In this note we study subplanes of order <i>q</i> of the projective plane <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>Π</mi><mo>=</mo><mi>P</mi><mi>G</mi><mrow><mo>(</mo> <mrow> <mn>2,</mn><msup> <mi>q</mi> <mn>3</mn> </msup> </mrow> <mo>)</mo></mrow></mrow> </math> and the ruled varieties <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msubsup> <mi>V</mi> <mn>2</mn> <mn>5</mn> </msubsup> </mrow> </math> of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <mi>Σ</mi><mo>=</mo><mi>P</mi><mi>G</mi><mrow><mo>(</mo> <mrow> <mn>6,</mn><mi>q</mi></mrow> <mo>)</mo></mrow></mrow> </math> using the spatial representation of Π in Σ, by fixing a hyperplane <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <msup> <mi>Σ</mi> <mo>′</mo> </msup> </math> with a regular spread of planes. First are shown some configurations of the affine <i>q</i>-subplanes. Then to prove that a variety <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msubsup> <mi>V</mi> <mn>2</mn> <mn>5</mn> </msubsup> </mrow> </math> of Σ represents a non-affine subplane of order <i>q</i> of Π, after having shown basic incidence properties of it, such a variety <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msubsup> <mi>V</mi> <mn>2</mn> <mn>5</mn> </msubsup> </mrow> </math> is constructed by choosing appropriately the two directrix curves in two complementary subspaces of Σ. The result can be translated into further incidence properties of the affine points of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msubsup> <mi>V</mi> <mn>2</mn> <mn>5</mn> </msubsup> </mrow> </math> . Then a maximal bundle of varieties <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msubsup> <mi>V</mi> <mn>2</mn> <mn>5</mn> </msubsup> </mrow> </math> having in common one directrix cubic curve is constructed.
It is known that a projective translation plane Π of order q r and kernel F = G F ( q ) can be represented by a 2r-dimensional projective space Σ = P G ( 2 r , q ) over F, fixing a hyperplane Σ ′ = P G ( 2 r − 1, q ) and a spread (partition) S of Σ ′ with ( r − 1 ) -dimensional subspaces (cf. [
The points of Π are represented by 1) the points of Σ \ Σ ′ (the affine points) and by 2) the elements of S (the points at infinity). The lines of Π are represented by 1) the r-dimensional subspaces S of Σ \ Σ ′ such that S ∩ Σ ′ belongs to S and by 2) the spread S . The translation line l ∞ of Π (the line at infinity) is represented by S (cf. Lemma 2.6).
If a subplane of Π meets l ∞ in a subline, then such a subplane is affine, if it meets l ∞ in one point is non-affine.
An affine subplane A of order q is represented by every transversal plane α to S , that is, by a plane α ⊂ Σ \ Σ ′ such that the line t = α ∩ Σ ′ meets q + 1 elements of S , t is a transversal line to S . In such a way l ∞ is a line of the projective completion of A . Of course all that holds also in case Π is the Desarguesian plane P G ( 2, q r ) when S is a regular spread (cf. [
Fix r = 3 so that Π = P G ( 2, q 3 ) , Σ = P G ( 6, q ) , Σ ′ = P G ( 5, q ) and S is a regular spread of planes.
About the affine subplanes of Π = P G ( 2, q 3 ) of order q (having the same subline at infinity) we prove there exist q 2 + q + 1 through one fixed affine point, while q 4 partition the affine points of Π (cf. Proposition 2.11, Theorem 2.12).
A variety V 2 5 of Σ is a ruled variety of P G ( 6, q ) with the minimum order directrix a conic and a maximum order directrix a skew cubic in a 3-dimensional subspace, the two curves lying in two complementary spaces (cf. [
We choose a conic in a plane π ∞ ∈ S and a cubic in a 3-dimensional subspace S 0 ∈ Σ \ Σ ′ with π 0 = S 0 ∩ Σ ′ ∈ S and π 0 ≠ π ∞ . Some fundamental incidence properties of V 2 5 are shown (cf. Paragraph 3.1). Then, if q ≡ 1 , ( mod 3 ) , we can prove that the variety V 2 5 represents a non-affine subplane Πq of order q of P G ( 2, q 3 ) (cf. Theorem 3.6).
