Subspaces intersecting each element of a regulus in one point, Andr\'e-Bruck-Bose representation and clubs

In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\cR$ of $(n-1)$-subspaces in $\PG(2n-1,q)$, with respect to the subgroup of $\PGL(2n,q)$ fixing $\cR$. Such results have consequences on several aspects of finite geometry. First of all, a necessary condition for an $(n-1)$-subspace $U$ and a regulus $\cR$ of $(n-1)$-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results in \cite{BaJa12} on the Andr\'e-Bruck-Bose representation of a $q$-subline in $\PG(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.


Introduction
The (n − 1)-dimensional projective projective space over the field F is denoted by PG(n − 1, F ) or PG(n − 1, q) if F is the finite field of order q (denoted by F q ). If L is an extension field F q , then the projective space defined by the F q -vector space induced by L d is denoted by PG q (L d ). For further notation and general definitions employed in this paper the reader is referred to [9,11,13]. For more information on Desarguesian spreads see [1]. This paper is structured as follows. In Section 2 subspaces which intersect each element of a regulus in one point are studied and a result from [6] is generalised. Section 3 contains one of the main results of this paper, determining the order of the normal rational curves obtained from n-dimensional subspaces on an external (n − 1)-dimensional subspace with respect to a regulus in PG(2n − 1, q), obtained from a point and a subline after applying the field reduction map to PG(1, q n ). This leads to a necessary condition on the existence of a Desarguesian spread containing a subspace and regulus (Corollary 3.4). The André-Bruck-Bose representation of sublines and subplanes of a finite projective plane is studied in Section 4 and improvements are obtained with respect to the known results [5,14,15,4]. The results from the first sections are then applied to the classification problem for clubs of rank three in PG(1, q n ) in Section 5. A study of the incidence structure of the clubs in PG(1, q n ) after field reduction yields to a partial classification, concluding that the orbits of clubs under PGL(2, q n ) are at least k − 1, where k stands for the number of divisors of n. The paper concludes with an appendix discussing a result motivated by Burau [6] for the complex numbers: the result is extended to general algebraically closed fields; a new proof is provided; and counterexamples are given to some of the arguments used in the original proof.

Subspaces intersecting each element of a regulus in one point
Let R be a regulus of subspaces in a projective space and let S be any subspace of R . Questions about the properties of the set of intersection points, which for reasons of simplicity of notation we will denote by S ∩ R, often turn up while investigating objects in finite geometry. If S intersects each element of the regulus R in a point, then the intersection S ∩ R is a normal rational curve, see Lemma 2.1. This was already pointed out in [6, p.173] with a proof originally intended for complex projective spaces, but actually holding in a more general setting. The notation of [6] will be partly adopted.
The Segre variety representing the Cartesian product PG(n, F ) × PG(m, F ) in PG((n + 1)(m + 1) − 1, F ) is denoted by S n,m,F . It is well known that S n,m,F contains two families S I n,m,F and S II n,m,F of maximal subspaces of dimensions n and m, respectively. When convenient, the notation S I or S II will be used for a subspace belonging to the first or second family. The points of S n,m,F may be represented as one-dimensional subspaces spanned by rank one (m+1)×(n+1) matrices. This is the standard example of a regular embedding of product spaces, see [16]. Note that in the finite case it is possible to embed product spaces in projective spaces of smaller dimension (see e.g. [7]). A regulus R of (n − 1)-dimensional subspaces can also be defined as S I n−1,1,F . Lemma 2.1. Let n > 1 be an integer, and F a field. Let S t be a t-subspace of PG(2n − 1, F ) intersecting each S I ∈ S I n−1,1,F in precisely one point. Define Φ = S t ∩ S n−1,1,F , and assume Φ = S t . Then |F | ≥ t and the following properties hold.
(i) The set Φ is a normal rational curve of order t.
(ii) Let Ξ I ∈ S I n−1,1,F . Then the set S(Φ, Ξ I ) of the intersections of Ξ I with all transversal lines l II such that l II ∩ Φ = ∅ is a normal rational curve of order t or t − 1 if |F | = t, and of order t − 1 if |F | > t.
(iii) If Φ is contained in a subvariety S t−1,1,F of S n−1,1,F , then homogeneous coordinates can be chosen such that Φ is represented parametrically by and S(Φ, Ξ I ), for z 0 , z 1 depending only on Ξ I , by Proof. (i), (iii) The proof in [6,Sect.41 no.3], which is offered for F = C, works exactly the same provided that |F | > t or, more generally, that Φ is contained in some subvariety S t−1,1,F of S n−1,1,F . In case |F | ≤ t, the size of Φ being |F | + 1 implies |F | = t, so Φ is just a set of t + 1 independent points in a subspace isomorphic to PG(t, t), hence Φ is a normal rational curve of order t.
