Rita Vincenti
Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy.
DOI: 10.4236/ojdm.2024.142003   PDF    HTML   XML   197 Downloads   738 Views   Citations

Abstract

In this note we study subplanes of order q of the projective plane $\Pi =PG\left(\mathrm{2,}{q}^3\right)$ and the ruled varieties $V_2^5$ of $\Sigma =PG\left(\mathrm{6,}q\right)$ using the spatial representation of Π in Σ, by fixing a hyperplane ${\Sigma }^{\prime }$ with a regular spread of planes. First are shown some configurations of the affine q-subplanes. Then to prove that a variety $V_2^5$ of Σ represents a non-affine subplane of order q of Π, after having shown basic incidence properties of it, such a variety $V_2^5$ 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 $V_2^5$ . Then a maximal bundle of varieties $V_2^5$ having in common one directrix cubic curve is constructed.

Keywords

Finite Geometry, Translation Planes, Spreads, Varieties

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1. Introduction

It is known that a projective translation plane Π of order $q^r$ and kernel $F=GF\left(q\right)$ can be represented by a 2r-dimensional projective space $\Sigma =PG\left(2r\mathrm{,}q\right)$ over F, fixing a hyperplane ${\Sigma }^{\prime }=PG\left(2r-\mathrm{1,}q\right)$ and a spread (partition) $\mathcal{S}$ of ${\Sigma }^{\prime }$ with $\left(r-1\right)$ -dimensional subspaces (cf. [1] and [2] ).

The points of Π are represented by 1) the points of $\Sigma \backslash {\Sigma }^{\prime }$ (the affine points) and by 2) the elements of $\mathcal{S}$ (the points at infinity). The lines of Π are represented by 1) the r-dimensional subspaces S of $\Sigma \backslash {\Sigma }^{\prime }$ such that $S\cap {\Sigma }^{\prime }$ belongs to $\mathcal{S}$ and by 2) the spread $\mathcal{S}$ . The translation line $l_{\infty }$ of Π (the line at infinity) is represented by $\mathcal{S}$ (cf. Lemma 2.6).

If a subplane of Π meets $l_{\infty }$ in a subline, then such a subplane is affine, if it meets $l_{\infty }$ in one point is non-affine.

An affine subplane $\mathcal{A}$ of order q is represented by every transversal plane α to $\mathcal{S}$ , that is, by a plane $\alpha \subset \Sigma \backslash {\Sigma }^{\prime }$ such that the line $t=\alpha \cap {\Sigma }^{\prime }$ meets $q+1$ elements of $\mathcal{S}$ , t is a transversal line to $\mathcal{S}$ . In such a way $l_{\infty }$ is a line of the projective completion of $\mathcal{A}$ . Of course all that holds also in case Π is the Desarguesian plane $PG\left(\mathrm{2,}{q}^r\right)$ when $\mathcal{S}$ is a regular spread (cf. [1] [2] [3] [4] [5] for $r=2$ ).

Fix $r=3$ so that $\Pi =PG\left(\mathrm{2,}{q}^3\right)$ , $\Sigma =PG\left(\mathrm{6,}q\right)$ , ${\Sigma }^{\prime }=PG\left(\mathrm{5,}q\right)$ and $\mathcal{S}$ is a regular spread of planes.

About the affine subplanes of $\Pi =PG\left(\mathrm{2,}{q}^3\right)$ 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 $PG\left(\mathrm{6,}q\right)$ 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. [6] , Capter 13, 8., 9.). The variety can be obtained by joining points of the two directrix curves corresponding via a projectivity.

We choose a conic in a plane ${\pi }_{\infty }\in \mathcal{S}$ and a cubic in a 3-dimensional subspace $S_0\in \Sigma \backslash {\Sigma }^{\prime }$ with ${\pi }_0=S_0\cap {\Sigma }^{\prime }\in \mathcal{S}$ and ${\pi }_0\ne {\pi }_{\infty }$ . Some fundamental incidence properties of $V_2^5$ are shown (cf. Paragraph 3.1). Then, if $q\equiv 1,\left(\mathrm{mod}3\right)$ , we can prove that the variety $V_2^5$ represents a non-affine subplane Π_q of order q of $PG\left(\mathrm{2,}{q}^3\right)$ (cf. Theorem 3.6).

The properties of Π_q 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 ${\mathcal{C}}_0$ , a maximal bundle of varieties $V_2^5$ having in common ${\mathcal{C}}_0$ is constructed (cf. Theorem 3.9).