The properties of Πq of being a plane, translate into further incidence properties of the affine points of V 2 5 (cf. Theorem 3.8).
By fixing the 3-subspace S 0 with the chosen directrix cubic curve C 0 , a maximal bundle of varieties V 2 5 having in common C 0 is constructed (cf. Theorem 3.9).
At the end is formulated the conjecture that a ruled variety V 2 2 r − 1 of P G ( 2 r , q ) represents a non affine subplane of order q of P G ( 2, q r ) , via the spatial representation.
Let F = G F ( q ) be a finite field, q = p s , p an odd prime. Denote F r + 1 the ( r + 1 ) -dimensional vector space over F, P G ( r , q ) = P r F r + 1 the r-dimensional projective space contraction of F r + 1 over F. Let F ¯ be the algebraic closure of the field F = G F ( q ) .
The geometry P G ( r , q ) is considered a sub-geometry of P G ( r , q ) ¯ , the projective geometry over F ¯ . We refer to the points of P G ( r , q ) as the rational points of P G ( r , q ) ¯ .
Denote S h or h-space with 1 ≤ h ≤ r − 1 a subspace of P G ( r , q ) of dimension h. A hyperplane S r − 1 will be denoted also by H, a plane by π. If A , B , C , ⋯ are subspaces denote A + B + C + ⋯ = 〈 A , B , C , ⋯ 〉 , the subspace generated by them. More simply, when A , B are points, AB denote the line defined by them.
Definition 2.1 A variety V u v of dimension u and of order v of P G ( r , q ) is the set of the rational points of a projective variety V ¯ u v of P G ( r , q ) ¯ defined by a finite set of polynomials with coefficients in the field F.
From [
Lemma 2.2 The ruled variety V 2 r − 1 of P G ( r , q ) is generated by the lines joining the corresponding points of two birationally (projectively) equivalent curves of order m and r − 1 − m , respectively, lying in two complementary subspaces of the same dimensions. As the directrix curves have no point in common, then the number of points of V 2 r − 1 is ( q + 1 ) 2 and the order is the sum of the orders of the curves.
Let Σ ′ be the projective space P G ( 2 r − 1, q ) over the field F = G F ( q ) , r > 1 an integer.
Definition 2.3 A spread of Σ ′ is a partition S with ( r − 1 ) -dimensional subspaces (that is, every point of Σ ′ lies in one element of S ). A regulus R of S is a collection of subspaces of S such that:
1) R contains at least 3 elements,
2) Every line meeting 3 elements of R , a transversal line, meets every element of R ,
3) Every point of a transversal line to R lies in one element of R ,
4) Every plane through a transversal line is a transversal plane to S .
Any three pairwise disjoint ( r − 1 ) -dimensional subspaces of S lie in a unique regulus. Any two distinct transversal lines are skew.
A spread is regular if for any three distinct elements of S , all the members of the unique regulus determined by them are in S .
Regular spreads represent Desarguesian planes P G ( 2, q r ) (cf. [
The construction of a regular spread can be described as follows.
Choose a coordinate system in Σ ′ so that for a point P of Σ ′ , P ≈ ( x , y ) = ( x 1 , x 2 , ⋯ , x r ; y 1 , y 2 , ⋯ , y r ) = F * ( x , y ) , F * = F \ { 0 } .
A regular spread of Σ ′ is given by the set { J ∞ = ( 0 , y ) | y ∈ F r } ∪ { J m = ( x , x m ) | x , m ∈ F r } where y = x m is the multiplication in the field F r . Such a multiplication can be represented also by y = x M with M a r × r matrix over F so that x M = x m . The set M = { M | x M = x m } is a field isomorphic to ( F r ) 2 , acting strictly transitively over F r .
In case of a projective plane over a skew-field, a spread can be constructed in the same way. The set of matrices is not a field, anyway it operates strictly transitively over F r (cf. [
Let Σ be the projective geometry P G ( 6, q ) .
Denote π ∞ and S0 a plane and a 3-space, respectively, of Σ. Assume π ∞ and S0 are complementary. Let C 2 be an irreducible conic of π ∞ , C 3 a skew cubic curve of S0. The curves C 2 and C 3 are projectively equivalent so that their points can be connected by a projectivity.
