B2]k∩Fk,m(Q,W ),
N2 := [H2, B1]k ∩ Fk,m(Q,W ). Then N1, N2 ∈ Ak,m, N∞2 = U = N∞1 .
If L1 = N2 we set M2 := L1. Assume that L1 ̸= N2. Note that L1, N2 ∈ T(B1) ∩
Fk,m(Q,W ) ∈ T ω . So, we set M2 := N2.
Observe that N2, L2 ⊂ [H2, V ]k ∩ Fk,m(Q,W ). So, the problem reduces to find a
required sequence of lines in the projective star S(H2). Let B1 ⊂ Y ′ ∈ Qr, B2 ⊂ Y ′′ ∈
Qr. There is a sequence Y2, . . . , Yt of elements of Qr such that Y ′ = Y2, Y ′′ = Yt, and
U ⊂ Yi, Ei := Yi ∩ Yi+1, dim(Ei) = r − 1 for i = 2, . . . , t − 1, t ≤ r + 1. Then
dim(Ei ∩W ) ≥ m + 1. From our assumption k + 1 ≤ r − 1 = dim(Ei). So, for every
i = 3, . . . , t− 1 one can find Di such that U ≺ Di ⊂ Ei−1 and dim(Di ∩W ) = m+ 1.
With Ni = [H2, Di]k we close our proof.
Corollary 6.8. Under assumptions of Lemma 6.7 the parallelism ∥ in M coincides with
the transitive closure of ∥v. Actually, it is the (r + 1)-th relational power ∥v ◦ · · · ◦ ∥v︸ ︷︷ ︸
(r+1) times
of
∥v, defined by (6.1), and therefore ∥ is definable in the incidence structure M.
As an immediate corollary we conclude with the following theorem.
Theorem 6.9. Assume the following
(1) w < n+m− 2k to assure that every line in Lωk,m+1 can be extended to a nontrivial
α-star of M (cf. Lemma 6.3),
(2) w < n+m+ 1− 2k to assure extendability of each improper point to an affine line
(cf. Corollary 2.5),
(3) m+1 > 0 or w < r−k to assure definability of Ak,m+1 in Ak,m+1(Q,W ) in terms
of its projective lines (cf. Proposition 5.2),
(4) k ≤ r − 2, to assure definability of parallelism in M (cf. Corollary 6.8).
Then Ak,m+1(Q,W ) is definable within Ak,m(Q,W ).
166 Ars Math. Contemp. 20 (2021) 151–170
In analogy to [17] in the fragment of Pk(Q) determined by R := Fk,m(Q,W ) ∪
Fk,m+1(Q,W ) (i.e. the points of M and the points of the “affine horizon” of M) we
distinguish two substructures corresponding to two possible sorts of lines. Let us set
Lτk,m :=
{
[H,B]k : H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W )
}
; it is seen that
Lτk,m = {L : L ∈ Ak,m}. Note evident relation:{
L : L is a line of Pk(Q), L ⊂ R
}
= Lαk,m ∪ Lωk,m ∪ Lαk,m+1 ∪ Lωk,m+1 ∪ Lτk,m. (6.2)
We define (write: −α = ω, −ω = α)
Nσ := ⟨R,Lσk,m ∪ Lτk,m ∪ L−σk,m+1⟩ with σ ∈ {α, ω}.
Evidently, Mσ can be embedded into Nσ . Intuitively, while the structure〈
R, {L : L is a line of Pk(Q), L ⊂ R}
〉
can be considered as a projective completion of M and, under specific assumptions, it is
definable in M, Nσ is a projective completion of Mσ .
To close this part it is worth to note the following analogue of Remark 5.3 and, at the
same time, an analogue of [17, Fact 3.1].
Remark 6.10. Assume (2) and (4) from Theorem 6.9.
(i) If m > 0 (cf. Remark 5.3) then the structure Nω is definable in Mω .
(ii) If for each affine line L = p(H,B) there is a maximal isotropic Y such that B ⊂ Y
and dim(W ∩ Y ) ≥ r − k + m − 3 (cf. Remark 5.3) and w < n + m − 2k (cf.