At the end is formulated the conjecture that a ruled variety $V_2^{2r-1}$ of $PG\left(2r\mathrm{,}q\right)$ represents a non affine subplane of order q of $PG\left(\mathrm{2,}{q}^r\right)$ , via the spatial representation.

2. Preliminary Notes and Results

Let $F=GF\left(q\right)$ be a finite field, $q=p^s$ , p an odd prime. Denote $F^{r+1}$ the $\left(r+1\right)$ -dimensional vector space over F, $PG\left(r\mathrm{,}q\right)=Pr{F}^{r+1}$ the r-dimensional projective space contraction of $F^{r+1}$ over F. Let $\bar{F}$ be the algebraic closure of the field $F=GF\left(q\right)$ .

The geometry $PG\left(r\mathrm{,}q\right)$ is considered a sub-geometry of $\overline{PG\left(r\mathrm{,}q\right)}$ , the projective geometry over $\bar{F}$ . We refer to the points of $PG\left(r\mathrm{,}q\right)$ as the rational points of $\overline{PG\left(r\mathrm{,}q\right)}$ .

Denote $S_h$ or h-space with $1\le h\le r-1$ a subspace of $PG\left(r\mathrm{,}q\right)$ of dimension h. A hyperplane $S_{r-1}$ will be denoted also by H, a plane by π. If $A\mathrm{,}B\mathrm{,}C\mathrm{,}\cdots$ are subspaces denote $A+B+C+\cdots =\langle A,B,C,\cdots \rangle$ , the subspace generated by them. More simply, when $A\mathrm{,}B$ 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 $PG\left(r\mathrm{,}q\right)$ is the set of the rational points of a projective variety ${\bar{V}}_u^v$ of $\overline{PG\left(r\mathrm{,}q\right)}$ defined by a finite set of polynomials with coefficients in the field F.

From [6] , pp. 290, 7., for $r\ge 4$ follows.

Lemma 2.2 The ruled variety $V_2^{r-1}$ of $PG\left(r\mathrm{,}q\right)$ 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 ${\left(q+1\right)}^2$ and the order is the sum of the orders of the curves.

Let ${\Sigma }^{\prime }$ be the projective space $PG\left(2r-\mathrm{1,}q\right)$ over the field $F=GF\left(q\right)$ , $r>1$ an integer.

Definition 2.3 A spread of ${\Sigma }^{\prime }$ is a partition $\mathcal{S}$ with $\left(r-1\right)$ -dimensional subspaces (that is, every point of ${\Sigma }^{\prime }$ lies in one element of $\mathcal{S}$ ). A regulus $\mathcal{R}$ of $\mathcal{S}$ is a collection of subspaces of $\mathcal{S}$ such that:

1) $\mathcal{R}$ contains at least 3 elements,

2) Every line meeting 3 elements of $\mathcal{R}$ , a transversal line, meets every element of $\mathcal{R}$ ,

3) Every point of a transversal line to $\mathcal{R}$ lies in one element of $\mathcal{R}$ ,

4) Every plane through a transversal line is a transversal plane to $\mathcal{S}$ .

Any three pairwise disjoint $\left(r-1\right)$ -dimensional subspaces of $\mathcal{S}$ lie in a unique regulus. Any two distinct transversal lines are skew.

A spread is regular if for any three distinct elements of $\mathcal{S}$ , all the members of the unique regulus determined by them are in $\mathcal{S}$ .

Regular spreads represent Desarguesian planes $PG\left(\mathrm{2,}{q}^r\right)$ (cf. [2] , pp. 162-163).

The construction of a regular spread can be described as follows.

Choose a coordinate system in ${\Sigma }^{\prime }$ so that for a point P of ${\Sigma }^{\prime }$ , $P\approx \left(x\mathrm{,}y\right)=\left(x_1\mathrm{,}{x}_2\mathrm{,}\cdots \mathrm{,}{x}_r\mathrm{<mo>\; ;\; </mo>}{y}_1\mathrm{,}{y}_2\mathrm{,}\cdots \mathrm{,}{y}_r\right)=F^{\mathrm{<mo>\; *\; </mo>}}\left(x\mathrm{,}y\right)$ , $F^{*}=F\backslash \left\{0\right\}$ .