(ii) The case |F | > t is proved in [6] immediately after the corollary at p. 175. If |F | ≤ t, then |F | = t and two cases are possible. If Φ is contained in some S t−1,1,F ⊆ S n−1,1,F , Burau's proof is still valid as was mentioned in case (ii); so, S(Φ, Ξ I ) is a normal rational curve of order t − 1 = |F | − 1. Otherwise S(Φ, Ξ I ) is an independent (t + 1)-set, hence a normal rational curve of order |F |. a. If {r 0 , r 1 , . . . , r t } is a frame of a hyperplane of PG(t−1, t) then Φ generates a t-dimensional Remark 2.3. By (1) and (2), the map α : Φ → S(Φ, Ξ I ) defined by the condition that X and X α are on a common line in S II n−1,1,F is related to a projectivity between the parametrizing projective lines. Such an α is also called a projectivity. 3 The order of normal rational curves contained in S n−1,1,q Here n ≥ 2 is an integer. The field reduction map F m,n,q from PG(m − 1, q n ) to PG(mn − 1, q) will also be denoted by F. If S is a set of points, in PG(m − 1, q n ), then F(S) is a set of subspaces, whose union, as a set of points will be denoted byF(S).
Proposition 3.1. Let b be a q-subline of PG(1, q n ), and let Θ ∈ b be a point of PG(1, q n ). Let 1, ζ and 1, ζ ′ be homogeneous coordinates of Θ with respect to two reference frames for b q n , each of which consists of three points of b. Then F q (ζ) = F q (ζ ′ ).
Proof. Homogeneous coordinates of a point in both reference frames, say (x 0 , By Proposition 3.1, the degree of a point over a q-subline b in a finite projective space PG(d, q n ), also equals the minimum integer m such that a subgeometry Σ ∼ = PG(d, q m ) exists containing both b and Θ. Proposition 3.2. Any n-subspace of PG(2n − 1, q) containing an (n − 1)-subspace S I ∈ S I n−1,1,q intersects S n−1,1,q in the union of S I and a line in S II n−1,1,q .
in a normal rational curve whose order is min{q, gives two equations in α, β ∈ L: This S(H ∩ S n−1,1,q , Ξ) is obtained by inversion from the line joining the points F (0, θ −1 ) and An important consequence of the above result answers the question of the existence of a Desarguesian spread containing a given regulus R and a subspace disjoint from R.
Corollary 3.4. If a regulus R = S n−1,1,q and an (n − 1)-dimensional subspace U , disjoint from R, in PG(2n − 1, q) are contained in a Desarguesian spread then there is an integer c such that any n-subspace H containing U intersects R in a normal rational curve of order c.
The following remark illustrates that this necessary condition is not always satisfied.

André-Bruck-Bose representation
The André-Bruck-Bose representation of a Desarguesian affine plane of order q n is related to the image of PG(2, q n ), under the field reduction map F, by means of the following straightforward result.
Proposition 4.1. Let D be the Desarguesian spread in PG(3n − 1, q) obtained after applying the field reduction map F to the set of points of PG(2, q n ), l ∞ a line in PG(2, q n ), and K a (2n)subspace of PG(3n − 1, q), containing the spread F(l ∞ ). Take PG(2, q n ) \ l ∞ and K \ F(l ∞ ) q as representatives of AG(2, q n ) and AG(2n, q), respectively. Then the map ϕ : AG(2, q n ) → AG(2n, q) defined by ϕ(X) = F(X) ∩ K for any X ∈ AG(2, q n ) is a bijection, mapping lines of AG(2, q n ) into n-subspaces of AG(2n, q) whose (n − 1)-subspaces at infinity belong to the spread F(l ∞ ).  The results in [4, Theorems 3.3 and 3.5] also characterize the normal rational curves arising from q-sublines in AG(2, q n ).

The notation in
In [5,14,15] for n = 2 and [4, Theorem 3.6 (a)(b)] for any n the André-Bruck-Bose representation of a q-subplane tangent to a line at the infinity is described. Further properties are stated in the following theorem: and a projectivity κ : C 0 → C 1 (in the sense of Remark 2.3), such that ϕ(B \ l ∞ ) is the ruled surface union of all lines XX κ for X ∈ C 0 .