Lemma 6.3), then the structure Nα is definable in Mα.
According to Corollary 2.5 and Lemma 6.3, under condition w < n + m − 2k each
point of Ak,m+1(Q,W ) is a direction of a line in M and each line of Ak,m+1(Q,W ) is a
direction of a plane in M. This observation leads to the following.
Proposition 6.11. If w < n + m − 2k, then the horizon Ak,m+1(Q,W ) of M can be
defined in terms of A.
Finally, the question arises whether the adjacency of M is definable purely in terms
of the geometry of A? Unfortunately, the answer is not straightforward. The reasoning
for spine spaces that justifies [17, Proposition 4.12], based on the fact that two distinct
stars or tops of A share no line on the horizon, cannot be adopted here without significant
alterations. Note that if Lω ∪ Lα = ∅, then practically A = M. Therefore we assume that
Lω ∪ Lα ̸= ∅.
Theorem 6.12. If the ground field of V is of odd characteristic, then the structure M can
be defined in terms of A.
Proof. The proof is divided into several steps. For distinct points U1, U2 of M we define
U1 ∼+ U2 :⇐⇒ U1, U2 ⊂ B for some B ∈ Fk+1,m+1(Q,W ),
U1 ∼− U2 :⇐⇒ H ⊂ U1, U2 for some H ∈ Fk−1,m(Q,W ),
U1 ∼ U2 :⇐⇒ U1 ∼+ U2 or U1 ∼− U2.
Note that U1 ∼+ U2 yields that either U1 λaf U2 or U1 λω U2, while U1 ∼− U2 yields that
either U1 λaf U2, U1 λα U2, or U1, U2 are not collinear in M.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 167
Step 1. The following conditions are equivalent.
(i) U1 ∼+ U2 or U1 ∼− U2.
(ii) There is a plane Π1 through U1 parallel to a plane Π2 through U2 in A.
Proof of Step 1. (i) =⇒ (ii): Assume that U1, U2 ⊂ B ∈ Fk+1,m+1(Q,W ), then T(B) is
a semiaffine space (of the form T ω) and one easily finds Π1,Π2 in it.
Next, assume that U1, U2 ⊃ H ∈ Fk−1,m(Q,W ). Set B := U1 + U2. If B ∈ Q
then L = p(H,B) is a line of M. Applying analogous reasoning we find Π1,Π2 in an
extension [H,Y ]k of the type Sα. If B /∈ Q then, in any case L is a line of the surrounding
Ak,m(V,W ). Let us restrict to the subspaces around H; they form a spine space in the
projective space P1(V/H) with the quadric Q(ξ/H) distinguished. Projective reasoning
proves that required planes Π1,Π2 exist.
(ii) =⇒ (i): Let Πi be parallel planes of A with Ui ∈ Πi, i = 1, 2. Let L0 = Π∞1 = Π∞2
be the improper line of Πi. Then L0 ∈ Lαk,m+1 or L0 ∈ Lωk,m+1. In the first case L0, U1, U2
are contained in the (unique) extension to a top T(B) with B ∈ Fk+1,m+1(Q,W ) and
therefore U1 ∼+ U2. In the second case extensions of Πi to maximal strong subspaces have
form [H,Yi]k (they have L0 in common), where H ∈ Fk−1,m(Q,W ). So, U1 ∼− U2. ♢
Let us write
M0 := Ak,m(V,W ) ↾ Fk,m(Q,W )
for the surrounding spine space with point set restricted to totally isotropic subspaces. Note
that the distinction between M and M0 consists in the range of their line sets. More pre-
cisely, for a line L = p(H,B) of M0 its base B needs not to be totally isotropic and
L is a line of M iff |L| ≥ 3.
Step 2. Let U1, U2 ∈ Fk,m(Q,W ) and U1 ̸= U2. The following conditions are equivalent.
(i) U1 ∼ U2.
(ii) U1, U2 are collinear in M0 with exception when the line L of M0 which joins them
has form L = p(H,B) where H ∈ Fk−1,m−1(Q,W ), B ∈ Fk+1,m+1(V,W ), and
B /∈ Q (i.e. L is an ω-line in Ak,m(V,W )).