A regular spread of ${\Sigma }^{\prime }$ is given by the set $\left\{J_{\infty }=\left(0,y\right)|y\in F^r\right\}\cup \left\{J_m=\left(x,xm\right)|x,m\in F^r\right\}$ where $y=xm$ is the multiplication in the field $F^r$ . Such a multiplication can be represented also by $y=xM$ with M a $r\times r$ matrix over F so that $xM=xm$ . The set $\mathcal{M}=\left\{M\mathrm{<mo>\; |\; </mo>}xM=xm\right\}$ is a field isomorphic to ${\left(F^r\right)}^2$ , 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. [1] and [2] ).

Let Σ be the projective geometry $PG\left(\mathrm{6,}q\right)$ .

Denote ${\pi }_{\infty }$ and S₀ a plane and a 3-space, respectively, of Σ. Assume ${\pi }_{\infty }$ and S₀ are complementary. Let ${\mathcal{C}}^2$ be an irreducible conic of ${\pi }_{\infty }$ , ${\mathcal{C}}^3$ a skew cubic curve of S₀. The curves ${\mathcal{C}}^2$ and ${\mathcal{C}}^3$ 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 ${\mathcal{C}}^2$ and of ${\mathcal{C}}^3$ .

Proof. See [6] , p. 291.

Corollary 2.5 The variety $V_2^5$ consists of ${\left(q+1\right)}^2$ points of the $q+1$ generatrix lines including the points of the minimum order directrix ${\mathcal{C}}^2$ and of the maximum order directrix ${\mathcal{C}}^3$ .

Note that the set of the $q+1$ generatrix lines partition the variety.

Choose a hyperplane ${\Sigma }^{\prime }=PG\left(\mathrm{5,}q\right)$ of Σ as the hyperplane at infinity.

Fix a coordinate system in Σ so that it is a coordinate system also for $\bar{\Sigma }$ . Denote a point $P\approx \left(x\mathrm{,}y\mathrm{,}t\right)=\left(x_1\mathrm{,}{x}_2\mathrm{,}{x}_3\mathrm{,}{y}_1\mathrm{,}{y}_2\mathrm{,}{y}_3\mathrm{,}t\right)\mathrm{<mo>\; :\; </mo>}={\bar{F}}^{\mathrm{<mo>\; *\; </mo>}}\left(x\mathrm{,}y\mathrm{,}t\right)$ , ${\bar{F}}^{*}=\bar{F}\backslash \left\{0\right\}$ . Let $t=0$ be the equation of ${\Sigma }^{\prime }$ .

P is a rational point if there exists $\left(x\mathrm{,}y\mathrm{,}t\right)\in F^7$ such that $P\approx \left(x\mathrm{,}y\mathrm{,}t\right)$ .

A variety V of Σ is the set of the rational points of $\bar{\Sigma }$ solutions of a finite set of polynomials of $F\left[x\mathrm{,}y\mathrm{,}t\right]$ .

Let $\Pi =PG\left(\mathrm{2,}{q}^3\right)$ be the Desarguesian plane over the field $GF\left(q^3\right)$ . Denote $l_{\infty }$ the line at infinity of Π. Represent Π in $\Sigma =PG\left(\mathrm{6,}q\right)$ by a regular spread $\mathcal{S}$ of planes of ${\Sigma }^{\prime }$ , with $\left|\mathcal{S}\right|=q^3+1$ .

More precisely define the following incidence structure ${\Pi }^{\prime }=\left(\mathcal{P}\mathrm{,}\mathcal{L}\mathrm{,}\mathcal{I}\right)$ (points, lines, incidence, respectively) where

$\mathcal{P}=\left\{P\in \Sigma \backslash {\Sigma }^{\prime }\right\}\cup \left\{\pi \in \mathcal{S}\right\}$ ,

$\mathcal{L}=\left\{{\mathcal{L}}_0=\left\{S_3\subset \Sigma \backslash {\Sigma }^{\prime }\mathrm{<mo>\; |\; </mo>}{S}_3\cap {\Sigma }^{\prime }\in \mathcal{S}\right\}\right\}\cup \left\{l_{\infty }=\mathcal{S}\right\}$ ,

$\mathcal{I}$ is defined as follows

if $P\in \Sigma \backslash {\Sigma }^{\prime }$ , $l\in {\mathcal{L}}_0$ then $P\mathcal{I}l\iff P\in l$ , no point of $\Sigma \backslash {\Sigma }^{\prime }$ incides $l_{\infty }$ , $\pi \mathcal{I}{l}_{\infty }$ for all $\pi \in \mathcal{S}$ , $\pi \mathcal{I}l$ where $l\in {\mathcal{L}}_0\iff l\cap {\Sigma }^{\prime }=\pi$ .