Proof. By Theorem 4.2, C 0 := ϕ(b) is a normal rational curve of order δ in the n-subspace ϕ( b q n \ l ∞ ), and for any P = ϕ(X) ∈ C 0 , the subline T X of B corresponds to an affine line P P κ with P κ ∈ F(T ) at infinity. Define C 1 = {P κ | P ∈ C 0 }.
By the field reduction map F = F 3,n,q , the subplane B is mapped to F(B) which is the set of all maximal subspaces of the first family in S n−1,2,q ⊂ PG(3n − 1, q). The vector homomorphism corresponds to a projective embedding g : PG(n − 1, q) × B → S n−1,2,q whose image is S n−1,2,q , and such that F(X) = (PG(n − 1, q) × X) g for any point X in B. It holds ϕ(B \ l ∞ ) = S n−1,2,q ∩ K \ F(T ). For any point U in B define Note that for any Y ∈ B, the restriction of κ U to F(Y ) is a projectivity. For any U ∈ b, using the notation from Lemma 2.1 it holds C κ U 0 = S(C 0 , F(U )), and as a consequence, C κ U 0 is a normal rational curve of order δ ′ as in (3). Now, since for any P ∈ C 0 , say P = (X P , Y P ) g , the points P , P κ and P κ T are on the plane (X P × B) g ∈ S II n−1,2,q , and P κ , P κ T ∈ F(T ), it follows that P κ = P κ T . It also follows that C 1 = C κ U κ T 0 = S(C 0 , F(U )) κ T , and hence C 1 is a normal rational curve of order δ ′ as in (3). Finally, κ U : C 0 → S(C 0 , F(U )) is a projectivity as defined in Remark 2.3, and hence so is κ.

On the classification of clubs
An F q -club (or simply a club) in PG(1, q n ) is an F q -linear set of rank three, having a point of weight two, called the head of the club. An F q -club has q 2 + 1 points, and the non-head points have weight one. From now on it will be assumed that n > 2. The next proposition is a straightforward consequence of the representation of linear sets as projections of subgeometries [12, Theorem 2].
Proposition 5.1. Let L be an F q -club in PG(1, q n ) ⊂ PG(2, q n ). Then there are a q-subplane Σ of PG(2, q n ), a q-subline b in Σ, and a point Θ ∈ b q n \ b, such that L is the projection of Σ from the center Θ onto the axis PG(1, q n ).
As before the notation F andF is used, where F = F 2,n,q denotes the field reduction map from PG(1, q n ) to PG(2n − 1, q).
Proposition 5.2. Let L be an F q -club of PG(1, q n ) with head Υ. ThenF(L) contains two collections of subspaces, say F 1 and F 2 , satisfying the following properties.
(i) The subspaces in F 1 are (n − 1)-dimensional, are pairwise disjoint, and any subspace in F 1 is disjoint from F(Υ).
(ii) Any subspace in F 2 is a plane and intersects F(Υ) in precisely a line.
(iii) Any point of F(Υ) belongs to exactly q + 1 planes in F 2 .
(iv) If L is not isomorphic to PG(1, q 2 ), and l is any line of PG(2n − 1, q) contained inF(L), then l is contained in F(Υ) or in a subspace in F 1 ∪ F 2 .
Proof. The assumptions imply the existence of Σ and a q-subline b in Σ as in Proposition 5.1. The assertions are a consequence of the fact thatF (Σ) is a Segre variety S n−1,2,q in PG(3n−1, q). Let p 1 : PG(2, q n ) \ Θ → PG(1, q n ) be the projection with center Θ, associated with The collections F 1 and F 2 are defined as follows: The assertion (i) is straightforward, as well as dim(V ) = 2 for any V ∈ F 2 . For any V II ∈ F(Σ) II , the intersection V II ∩ F (b) q is a line, and this with p −1 2 (F(Υ)) = F (b) q \ F(Θ) implies the second assertion in (ii). Next, let P be a point in F(Υ). A plane V = p 2 (V II ) contains P if, and only if, V II intersects the n-subspace F(Θ), P q , that is, V II intersects the normal rational curve S n−1,2,q ∩ F(Θ), P q ; this implies (iii).
Assume that a line l ⊂F(L) exists which is neither contained in F(Υ), nor in a T ∈ F 1 ∪ F 2 . Let Q be a point in l \ F(Υ), and let V ∈ F 2 such that Q ∈ V . It holds L = B(V ). Then B(l) is a q-subline of L. Suppose that a line l ′ in V exists such that B(l ′ ) = B(l). Since B(Q) = B(Q ′ ) for any Q ′ ∈ V , Q ′ = Q, the line l ′ contains Q. Then l, l ′ are two distinct transversal lines in B(l) II , a contradiction. Hence B(l ′ ) = B(l) for any line l ′ in V , that is, B(l) is a so-called irregular subline [8]. By [8,Corollary 13], no irregular subline exists in L, and this contradiction implies (iv ).