Lemma 2.4 A variety V 2 5 of Σ is obtained by joining the corresponding points of C 2 and of C 3 .
Proof. See [
Corollary 2.5 The variety V 2 5 consists of ( q + 1 ) 2 points of the q + 1 generatrix lines including the points of the minimum order directrix C 2 and of the maximum order directrix C 3 .
Note that the set of the q + 1 generatrix lines partition the variety.
Choose a hyperplane Σ ′ = P G ( 5, q ) of Σ as the hyperplane at infinity.
Fix a coordinate system in Σ so that it is a coordinate system also for Σ ¯ . Denote a point P ≈ ( x , y , t ) = ( x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , t ) : = F ¯ * ( x , y , t ) , F ¯ * = F ¯ \ { 0 } . Let t = 0 be the equation of Σ ′ .
P is a rational point if there exists ( x , y , t ) ∈ F 7 such that P ≈ ( x , y , t ) .
A variety V of Σ is the set of the rational points of Σ ¯ solutions of a finite set of polynomials of F [ x , y , t ] .
Let Π = P G ( 2, q 3 ) be the Desarguesian plane over the field G F ( q 3 ) . Denote l ∞ the line at infinity of Π. Represent Π in Σ = P G ( 6, q ) by a regular spread S of planes of Σ ′ , with | S | = q 3 + 1 .
More precisely define the following incidence structure Π ′ = ( P , L , I ) (points, lines, incidence, respectively) where
P = { P ∈ Σ \ Σ ′ } ∪ { π ∈ S } ,
L = { L 0 = { S 3 ⊂ Σ \ Σ ′ | S 3 ∩ Σ ′ ∈ S } } ∪ { l ∞ = S } ,
I is defined as follows
if P ∈ Σ \ Σ ′ , l ∈ L 0 then P I l ⇔ P ∈ l , no point of Σ \ Σ ′ incides l ∞ , π I l ∞ for all π ∈ S , π I l where l ∈ L 0 ⇔ l ∩ Σ ′ = π .
From [
Lemma 2.6 Π ′ ≅ Π .
In short, the affine points of Π are represented by the q 6 affine points of Σ \ Σ ′ , the points at infinity by the q 3 + 1 planes of S . The affine lines of Π are represented by the 3-spaces S of Σ \ Σ ′ such that the plane S ∩ Σ ′ belongs to S , the line at infinity l ∞ by the spread S .
Definition 2.7 A subplane π ′ = ( P ′ , L ′ , I ′ ) of a plane π = ( P , L , I ) is a subgeometry of π, that is, an incidence substructure for which P ′ ⊂ P , for each line l ′ ⊂ L ′ , there exists a line l ∈ L such that l ′ ⊂ L and I ′ = I .
Definition 2.8 A subplane of Π = P G ( 2, q 3 ) of order q is affine if it meets the line l ∞ of Π in a subline consisting of q + 1 points, it is non-affine if it meets the line l ∞ in one point.
Let r be any transversal line to S . As S is regular, the q + 1 planes that r meets form a regulus R ⊂ S (cf. [
It is easy to prove the following.
Proposition 2.9 The plane α is isomorphic to a subplane π ≅ P G ( 2, q ) of Π whose points at infinity are represented by the q + 1 planes of R , the lines of π being represented by the sublines intersections of α with the 3-spaces of Σ through the planes of R . As the line at infinity of π is a subline of the infinity line of Π, then π is an affine subplane.
To construct transversal lines to R in Σ ′ in a synthetic way the procedure is similar to the one used for dimension 3.
Proposition 2.10 The transversal lines to R are q 2 + q + 1 .
Proof. Denote π 1 , π 2 , π 3 three planes of the regulus R . Fix a point P ∈ π 1 and denote S = 〈 P , π 2 〉 the 3-space of Σ ′ direct sum of P and π 2 and S ′ = 〈 P , π 3 〉 the 3-space of Σ ′ direct sum of P and π 3 . Lying in a 5-dimensional subspace, then S ∩ S ′ = r is a line. As a line of S, r meets π 2 in a point, as a line of S ′ , r meets π 3 in a point. Therefore r is a transversal line to the planes π 1 , π 2 , π 3 . As π 1 , π 2 , π 3 belong to the regulus R , the line r meets each of the q + 1 elements of R . In such a way one can construct a transversal line for every point P chosen in π 1 , that is, q 2 + q + 1 .
