Subspaces intersecting each element of a regulus in one point, André-Bruck-Bose representation and clubs
aa r X i v : . [ m a t h . C O ] S e p Subspaces intersecting each element of a regulus in one point,Andr´e-Bruck-Bose representation and clubs
Michel Lavrauw ∗ and Corrado Zanella † August 16, 2018
Abstract
In this paper results are proved with applications to the orbits of ( n − R of ( n − n − , q ), with respect tothe subgroup of PGL(2 n, q ) fixing R . Such results have consequences on several aspects offinite geometry. First of all, a necessary condition for an ( n − U and a regulus R of ( n − q -sublinein PG(2 , q n ). Furthermore, the results in this paper are applied to the classification of linearsets, in particular clubs.A.M.S. CLASSIFICATION: 51E20KEY WORDS: club; linear set; subplane; Andr´e-Bruck-Bose representation; Segre variety The ( n − F is denoted by PG( n − , F )or PG( n − , 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 aregulus in one point are studied and a result from [6] is generalised. Section 3 contains one ofthe main results of this paper, determining the order of the normal rational curves obtainedfrom n -dimensional subspaces on an external ( n − n − , q ), obtained from a point and a subline after applying the field reductionmap to PG(1 , q n ). This leads to a necessary condition on the existence of a Desarguesian spreadcontaining a subspace and regulus (Corollary 3.4). The Andr´e-Bruck-Bose representation of ∗ Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit`a di Padova, Stradella S. Nicola 3,36100 Vicenza, Italy, e-mail: [email protected]. The research of this author is supported by the Re-search Foundation Flanders-Belgium (FWO-Vlaanderen) and by a Progetto di Ateneo from Universit`a di Padova(CPDA113797/11). † Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Universit`a di Padova, Stradella S. Nicola 3,36100 Vicenza, Italy, e-mail: [email protected]. The research of this author is supported by the ItalianMinistry of Education, University and Research (PRIN 2012 project “Strutture geometriche, combinatoria e loroapplicazioni”). , q n ) in Section 5. Astudy of the incidence structure of the clubs in PG(1 , q n ) after field reduction yields to a partialclassification, 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 resultmotivated by Burau [6] for the complex numbers: the result is extended to general algebraicallyclosed fields; a new proof is provided; and counterexamples are given to some of the argumentsused in the original proof. Let R be a regulus of subspaces in a projective space and let S be any subspace of hRi . Questionsabout the properties of the set of intersection points, which for reasons of simplicity of notationwe 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 normalrational curve, see Lemma 2.1. This was already pointed out in [6, p.173] with a proof originallyintended for complex projective spaces, but actually holding in a more general setting. Thenotation 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) − , F ) is denoted by S n,m,F . It is well known that S n,m,F contains two families S In,m,F and S IIn,m,F of maximal subspaces of dimensions n and m , respectively. When convenient, thenotation S I or S II will be used for a subspace belonging to the first or second family. The pointsof 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]. Notethat in the finite case it is possible to embed product spaces in projective spaces of smallerdimension (see e.g. [7]). A regulus R of ( n − S In − , ,F . Lemma . Let n > be an integer, and F a field. Let S t be a t -subspace of PG(2 n − , F ) intersecting each S I ∈ S In − , ,F in precisely one point. Define Φ = S t ∩ S n − , ,F , and assume h Φ i = 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 In − , ,F . Then the set S (Φ , Ξ I ) of the intersections of Ξ I with all transversallines l II such that l II ∩ Φ = ∅ is a normal rational curve of order t or t − if | F | = t , andof order t − if | F | > t .