Proof of Step 2. (i) =⇒ (ii): It is clear that U1, U2 are collinear in the surrounding Grass-
mann space. If U1 ∼+ U2, then they lie on an affine or ω-line in M by Table 1, while if
U1 ∼− U2, then they lie on an affine or α-line in M0.
(ii) =⇒ (i): Now, let U1, U2 be collinear in M0. Hence U1, U2 ∈ p(H,B) for suitable
H,B. If dim(B∩W ) = m, then dim(H∩W ) = m and thus U1 ∼− U2. If dim(B∩W ) =
m+1, then two cases arise: dim(H ∩W ) = m,m− 1. In the former we have U1 ∼− U2.
In the later H ∈ Fk−1,m−1(Q,W ) and B ∈ Fk+1,m+1(V,W ). If B ∈ Q, then U1 ∼+ U2,
otherwise we get the excluded case. ♢
Step 3. A set X of points of A is a maximal at least 3-element ∼-clique iff X has one of
the following forms:
(a) X = T(B) for some B ∈ Fk+1,m+1(Q,W ),
(b) X = T(B) for some B ∈ Fk+1,m(Q,W ),
168 Ars Math. Contemp. 20 (2021) 151–170
(c) X = S(H) for some H ∈ Fk−1,m(Q,W ), or
(d) X = [H,Y ] ∩ Fk,m(Q,W ) for some H ∈ Fk−1,m−1(Q,W ) and H ⊂ Y ∈ Qr.
Proof of Step 3. It is easy to verify that sets defined in (a) – (d) are maximal ∼-cliques.
Now, let X be a maximal at least 3-element ∼-clique. In view of Step 2, X is a subset of
a clique in M0. So, we need general tops T0(B) = [Θ, B]k for B ∈ Subk+1(V) and stars
S0(H) = [H,V ]k for H ∈ Subk−1(V). Let us examine the following four cases:
X ⊆ T0(B), B ∈ Fk+1,m+1(V,W )
If B ∈ Q, then any two points of M0 in T(B) are ∼+-adjacent and thus X =
T(B) ∩ Fk,m(Q,W ) is a ∼-clique as in (a). If B /∈ Q, then |X| ≤ 2 by [19,
Proposition 4.4], a contradiction.
X ⊆ T0(B), B ∈ Fk+1,m(V,W )
Since |X| ≥ 3 we have B ∈ Q by [19, Proposition 4.4]. Any two points of M0 in
T(B) are ∼−-adjacent, so X = T(B) ∩ Fk,m(Q,W ) has form (b).
X ⊆ S0(H), H ∈ Fk−1,m(V,W )
Note that H ∈ Q as X is nonempty. This implies that any two points of M0 in S(H)
are ∼−-adjacent. Consequently, X = S(H) ∩ Fk,m(Q,W ) has form (c).
X ⊆ S0(H), H ∈ Fk−1,m−1(V,W )
As above H ∈ Q. The points of M0 in S0(H) are ∼-adjacent iff they are ∼+-
adjacent i.e. they are collinear in the surrounding polar Grassmann space where the
appropriate clique has form [H,Y ]k for some Y ∈ Qr (cf. [7, Section 3]). Hence
X = [H,Y ]k ∩ Fk,m(Q,W ) has form (d).
That way we obtain the desired list (a) – (d). ♢
Note that the λaf-cliques are essentially smaller than ∼-cliques.
Step 4. At least 3-element minimal intersections of the maximal ∼-cliques are lines of M.
Proof of Step 4. Let Kx be the family of cliques of the form (x) defined in Step 3. Let
X1, X2 be two distinct ∼-cliques and Z = X1 ∩X2. If X1, X2 ∈ K(a) ∪K(b), X1, X2 ∈
K(c), or X1 ∈ K(b) ∪ K(c) and X2 ∈ K(d), then Z contains at most a single point. If
X1 ∈ K(a) and X2 ∈ K(c), then Z is an affine line of M. If X1 ∈ K(b) and X2 ∈ K(c), then
Z is an α-line of M. If either X1 ∈ K(a) and X2 ∈ K(d) or X1, X2 ∈ K(d), then at least
3-element minimal Z is an ω-line of M. ♢
It is evident that every projective line of M can be presented as the intersection of
cliques enumerated in Step 3. So, applying Step 4 we get the line set of M recovered
which makes the proof of Theorem 6.12 complete.