Proposition 2.11 The cardinality of the set of all affine subplanes of Π isomorphic to P G ( 2, q ) having the same subline of q + 1 points at infinity and containing one affine point is q 2 + q + 1 .
Proof. Let r 0 be a transversal line to R , α 0 ⊃ r 0 a transversal plane, O an affine point of α 0 . Denote { r i | i = 0 , ⋯ , q 2 + q } the transversal lines to the regulus R . Each of the q 2 + q + 1 planes α i = 〈 O , r i 〉 represents an affine subplane π i of Π, π i ≅ P G ( 2, q ) (cf. Proposition 2.9).
Choose and fix a transversal line r. Consider the bundle (r) of the planes of Σ \ Σ ′ having the line r as axis. Each plane α ∈ ( r ) is isomorphic to P G ( 2, q ) (cf. Proposition 2.9) and it is an affine subplane of Π having the same subline of q + 1 points at infinity.
Theorem 2.12 The planes of (r) partition the q 6 affine points of Π.
Proof. The planes of (r) through the transversal line r are parallel, therefore they have no affine point in common otherwise they would coincide. Each such a plane contains q 2 affine points.
To prove the statement it is appropriate to calculate the number of the planes of Σ \ Σ ′ of the bundle through the line r.
It is known that the number of k-subspaces in a n-space (vector notation) is
[ n | k ] = ( q n − 1 ) ( q n − q ) ⋯ ( q n − q k − 1 ) ( q k − 1 ) ( q k − q ) ⋯ ( q k − q k − 1 ) .
The number h of the planes through a line in P G ( 6, q ) \ P G ( 5, q ) equals the number of 3-spaces in a 4-space, minus the number of the planes through a line in P G ( 5, q ) , which equals the number of the planes in a 3-space.
Therefore, after simplification of the two ratios, follows
h = q 5 − 1 q − 1 − q 4 − 1 q − 1 = q 4 .
More simply, as a line and an independent point define a plane, fixed the line r, there are q 6 choices for a point in Σ \ Σ ′ to get the plane 〈 r , P 〉 ⊂ Σ \ Σ ′ , this number to be divided by q 2 , which equals the choices of an affine point on a same plane, hence again h = q 4 .
As each plane of (r) contains q 2 points of Σ \ Σ ′ and | ( r ) | = q 4 , then the total number of points of Σ \ Σ ′ covered by them is q 6 , which are all the affine points of Π.
Denote Σ = P G ( 6 , q ) , Σ ′ = P G ( 5, q ) a hyperplane of Σ, S a regular spread of Σ ′ . Choose and fix a plane π ∞ ∈ S and a 3-space S 0 in Σ \ Σ ′ such that S 0 ∩ Σ ′ = π 0 ∈ S \ π ∞ . Then choose and fix:
1) A non-degenerate conic C 2 ⊂ π ∞ ,
2) A skew cubic curve C 3 in S 0 such that C 3 ∩ π 0 = ∅ (cf. [
Let Λ : C 2 → C 3 be a projectivity. Represent C 2 = { G i ∞ | i = 1 , ⋯ , q + 1 } , C 3 = { G i = Λ G i ∞ | i = 1 , ⋯ , q + 1 } . Denote V 2 5 the variety arising by connecting corresponding points of C 2 and C 3 via Λ (cf. [
Let H be any hyperplane. In a suitable complexification of Σ, H ∩ V 2 5 is a curve of order 5 (cf. [
Proposition 3.1 The variety V 2 5 consists of q + 1 skew affine generatrix lines and of q 2 + q affine points.
1) A directrix curve C ≠ C 2 cut by a hyperplane on V 2 5 cannot lie in a plane. The conic C 2 is the unique minimum order 2 directrix.
2) The 3-space joining two generatrix lines cannot contain the plane π ∞ .
3) The 3-space joining one generatrix line and the plane π ∞ meets no other generatrix.