(iii) If Φ is contained in a subvariety S t − , ,F of S n − , ,F , then homogeneous coordinates canbe chosen such that Φ is represented parametrically by (cid:28)(cid:18) y t y t − y . . . y y t − y t − y y t − y . . . y t (cid:19)(cid:29) , ( y , y ) ∈ ( F ) ∗ , (1) and S (Φ , Ξ I ) , for z , z depending only on Ξ I , by (cid:28)(cid:18) y t − z y t − y z . . . y t − z y t − z y t − y z . . . y t − z (cid:19)(cid:29) , ( y , y ) ∈ ( F ) ∗ . (2)2 roof. (i), (iii) The proof in [6, Sect.41 no.3], which is offered for F = C , works exactly thesame provided that | F | > t or, more generally, that Φ is contained in some subvariety S t − , ,F of S n − , ,F . In case | F | ≤ t , the size of Φ being | F | + 1 implies | F | = t , so Φ is just a set of t + 1independent points in a subspace isomorphic to PG( t, t ), hence Φ is a normal rational curve oforder 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 − , ,F ⊆ S n − , ,F , Burau’sproof is still valid as was mentioned in case (ii); so, S (Φ , Ξ I ) is a normal rational curve of order t − | F | −
1. Otherwise S (Φ , Ξ I ) is an independent ( t + 1)-set, hence a normal rational curveof order | F | . Remark . If | F | = t both cases in Lemma 2.1 (ii) can occur. The following two examplesuse the Segre embedding σ = σ t − , ,F of the product space PG( t − , t ) × PG(1 , t ) in PG(2 t − , t ).Let { s , s , . . . , s t } be the set of points on PG(1 , t ) and suppose { r , r , . . . , r t } is a set of t + 1points in PG( t − , t ). Put Ξ I = σ (PG(1 , t ) × s ) and Φ := { σ ( r i × s i ) : i = 0 , , . . . , t } . ThenΦ consists of t + 1 points on the Segre variety S t − , ,F . Depending on the set { r , r , . . . , r t } oneobtains the two cases described in Lemma 2.1 (ii) .a. If { r , r , . . . , r t } is a frame of a hyperplane of PG( t − , t ) then Φ generates a t -dimensionalsubspace of PG(2 t − , t ) intersecting S t − , ,F in Φ and S (Φ , Ξ I ) is a normal rational curveof order t − { r , r , . . . , r t } generates PG( t − , t ) then Φ generates a t -dimensional subspace ofPG(2 t − , t ) intersecting S t − , ,F in Φ and S (Φ , Ξ I ) is a normal rational curve of order t . Remark . 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 IIn − , ,F is related to a projectivity between the parametrizingprojective lines. Such an α is also called a projectivity . S n − , ,q Here n ≥ F m,n,q from PG( m − , q n ) to PG( mn − , q )will also be denoted by F . If S is a set of points, in PG( m − , q n ), then F ( S ) is a set ofsubspaces, whose union, as a set of points will be denoted by ˜ F ( S ). The F q h -span of a subset b of PG( d, q n ) is denoted by h b i q h . Proposition . Let b be a q -subline of PG(1 , q n ) , and let Θ b be a point of PG(1 , q n ) . Let , ζ and , ζ ′ be homogeneous coordinates of Θ with respect to two reference frames for h b i 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 , x ) and ( x ′ , x ′ ),are related by an equation of the form ρ ( x ′ x ′ ) T = A ( x x ) T , ρ ∈ F ∗ q n , A ∈ GL(2 , q ). Hence( ρ ρζ ′ ) T = A (1 ζ ) T and this implies ζ ′ ∈ F q ( ζ ). The proof of ζ ∈ F q ( ζ ′ ) is similar.By Proposition 3.1, the degree of a point over a q -subline b in a finite projective space PG( d, q n ),[Θ : b ] = [ F q ( ζ ) : F q ] for Θ ∈ h b i q n \ b , [Θ : b ] = 1 for Θ ∈ b , is well-defined. This [Θ : b ] alsoequals the minimum integer m such that a subgeometry Σ ∼ = PG( d, q m ) exists containing both b and Θ. 3 roposition . Any n -subspace of PG(2 n − , q ) containing an ( n − -subspace S I ∈ S In − , ,q intersects S n − , ,q in the union of S I and a line in S IIn − , ,q . Theorem . Let b be a q -subline of PG(1 , q n ) , and Θ b a point of PG(1 , q n ) . Then in PG(2 n − , q ) any n -subspace H containing F (Θ) intersects the Segre variety S n − , ,q = ˜ F ( b ) ,in a normal rational curve whose order is min { q, [Θ : b ] } .Proof. Set L = F q n , F = F q . Without loss of generality, PG(2 n − , q ) = PG q ( L ), F ( b ) = { L ( x, y ) | ( x, y ) ∈ ( F ) ∗ } , and Θ = L (1 , ξ ) with [ F ( ξ ) : F ] = [Θ : b ]. The n -subspace H intersects L (1 ,
0) in one point Y of the form Y = F ( θ, θ ∈ L ∗ . For any x ∈ F , seeking forthe intersection hF (Θ) , Y i q ∩ L ( x, h L (1 , ξ ) , F ( θ, i q ∩ L ( x, α, β ∈ L : α + θ = βx, αξ = β, whence β = θ ( x − ξ − ) − . The intersection point is then F (cid:0) xθ ( x − ξ − ) − , θ ( x − ξ − ) − (cid:1) . So,for Ξ = L (0 , S IIn − , ,q which meet H is S ( H ∩ S n − , ,q , Ξ) = { F (0 , θ ( x − ξ − ) − ) | x ∈ F q } ∪ { F (0 , θ ) } . This S ( H ∩ S n − , ,q , Ξ) is obtained by inversion from the line joining the points F (0 , θ − ) and F (0 , θ − ξ − ). By [10, Theorem 5], C Y is a normal rational curve of order δ ′ = min { q, [ F ( ξ − ) : F ] − } = min { q, [Θ : b ] − } . Now apply lemma 2.1 for S t = hH ∩ S n − , ,q i q :if t ≥ q , then t = q and δ ′ = q or δ ′ = q −
1, so [Θ : b ] ≥ q and t = min { q, [Θ : b ] } . If on the contrary t < q , then t − δ ′ = [Θ : b ] −
1, so t = [Θ : b ] and t = min { q, [Θ : b ] } again.An important consequence of the above result answers the question of the existence of a Desar-guesian spread containing a given regulus R and a subspace disjoint from R . Corollary . If a regulus R = S n − , ,q and an ( n − -dimensional subspace U , disjoint from R , in PG(2 n − , q ) are contained in a Desarguesian spread then there is an integer c such thatany 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.
Remark . For n > by using the package FinInG [2] of GAP [3] examples can be given of ( n − -subspaces disjoint from S n − , ,q contained in n -subspaces intersecting the Segre varietyin normal rational curves of distinct orders. We include one explicit example. Let q = 4 , F q = F ( ω ) , with ω + ω + 1 = 0 . Let R be the regulus of -dimensional subspaces of PG(7 , obtained from the standard subline PG(1 , q ) in PG(1 , q ) , and put S := h (1 , , , , ω , , , , (0 , , , , , ω , , ω ) , (0 , , , , , ω, , ω ) , (0 , , , , ω , ω , ω, i . Then S is a three-dimensional subspace disjoint from the regulus R . Moreover, the 4-dimensionalsubspace h S , (1 , , , , , , , i intersects the regulus R in a normal rational curve of degree4, while the 4-dimensional subspace h S , (0 , , , ω , , , , i intersects R in a conic. For x, y ∈ L , F ( x, y ) = h ( x, y ) i q , and L ( x, y ) = h ( x, y ) i q n . Andr´e-Bruck-Bose representation
The Andr´e-Bruck-Bose representation of a Desarguesian affine plane of order q n is related to theimage of PG(2 , q n ), under the field reduction map F , by means of the following straightforwardresult. Proposition . Let D be the Desarguesian spread in PG(3 n − , q ) obtained after applyingthe 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 (2 n ) -subspace of PG(3 n − , q ) , containing the spread F ( l ∞ ) . Take PG(2 , q n ) \ l ∞ and K \ hF ( l ∞ ) i q as representatives of AG(2 , q n ) and AG(2 n, q ) , respectively. Then the map ϕ : AG(2 , q n ) → AG(2 n, 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(2 n, q ) whose ( n − -subspaces at infinity belong to the spread F ( l ∞ ) . The notation in Proposition 4.1 is assumed to hold in the whole section. The following resultimproves [4, Theorems 3.3 and 3.5], by determining the order of the involved normal rationalcurves.