Proposition 5.3. Let L be an F q -club with head Υ. Let Θ be the point and b be the subline as defined in Proposition 5.1. Then for any point X in F(Υ), the intersection lines of F(Υ) with any q distinct planes in F 2 containing X span an s-dimensional subspace, where Proof. Let p 2 be the projection map as defined in the proof of Proposition 5.2, X = p 2 (P ), and H = F(Θ), P q . For any plane V = p 2 (V II ), it holds X ∈ V if, and only if V II ∩ H = ∅. Theorem 5.4. Let I n,q be the set of integers h dividing n and such that 1 < h < q. For any h ∈ I n,q , let L h be the linear set obtained by projecting a q-subplane Σ of PG(2, q n ) from a point Θ h collinear with a q-subline b in Σ and such that [Θ h : b] = h. Then the set Λ = {L h | h ∈ I n,q } contains F q -clubs in PG(1, q n ) all belonging to distinct orbits under PGL(2, q n ).
Proof. If n is odd, then no club is isomorphic to PG(1, q 2 ). So, by Proposition 5.2 (iv), the families F 1 and F 2 are uniquely determined. The thesis is a consequence of Proposition 5.3, taking into account that if L and L ′ are projectively equivalent, thenF(L) andF(L ′ ) are projectively equivalent in PG(2n − 1, q).
In order to deal with the case n even, it is enough to show that in Λ at most one club is isomorphic to PG(1, q 2 ). So assume L h ∼ = PG(1, q 2 ). ThenF (L h ) has a partition P 1 in (n − 1)subspaces, and a partition P 2 in 3-subspaces. From [8,Lemma 11] it can be deduced that any line contained inF(L h ) is contained in an element of P 1 or P 2 . The intersections of a subspace U of a family P i with the elements of the other family form a line spread of U . Hence all planes in F 2 are contained in 3-subspaces of P 2 , and all planes of F 2 through a point X in F(Υ) meet F(Υ) in the same line. By Proposition 5.3 this implies h = 2.
A Appendix: On a result in [6] In [6, p.175] the following result (Korollar) is stated for F = C.
Corollary A.1. Let F be an algebraically closed field. If an s-subspace S s of PG(2s − 1, F ) meets all S I ∈ S I s−1,1,F only in points, then such points span S s .
In [6] the previous result is seemingly proved using methods valid in any field with enough elements. However such a generalisation would contradict Theorem 3.3. In the opinion of the authors the proof in [6] is obtained using an erroneous argument. As a matter of fact, it is claimed in the proof at page 174 that the assumption Φ = S s is not used. However the contradiction S s ⊂ S s−2,1,C is inferred from Φ ⊂ S s−2,1,C .
A further counterexample, which exists whenever a hyperbolic quadric Q + (3, F ) in a threedimensional projective space admits an external line (a condition which is not met when the field F is algebraically closed) is the following. If ℓ is the line corresponding to the two-dimensional vector space e 1 ⊗ e ′ 1 , e ′ 2 and m is a line external to the hyperbolic quadric obtained by the intersection of the Segre variety S 2,1,F with the 3-space corresponding to the vector space e 2 , e 3 ⊗ e ′ 1 , e ′ 2 , then the 3-dimensional subspace ℓ, m intersects S 2,1,F in the line ℓ belonging to S II 2,1,F . For the sake of completeness, a proof for corollary A.1 is given.
Next, let C I ∈ S I s−1,1,F be such that π(P ) ∈ C I , and Q the point in S t−1,1,F such that Q, P , and π(P ) are collinear. If Q ∈ S t , then π(P ) ∈ S s , a contradiction; also Q ∈ C I leads to a contradiction (since it implies P ∈ C I ). So Q ∈ S t ∪ C I and by a dimension argument two points Q 1 ∈ C I \ S t and Q 2 ∈ S t \ C I exist such that Q, Q 1 and Q 2 are collinear: they are on the unique line through Q meeting both C I ∩ S t−1,1,F and a (t − 1)subspace of S t disjoint from C I . The plane P, Q 1 , Q 2 contains the lines P Q 2 ⊂ S s and π(P )Q 1 ⊂ S s−1,1,F which meet outside S t−1,1,F . This is again a contradiction.