4) Three generatrices g 1 , g 2 , g 3 are joint by a hyperplane H that contains the plane π ∞ , so that H ∩ V 2 5 = { g 1 , g 2 , g 3 } ∪ C 2 .
5) A hyperplane contains neither a fixed directrix, nor a fixed generatrix.
Proof. Let g , g ′ be two generatrix lines. Denote G ∞ = g ∩ π ∞ , G ′ ∞ = g ′ ∩ π ∞ , with G ∞ , G ′ ∞ ∈ C 2 , G = g ∩ C 3 , G ′ = g ′ ∩ C 3 , G , G ′ ∈ S 0 . Let r be the line G ∞ G ′ ∞ , r ⊂ π ∞ , r ′ the line G G ′ , r ′ ⊂ S 0 . Assume g , g ′ meet in a point. Hence they define a plane τ = 〈 r , r ′ 〉 so that r ∩ r ′ = P with P ∈ π ∞ ∩ S 0 , a contradiction.
As C 3 has no points in π 0 , then all the q + 1 generatrices are affine and the points of them are affine except the points of C 2 , therefore the affine points of V 2 5 are q ( q + 1 ) = q 2 + q .
1) Assume a hyperplane H meets V 2 5 in a directrix curve C ≠ C 2 lying in a plane τ . Then V 2 5 is contained at most in the 5-space generated by τ and π ∞ , a contradiction. The conic C 2 is the unique minimum order 2 directrix, otherwise the variety generated by the two conics would have order at most 4.
2) Assume there exists a 3-dimensional subspace S containing π ∞ and two generatrix lines, g 1 , g 2 . Denote G i = g i ∩ C 3 , i = 1,2 and G 1 G 2 ∩ Σ ′ = G . The line G 1 G 2 belongs to S 0 and to S, so that the point G is a common point of π ∞ and π 0 , a contradiction.
3) Let S = 〈 g , π ∞ 〉 , with g ∈ G , be a 3-space. If S ∩ g ′ ≠ ∅ with g ′ ≠ g , g ′ ∈ G , then g ′ ⊂ S , so that S contains two generatrix lines and the plane π ∞ , a contradiction to (2).
4) Assume three generatrices g 1 , g 2 , g 3 are joint by a 4-space S ′ . As S ′ contains the three points G 1 ∞ , G 2 ∞ , G 3 ∞ ∈ C 2 , G i ∞ = g i ∩ C 2 , i = 1,2,3 , then π ∞ ⊂ S ′ and C 2 ⊂ S ′ . As S ′ cannot contain V 2 5 , a hyperplane H ⊃ S ′ and through a further point P ∈ V 2 5 \ S ′ should contain also the generatrix g P through P. Hence H would meet V 2 5 in 4 generatrix lines and a conic, that is, in a variety of order 6, a contradiction (cf. [
5) Let g , g ′ two generatrices of V 2 5 . Denote S the 3-space containing them. Let H be a generic hyperplane with H ⊃ S and assume H contains a fix directrix C . Let P be a point of V 2 5 , P ∈ H \ C . Denote S 4 = 〈 S , P 〉 . Then every hyperplane containing S 4 and S 4 itself, would contain the generatrix g P through P, so that g , g ′ , g P ⊂ S 4 , a contradiction to (4). An analogous contradiction is reached if we assume a generic hyperplane containing g , g ′ contained a fix generatrix (cf. [
The following propositions are a rereading in the current case of [
Proposition 3.2 A hyperplane H containing two generatrices, contains a residual cubic curve C lying in a 3-space S ⊂ H , S skew to π ∞ . C is irreducible and is a directrix.
Proof. In [
Set m = 3 . If a hyperplane contains 5 − m = 2 generatrix lines, then it meets V 2 5 in a residual cubic curve C and viceversa.
Assume the cubic C exists in a plane π. Let H be a hyperplane containing π and three further points P , Q , R ∈ V 2 5 \ π . Then H contains also the 3 generatrix lines g P , g Q , g R as all the generatrix meet C . In such a way H ∩ V 2 5 ⊃ { C , g P , g Q , g R } , that is a curve of order 6, a contradiction. Hence C lies in a 3-space S.