Theorem . Let b be a q -subline of PG(2 , q n ) , not contained in l ∞ . Set Θ = h b i q n ∩ l ∞ . Thenthe Andr´e-Bruck-Bose representation ϕ ( b \ l ∞ ) is the affine part of a normal rational curve whoseorder is δ = min { q, [Θ : b ] } . More precisely, if δ = 1 , then ϕ ( b \ l ∞ ) is an affine line; if δ > ,then b ∩ l ∞ = ∅ , and ϕ ( b ) is a normal rational curve with no points at infinity.Proof. The intersection H = hF ( b ) i q ∩ K is an n -space containing F (Θ), and contained in thespan of the Segre variety S n − , ,q = ˜ F ( b ). The result follows from Proposition 3.2 and Theorem3.3.The results in [4, Theorems 3.3 and 3.5] also characterize the normal rational curves arisingfrom q -sublines in AG(2 , q n ).In [5, 14, 15] for n = 2 and [4, Theorem 3.6 (a)(b)] for any n the Andr´e-Bruck-Bose representationof a q -subplane tangent to a line at the infinity is described. Further properties are stated inthe following theorem: Theorem . Let B be a q -subplane of PG(2 , q n ) that is tangent to l ∞ at the point T . Let b be a line of B not through T , Θ = h b i q n ∩ l ∞ , and δ = min { q, [Θ : b ] } . Then there are a normalrational curve C of order δ in the n -subspace ϕ ( h b i q n ) , a normal rational curve C ⊂ F ( T ) oforder δ ′ , with δ ′ (cid:26) = [Θ : b ] − for q > [Θ : b ] ∈ { q − , q } otherwise, (3) and a projectivity κ : C → C (in the sense of Remark 2.3), such that ϕ ( B \ l ∞ ) is the ruledsurface union of all lines XX κ for X ∈ C .Proof. By Theorem 4.2, C := ϕ ( b ) is a normal rational curve of order δ in the n -subspace ϕ ( h b i q n \ l ∞ ), and for any P = ϕ ( X ) ∈ C , the subline T X of B corresponds to an affine line P P κ with P κ ∈ F ( T ) at infinity. Define C = { P κ | P ∈ C } .By the field reduction map F = F ,n,q , the subplane B is mapped to F ( B ) which is the set of allmaximal subspaces of the first family in S n − , ,q ⊂ PG(3 n − , q ). The vector homomorphism( λ, v ) ∈ F q n × F q λ ⊗ F q v g : PG( n − , q ) × B → S n − , ,q whose image is S n − , ,q ,and such that F ( X ) = (PG( n − , q ) × X ) g for any point X in B . It holds ϕ ( B \ l ∞ ) = S n − , ,q ∩ K \ F ( T ). For any point U in B define κ U : ( X, Y ) g ∈ S n − , ,q ( X, U ) g ∈ F ( U ) . Note that for any Y ∈ B , the restriction of κ U to F ( Y ) is a projectivity. For any U ∈ b , usingthe notation from Lemma 2.1 it holds C κ U = S ( C , F ( U )), and as a consequence, C κ U is a normalrational curve of order δ ′ as in (3). Now, since for any P ∈ C , 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 IIn − , ,q , and P κ , P κ T ∈ F ( T ), it follows that P κ = P κ T . It also follows that C = C κ U κ T = S ( C , F ( U )) κ T , and hence C is a normal rationalcurve of order δ ′ as in (3). Finally, κ U : C → S ( C , F ( U )) is a projectivity as defined in Remark2.3, and hence so is κ . An F q -club (or simply a club) in PG(1 , q n ) is an F q -linear set of rank three, having a pointof weight two, called the head of the club. An F q -club has q + 1 points, and the non-headpoints have weight one. From now on it will be assumed that n >
2. The next proposition is astraightforward consequence of the representation of linear sets as projections of subgeometries[12, Theorem 2].