Remark 6.13. The horizon of a star in M may have strange properties. Assume that
W ∈ Q and let H ∈ Qk−1, H ⊂ W⊥. Set m := dim(H ∩ W ). This means that
k− 1 +w−m < r. Then there is an Y0 ∈ Qr such that H ∪W ⊂ Y0. So, Y0 ∩W = W .
Write S0 = [H,Y0]k ∩Fk,m(Q,W ). Then S∞0 = [H,H +W ]k. Take any S = [H,Y ]k ∩
Fk,m(Q,W ) contained in S(H). Then S∞ = [H,H+(W ∩Y )]k ⊂ [H,H+W ]k = S∞0 .
So, in this case
S(H)∞ is the projective space [H,H +W ]k contained in M∞.
K. Petelczyc et al.: Geometry of the parallelism in polar spine spaces and their line reducts 169
Nevertheless, S(H) contains affine subspaces of different dimensions.
Note that in this case k−m−1 = dim(T(B)∞) = dim(S(H)∞) = w−m−1 yields
w = k, so horizons of stars and tops may have equal dimensions only when M consists of
points in Qk that are at the fixed distance k −m from the fixed point W .
Remark 6.14. Theorem 6.12 for polar spine spaces and its counterpart [17, Proposi-
tion 4.12] for spine spaces both say that the respective geometry depends only on affine
lines together with parallelism, that is, projective lines can be recovered using affine line
structure. However, the idea of the proof presented in this paper is more general than that
in [17] because it does not rely on specific horizons and intersections of stars which are
completely different in M and in Ak,m(V,W ). As such it can be applied for spine spaces
and is expected to give less complex reasonings.
7 Classifications
Table 1: The classification of lines in a polar spine space Ak,m(Q,W ).
Class Representative line g = p(H,B) ∩ Fk,m(Q,W ) g∞
Ak,m(Q,W ) H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m+1(Q,W ) H + (B ∩W )
Lαk,m(Q,W ) H ∈ Fk−1,m(Q,W ), B ∈ Fk+1,m(Q,W ) –
Lωk,m(Q,W ) H ∈ Fk−1,m−1(Q,W ), B ∈ Fk+1,m+1(Q,W ) –
Each strong subspace X of a polar spine space is a slit space, that is a projective space P
with a subspace D removed. In the extremes D can be void, then X is basically a projective
space, or a hyperplane, then X is an affine space.
Table 2: The classification of stars and tops in a polar spine space Ak,m(Q,W ).
Class Representative subspace
dim(P) D dim(D)
Sωk,m(Q,W )
[H, (H +W ) ∩ Y ]k : H ∈ Fk−1,m−1(Q,W ), Y ∈ Qr, H ⊂ Y
dim(W ∩ Y )−m ∅ -1
Sαk,m(Q,W )
[H,Y ]k ∩ Fk,m(Q,W ) : H ∈ Fk−1,m(Q,W ), Y ∈ Qr, H ⊂ Y
r − k [H, (H +W ) ∩ Y ]k dim(W ∩ Y )−m− 1
T αk,m(Q,W )
[B ∩W,B]k : B ∈ Fk+1,m(Q,W )
k −m ∅ -1
T ωk,m(Q,W )
[Θ, B]k ∩ Fk,m(Q,W ) : B ∈ Fk+1,m+1(Q,W )
k [B ∩W,B]k k −m− 1
170 Ars Math. Contemp. 20 (2021) 151–170
ORCID iDs
Krzysztof Petelczyc https://orcid.org/0000-0003-0500-9699
Krzysztof Prażmowski https://orcid.org/0000-0002-5352-5973
Mariusz Żynel https://orcid.org/0000-0001-9297-4774
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