If such a space S met π ∞ , then a hyperplane H ⊇ 〈 S , π ∞ 〉 would contain V 2 5 , a contradiction, therefore S ∩ π ∞ = ∅ .
Assume a cubic curve C is reducible. Of course, the unique possibility is that C consists of at most 3 generatrix lines. In such a case C would meet the conic C 2 and then the plane π ∞ , a contradiction with S ∩ π ∞ = ∅ .
Each irreducible curve of order 3 lying in V 2 5 , meets each generatrix lines (as they partition V 2 5 ) that is, it is a directrix curve.
Corollary 3.3 All the directrix cubic curves are obtaining by cutting V 2 5 with the hyperplanes through any two generatrix lines. The maximal hyperplane section of V 2 5 consists 4 q + 1 points.
Proof. An irreducible cubic curve C ⊂ V 2 5 is a rational normal curve that is, it lies in a 3-space S (cf. Proposition 3.2). If C ⊂ V 2 5 is a cubic curve, for any two generatrix lines g , g ′ ∈ G is uniquely defined the hyperplane H = 〈 g , g ′ , S 〉 .
Let H be a hyperplane. If H ∩ V 2 5 were an irreducible curve of order 5, then | H ∩ V 2 5 | = q + 1 . If H ∩ V 2 5 = { g 1 , g 2 , C 3 } , g 1 , g 2 ∈ G , then | H ∩ V 2 5 | = 3 q + 1 . If H ∩ V 2 5 = { g 1 , g 2 , g 3 , C 2 } , g 1 , g 2 , g 3 ∈ G , then | H ∩ V 2 5 | = 4 q + 1 .
Proposition 3.4 1) No two directrix cubic curves belong to a same 3-space.
2) Two directrix cubic curves meet in at most one point.
Proof. 1) Assume two directrix cubic curves C , C ′ ⊂ V 2 5 belong to a same 3-space S. Then any hyperplane H ⊃ S meets V 2 5 in a curve of order at least 6, a contradiction.
2) Let C and C ′ be two cubic curves with C ⊂ S , C ′ ⊂ S ′ , where S , S ′ are 3-spaces, S ≠ S ′ from (1). Assume the curves have at least 2 points in common, P , Q ∈ C ∩ C ′ , P ≠ Q . Then S ∩ S ′ ⊃ P Q so that the hyperplane H = 〈 S , S ′ 〉 meets V 2 5 in a curve of order 6, a contradiction.
Denote F = G F ( q ) , Σ = P G ( 6, q ) . Let Σ ′ = P G ( 5, q ) be a hyperplane of Σ, S a regular spread of Σ ′ .
Choose a coordinate system ( x , y , t ) = ( x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , t ) in Σ so that t = 0 represents Σ ′ , ( x , y ) are internal coordinates for Σ ′ and for a point P ∈ Σ \ Σ ′ , P ≈ ( x , y , t ) = ( x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , t ) = F * ( x , y , t ) , F * = F \ { 0 } .
The spread S can be represented as follows
S = { π ∞ = ( 0, y ) | y ∈ F 3 } ∪ { π m = ( x , x m ) | x , m ∈ F 3 }
where y = x m is the multiplication in the field F 3 . Such a multiplication can be represented also by y = x M with M a 3 × 3 matrix over F so that x M = x m . The set M = { M | x M = x m } is a field isomorphic to ( F 3 ) 2 , strictly transitive over F 3 .
Denote R = { π ∞ , π k | k ∈ F } the regulus of S represented by the scalar matrices k I .
From now on we choose q ≡ 1 ( mod 3 ) , so that in F = G F ( q ) q − 1 3 elements are cubes, while the remaining ones are non cubes.
Let C ∞ be an irreducible conic of π ∞ and C 0 a skew cubic curve in the 3-space S 0 of Σ \ Σ ′ through π 0 so that C 0 ∩ π 0 = ∅ (cf. [
C ∞ = { ( 0,0,0,1, λ , λ 2 ,0 ) | λ ∈ G F ( q ) } ∪ { O ′ = ( 0,0,0,0,0,1,0 ) }
C 0 = { ( 1 , λ , λ 2 , 0 , 0 , 0 , s − λ 3 ) | λ ∈ G F ( q ) } ∪ { O = ( 0 , 0 , 0 , 0 , 0 , 0 , 1 ) } ,
where s ∈ G F ( q ) is a non cube.