From [1] [2] [3] and [4] pp. 38-39 follows.

Lemma 2.6 ${\Pi }^{\prime }\cong \Pi$ .

In short, the affine points of Π are represented by the $q^6$ affine points of $\Sigma \backslash {\Sigma }^{\prime }$ , the points at infinity by the $q^3+1$ planes of $\mathcal{S}$ . The affine lines of Π are represented by the 3-spaces S of $\Sigma \backslash {\Sigma }^{\prime }$ such that the plane $S\cap {\Sigma }^{\prime }$ belongs to $\mathcal{S}$ , the line at infinity $l_{\infty }$ by the spread $\mathcal{S}$ .

Definition 2.7 A subplane ${\pi }^{\prime }=\left(P^{\prime }\mathrm{,}{L}^{\prime }\mathrm{,}{I}^{\prime }\right)$ of a plane $\pi =\left(P\mathrm{,}L\mathrm{,}I\right)$ is a subgeometry of π, that is, an incidence substructure for which $P^{\prime }\subset P$ , for each line $l^{\prime }\subset L^{\prime }$ , there exists a line $l\in L$ such that $l^{\prime }\subset L$ and $I^{\prime }=I$ .

Definition 2.8 A subplane of $\Pi =PG\left(\mathrm{2,}{q}^3\right)$ of order q is affine if it meets the line $l_{\infty }$ of Π in a subline consisting of $q+1$ points, it is non-affine if it meets the line $l_{\infty }$ in one point.

Let r be any transversal line to $\mathcal{S}$ . As $\mathcal{S}$ is regular, the $q+1$ planes that r meets form a regulus $\mathcal{R}\subset \mathcal{S}$ (cf. [2] , Lemma 12.2). Choose and fix a transversal plane α through the line r.

It is easy to prove the following.

Proposition 2.9 The plane α is isomorphic to a subplane $\pi \cong PG\left(\mathrm{2,}q\right)$ of Π whose points at infinity are represented by the $q+1$ planes of $\mathcal{R}$ , the lines of π being represented by the sublines intersections of α with the 3-spaces of Σ through the planes of $\mathcal{R}$ . 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 $\mathcal{R}$ in ${\Sigma }^{\prime }$ in a synthetic way the procedure is similar to the one used for dimension 3.

Proposition 2.10 The transversal lines to $\mathcal{R}$ are $q^2+q+1$ .

Proof. Denote ${\pi }_1\mathrm{,}{\pi }_2\mathrm{,}{\pi }_3$ three planes of the regulus $\mathcal{R}$ . Fix a point $P\in {\pi }_1$ and denote $S=\langle P\mathrm{,}{\pi }_2\rangle$ the 3-space of ${\Sigma }^{\prime }$ direct sum of P and ${\pi }_2$ and $S^{\prime }=\langle P\mathrm{,}{\pi }_3\rangle$ the 3-space of ${\Sigma }^{\prime }$ direct sum of P and ${\pi }_3$ . Lying in a 5-dimensional subspace, then $S\cap S^{\prime }=r$ is a line. As a line of S, r meets ${\pi }_2$ in a point, as a line of $S^{\prime }$ , r meets ${\pi }_3$ in a point. Therefore r is a transversal line to the planes ${\pi }_1\mathrm{,}{\pi }_2\mathrm{,}{\pi }_3$ . As ${\pi }_1\mathrm{,}{\pi }_2\mathrm{,}{\pi }_3$ belong to the regulus $\mathcal{R}$ , the line r meets each of the $q+1$ elements of $\mathcal{R}$ . In such a way one can construct a transversal line for every point P chosen in ${\pi }_1$ , that is, $q^2+q+1$ .

Proposition 2.11 The cardinality of the set of all affine subplanes of Π isomorphic to $PG\left(\mathrm{2,}q\right)$ 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 $\mathcal{R}$ , ${\alpha }_0\supset r_0$ a transversal plane, O an affine point of ${\alpha }_0$ . Denote $\left\{r_i|i=0,\cdots ,q^2+q\right\}$ the transversal lines to the regulus $\mathcal{R}$ . Each of the $q^2+q+1$ planes ${\alpha }_i=\langle O\mathrm{,}{r}_i\rangle$ represents an affine subplane ${\pi }_i$ of Π, ${\pi }_i\cong PG\left(\mathrm{2,}q\right)$ (cf. Proposition 2.9).