Proposition . 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 ∈ h b i 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 and ˜ F is used, where F = F ,n,q denotes the field reduction map fromPG(1 , q n ) to PG(2 n − , q ). Proposition . Let L be an F q -club of PG(1 , q n ) with head Υ . Then ˜ F ( L ) contains twocollections of subspaces, say F and F , satisfying the following properties.(i) The subspaces in F are ( n − -dimensional, are pairwise disjoint, and any subspace in F is disjoint from F (Υ) .(ii) Any subspace in F 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 .(iv) If L is not isomorphic to PG(1 , q ) , and l is any line of PG(2 n − , q ) contained in ˜ F ( L ) ,then l is contained in F (Υ) or in a subspace in F ∪ F .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 that ˜ F (Σ) is a Segre variety S n − , ,q in PG(3 n − , q ).Let p : PG(2 , q n ) \ Θ → PG(1 , q n )be the projection with center Θ, associated with p : PG(3 n − , q ) \ F (Θ) → PG(2 n − , q ) . F and F are defined as follows: F = {F ( p ( X )) | X ∈ Σ \ b } = F ( L ) \ F (Υ) , F = { p ( V II ) | V II ∈ ˜ F (Σ) II } . The assertion (i) is straightforward, as well as dim( V ) = 2 for any V ∈ F . For any V II ∈ ˜ F (Σ) II , the intersection V II ∩ h ˜ F ( b ) i q is a line, and this with p − ( F (Υ)) = h ˜ F ( b ) i q \ F (Θ)implies the second assertion in (ii) . Next, let P be a point in F (Υ). A plane V = p ( V II )contains P if, and only if, V II intersects the n -subspace hF (Θ) , P i q , that is, V II intersects thenormal rational curve S n − , ,q ∩ hF (Θ) , P i q ; this implies (iii) .Assume that a line l ⊂ ˜ F ( L ) exists which is neither contained in F (Υ), nor in a T ∈ F ∪ F .Let Q be a point in l \ F (Υ), and let V ∈ F such that Q ∈ V . It holds L = B ( V ). Then B ( l ) isa 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 linesin 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 contradictionimplies (iv ). Proposition . Let L be an F q -club with head Υ . Let Θ be the point and b be the subline asdefined in Proposition 5.1. Then for any point X in F (Υ) , the intersection lines of F (Υ) withany q distinct planes in F containing X span an s -dimensional subspace, where(i) s = [Θ : b ] − if q > [Θ : b ] ;(ii) s ∈ { q − , q } if q ≤ [Θ : b ] .Proof. Let p be the projection map as defined in the proof of Proposition 5.2, X = p ( P ), and H = hF (Θ) , P i q . For any plane V = p ( V II ), it holds X ∈ V if, and only if V II ∩ H 6 = ∅ . Theintersection H ∩ ˜ F ( b ) is a normal rational curve of order min { q, [Θ : b ] } (cf. Theorem 3.3). Let V = p ( V II ) be the unique plane of F through X distinct from the q planes chosen in theassumptions (cf. Proposition 5.2). Let Q = ˜ F ( b ) ∩ V II ; B ( Q ) is an ( n − F ( b ) I .Such B ( Q ) is mapped onto B ( X ) = F (Υ) by p . Assume V i = p ( V IIi ), i = 1 , , . . . , q , are the q planes chosen in the assumptions. Any V IIi , i = 1 , , . . . , q , intersects H , hence V IIi ∩ B ( Q ) isthe intersection of B ( Q ) with a transversal line of ˜ F ( b ) intersecting the normal rational curve H ∩ ˜ F ( b ). By Lemma 2.1 (ii) , the set S = { V IIi ∩ B ( Q ) | i = 1 , , . . . , q } ∪ { Q } is a normal rational curve of order s where s takes the values as stated in ( i ) and ( ii ). Since V i ∩F (Υ) is the line through X and a point of p ( S ), distinct from X , the span of the intersectionlines is the same as the span of p ( S ). Theorem . Let I n,q be the set of integers h dividing n and such that < 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 ). So, by Proposition 5.2 (iv) , thefamilies F and F are uniquely determined. The thesis is a consequence of Proposition 5.3,taking into account that if L and L ′ are projectively equivalent, then ˜ F ( L ) and ˜ F ( L ′ ) areprojectively equivalent in PG(2 n − , q ). 7n order to deal with the case n even, it is enough to show that in Λ at most one club isisomorphic to PG(1 , q ). So assume L h ∼ = PG(1 , q ). Then ˜ F ( L h ) has a partition P in ( n − P in 3-subspaces. From [8, Lemma 11] it can be deduced that anyline contained in ˜ F ( L h ) is contained in an element of P or P . 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 planesin F are contained in 3-subspaces of P , and all planes of F through a point X in F (Υ) meet F (Υ) in the same line. By Proposition 5.3 this implies h = 2. Acknowledgement.