The two curves are referred through a projectivity Λ : C 0 → C ∞ represented by having inserted the same parameter λ for which it is agreed that the points are considered corresponding to each other, plus Λ ( O ) = O ′ .
Denote V the ruled variety V 2 5 defined by C ∞ and C 0 . The curves C ∞ and C 0 are directrix curves of V , the set G of the lines connecting corresponding points are the generatrix lines of V .
Let us consider the affinity φ of Σ represented by the following 6 × 6 matrix in 3 × 3 blocks
M φ = ( I k I 0 I )
so that the extended projectivity φ ¯ is represented by the 7 × 7 matrix M φ ¯ obtained from M φ by adding the vector ( 0,0,0,0,0,0,1 ) as the 7th column and the 7th row.
Theorem 3.5 1) For every point P ∈ O O ′ \ { O ′ } there exists a bundle C P of q cubic curves on the variety V = V 2 5 having the point P in common, each curve lying in one 3-space intersecting a plane of R \ π ∞ . Each bundle cover the q 2 points of V \ O O ′ .
2) Such cubic curves are q 2 .
Proof. 1) For each point A = ( 0 , a , 0 ) = ( 0 , 0 , 0 , a 1 , a 2 , a 3 , 0 ) ∈ π ∞ it is φ ¯ ( A ) : = ( A ) M φ ¯ = A , that is, π ∞ is pointwise fixed. For a point B = ( b , 0 , 0 ) = ( b 1 , b 2 , b 3 , 0 , 0 , 0 , 0 ) ∈ π 0 it is φ ¯ ( B ) : = ( B ) M φ ¯ = ( b , k b , 0 ) , that is, φ ¯ ( π 0 ) = π k , and φ ¯ ( O ) = O . Hence φ ¯ ( S 0 ) is a 3-space S k through O with S k ∩ Σ ′ = π k . The cubic C 0 ⊂ S 0 is mapped onto a cubic C k ⊂ S k with O ∈ C k and C k ∩ π k = ∅ . Therefore there exists a bundle C0 of q cubic curves through O collecting the q 2 points of V \ O O ′ .
Let P = ( 0,0,0,0,0, h ,1 ) be a point of O O ′ \ { O , O ′ } and denote τ h the associated translation. Therefore τ h ( O ) = P and τ h ( C 0 ) = C P .
2) The cardinality of { C P | P ∈ O O ′ \ { O ′ } } is q 2 as for each point P ∈ O O ′ \ { O ′ } it is | C P | = q and the points of O O ′ \ { O ′ } are q.
Note that, chosen π ∞ and S 0 , the variety V 2 5 selections in the spread S the regulus R to which π ∞ and π 0 belong.
Denote Π the projective plane P G ( 2, q 3 ) . Represent Π in Σ, Π = ( P , L , I ) as in Lemma 2.6.
Denote V ′ the set of the q 2 affine points of V = V 2 5 .
Let Π q = ( P ′ , L ′ , I ′ ) be the incidence substructure of Π defined as follows:
P ′ = { P ∈ V ′ } ∪ { π ∞ } ,
L ′ = { C ∈ C P | P ∈ O O ′ \ { O ′ } } ∪ G ,
I ′ is defined as follows
I ′ = I restricted to the affine points and lines, π ∞ I ′ g for all g ∈ G .
Theorem 3.6 Πq is a non-affine subplane of Π of order q.
Proof. It is known from [
( 1 3 − 2 − 6 2 )
where
1: the number of the points is q 2 + q + 1 ,
2: the number of the lines is q 2 + q + 1 ,
3: each line contains q + 1 points,
62: two lines meet in at most one point,
then the structure is a projective plane of order q.
The affine points of V are the affine points of the q + 1 generatrix lines of G , that is, they are q ( q + 1 ) = q 2 + q to which the point at infinity π ∞ has to be added. Hence | P ′ | = q 2 + q + 1 , that is, 1 - holds.
From Theorem 3.5 follows | { C ∈ C P | P ∈ O O ′ \ { O ′ } } | = q 2 . As | G | = q + 1 then | L ′ | = q 2 + q + 1 , that is, 2 - holds.
Each cubic curve of CP has as many points as C 0 has, that is q + 1 . Each generatrix line g ∈ G has q affine points and the point ad infinity π ∞ , hence 3 - holds.
From Proposition 3.4, (2) follows that two cubic curves meet in at most one point. Moreover each cubic curve, being a directrix, meets a generatrix line in one point. Two generatrix lines meet only in the point π ∞ . Hence 62 holds.
To end proving that Πq is a subplane of Π it needs to verify that Πq is a subgeometry of Π (cf. Definition 2.7). Its set of points is clearly a subset of the points of Π. Moreover, every line g ∈ G is contained in a unique 3-space S = 〈 g , π ∞ 〉 which meets no other generatrix (cf. Proposition 3.1, 3)) and every cubic of CP lies in a unique 3-space (cf. Proposition 3.4, 1)) meeting Σ ′ in a plane of R (cf. Theorem 3.5, 1)).
From Theorem 3.6 follows.
Corollary 3.7 Let P, Q be two points of V ′ . If PQ is not a generatrix line then P, Q belong to one directrix cubic, if P ∈ V ′ and Q = π ∞ then the line PQ of Πq is the generatrix g P .
The properties of Πq of being a plane can be translated into further incidence properties of the affine points of V 2 5 .
Theorem 3.8 Let P, Q be two points of V 2 5 \ C ∞ . Then P, Q are joined by one generatrix line or by one directrix cubic C ⊂ S , where S is a 3-space with S ∩ Σ ′ = π k ∈ R \ π ∞ . Every two directrix cubic curves of V 2 5 \ C ∞ meet in one point.
In Theorem 3.5, 1), is shown that the variety V selects a regulus R to which both π ∞ , π 0 belong. Denote π ∞ : = π 1 ∞ , C ∞ : = C 1 ∞ . R : = R 1 . Fix the directrix cubic curve C 0 ⊂ S 0 .
Theorem 3.9 There exists a bundle B of varieties V 2 5 with the cubic C 0 as directrix, any to varieties having C 0 in common, | B | = q 2 .
Proof. The construction is done step by step, by choosing at each step a plane of the spread S out the regulus identified by the variety of the previous step, and a directrix conic in it.
Step 1 - Construct the variety V 1 = V 2 5 starting from the conic C 1 ∞ and the cubic C 0 ⊂ S 0 . In S \ R 1 are q 3 − q possible choices for the next step.
Step 2 - Choose a plane π 2 ∞ ∈ S \ R 1 . Fix a conic C 2 ∞ in it and construct the variety V 2 = V 2 5 starting from the conic C 2 ∞ and the cubic C 0 ⊂ S 0 . Let R 2 be the regulus of S to which π 2 ∞ and π 0 belong. In S \ { R 1 , R 2 } are q 3 − 2 q possible choices for the next step.
Step 3 - Choose a plane π 3 ∞ ∈ S \ { R 1 , R 2 } . Fix a conic C 3 ∞ in it and construct the variety V 3 = V 2 5 starting from the conic C 3 ∞ and C 0 ⊂ S 0 . Let R 3 be the regulus of S to which π 3 ∞ and π 0 belong. In S \ { R 1 , R 2 , R 3 } are q 3 − 3 q possible choices for the next step. And so on.
The procedure ends evidently at the q2-th step. Therefore B = { V i | i = 1,2, ⋯ , q 2 } and | B | = q 2 .
Conjecture - A variety V 2 2 r − 1 of P G ( 2 r , q ) represents a non-affine subplane of order q of P G ( 2, q r ) via the spatial representation. This is partially addressed in Theorem 11 of the following paper: M. Lavrauw, C. Zanella, Subspaces intersecting each element of a regulus in one point, Andr-Bruck-Bose Representation and Clubs, Electron. J. Combin. 23 (2016), Paper 1.37, pp. 1-11.
The author declares no conflicts of interest regarding the publication of this paper.
Vincenti, R. (2024) Subplanes of and the Ruled Varieties of . Open Journal of Discrete Mathematics, 14, 16-27. https://doi.org/10.4236/ojdm.2024.142003