The authors thank Hans Havlicek for his helpful remarks in the prepara-tion of this paper.
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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(2 s − , F ) meets all S I ∈ S Is − , ,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 enoughelements. However such a generalisation would contradict Theorem 3.3. In the opinion of theauthors the proof in [6] is obtained using an erroneous argument. As a matter of fact, it isclaimed in the proof at page 174 that the assumption h Φ i = S s is not used. However thecontradiction S s ⊂ hS s − , , C i is inferred from Φ ⊂ S s − , , C .A further counterexample, which exists whenever a hyperbolic quadric Q + (3 , F ) in a three-dimensional 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-dimensionalvector space h e i ⊗ h e ′ , e ′ i and m is a line external to the hyperbolic quadric obtained bythe intersection of the Segre variety S , ,F with the 3-space corresponding to the vector space h e , e i ⊗ h e ′ , e ′ i , then the 3-dimensional subspace h ℓ, m i intersects S , ,F in the line ℓ belongingto S II , ,F .For the sake of completeness, a proof for corollary A.1 is given. Proof of corollary A.1.
Define S t = h S s ∩ S s − , ,F i , t = dim S t (4)and suppose t < s . It is proved in [6, p.173 (6)] that S t ⊂ hS t − , ,F i for some S t − , ,F ⊂ S s − , ,F .Note that S s ∩ hS t − , ,F i = S t ; otherwise, comparing dimensions, S s would intersect each S I ∈S t − , ,F in more than one point. Now choose • a subspace S s − t − ⊂ S s such that S s − t − ∩ hS t − , ,F i = ∅ ; • a Segre variety S s − t − , ,F ⊂ S s − , ,F , such that hS s − t − , ,F i ∩ hS t − , ,F i = ∅ ; • two distinct A I , B I ∈ S Is − t − , ,F .Since hS s − t − , ,F i and hS t − , ,F i are complementary subspaces of hS s − , ,F i , a projection map π : hS s − , ,F i \ hS t − , ,F i → hS s − t − , ,F i is defined by π ( P ) = h P ∪ S t − , ,F i ∩ hS s − t − , ,F i .Now suppose π ( S s − t − ) ∩ S s − t − , ,F = ∅ . In hS s − t − , ,F i consider9 the regulus R corresponding to S Is − t − , ,F , and the projectivity κ : A I → B I such that,for any P ∈ A I , the line h P, κ ( P ) i belongs to S IIs − t − , ,F ; • the regulus R ′ containing A I , B I and π ( S s − t − ), and the projectivity κ ′ : A I → B I suchthat, for any P ∈ A I , the line h P, κ ′ ( P ) i is a transversal line of R ′ .Since F is an algebraically closed field, κ ′− ◦ κ has a fixed point P . Therefore κ ( P ) = κ ′ ( P ), so R and R ′ have a common transversal. This contradicts π ( S s − t − ) ∩ S s − t − , ,F = ∅ . So, a point P ∈ S s − t − exists such that π ( P ) ∈ S s − t − , ,F .Next, let C I ∈ S Is − , ,F be such that π ( P ) ∈ C I , and Q the point in hS t − , ,F i 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 toa contradiction (since it implies P ∈ C I ). So Q S t ∪ C I and by a dimension argument twopoints Q ∈ C I \ S t and Q ∈ S t \ C I exist such that Q , Q and Q are collinear: they are onthe unique line through Q meeting both C I ∩ hS t − , ,F i and a ( t − S t disjoint from C I .The plane h P, Q , Q i contains the lines P Q ⊂ S s and π ( P ) Q ⊂ S s − , ,F which meet outside hS t − , ,F ii