Choose and fix a transversal line r. Consider the bundle (r) of the planes of $\Sigma \backslash {\Sigma }^{\prime }$ having the line r as axis. Each plane $\alpha \in \left(r\right)$ is isomorphic to $PG\left(\mathrm{2,}q\right)$ (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 $\Sigma \backslash {\Sigma }^{\prime }$ of the bundle through the line r.

It is known that the number of k-subspaces in a n-space (vector notation) is

$\left[n\mathrm{<mo>\; |\; </mo>}k\right]=\frac{\left(q^n-1\right)\left(q^n-q\right)\cdots \left(q^n-q^{k-1}\right)}{\left(q^k-1\right)\left(q^k-q\right)\cdots \left(q^k-q^{k-1}\right)}\mathrm{.}$

The number h of the planes through a line in $PG\left(\mathrm{6,}q\right)\backslash PG\left(\mathrm{5,}q\right)$ equals the number of 3-spaces in a 4-space, minus the number of the planes through a line in $PG\left(\mathrm{5,}q\right)$ , which equals the number of the planes in a 3-space.

Therefore, after simplification of the two ratios, follows

$h=\frac{q^5-1}{q-1}-\frac{q^4-1}{q-1}=q^4.$

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 $\Sigma \backslash {\Sigma }^{\prime }$ to get the plane $\langle r\mathrm{,}P\rangle \subset \Sigma \backslash {\Sigma }^{\prime }$ , 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 $\Sigma \backslash {\Sigma }^{\prime }$ and $\left|\left(r\right)\right|=q^4$ , then the total number of points of $\Sigma \backslash {\Sigma }^{\prime }$ covered by them is $q^6$ , which are all the affine points of Π.

3. Main Results

3.1. The Variety $V_2^5$ and Its Sections

Denote $\Sigma =PG\left(6,q\right)$ , ${\Sigma }^{\prime }=PG\left(\mathrm{5,}q\right)$ a hyperplane of Σ, $\mathcal{S}$ a regular spread of ${\Sigma }^{\prime }$ . Choose and fix a plane ${\pi }_{\infty }\in \mathcal{S}$ and a 3-space $S_0$ in $\Sigma \backslash {\Sigma }^{\prime }$ such that $S_0\cap {\Sigma }^{\prime }={\pi }_0\in \mathcal{S}\backslash {\pi }_{\infty }$ . Then choose and fix:

1) A non-degenerate conic ${\mathcal{C}}^2\subset {\pi }_{\infty }$ ,

2) A skew cubic curve ${\mathcal{C}}^3$ in $S_0$ such that ${\mathcal{C}}^3\cap {\pi }_0=\varnothing$ (cf. [7] , p. 234, Corollary 4, ${\mathcal{N}}_5$ ).

Let $\Lambda \mathrm{<mo>\; :\; </mo>}{\mathcal{C}}^2\to {\mathcal{C}}^3$ be a projectivity. Represent ${\mathcal{C}}^2=\left\{G_{i_{\infty }}|i=1,\cdots ,q+1\right\}$ , ${\mathcal{C}}^3=\left\{G_i=\Lambda G_{i_{\infty }}|i=1,\cdots ,q+1\right\}$ . Denote $V_2^5$ the variety arising by connecting corresponding points of ${\mathcal{C}}^2$ and ${\mathcal{C}}^3$ via Λ (cf. [6] , p. 291). The curves ${\mathcal{C}}^2$ and ${\mathcal{C}}^3$ are directrix curves of $V_2^5$ , the set $\mathcal{G}=\left\{g_i=G_i{G}_{i_{\infty }}|i=1,\cdots ,q+1\right\}$ is the set of the generatrix lines of $V_2^5$ . The set $\mathcal{G}$ partitions the variety.

Let H be any hyperplane. In a suitable complexification of Σ, $H\cap V_2^5$ is a curve of order 5 (cf. [6] , p. 288, 5.).

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 $\mathcal{C}\ne {\mathcal{C}}^2$ cut by a hyperplane on $V_2^5$ cannot lie in a plane. The conic ${\mathcal{C}}^2$ is the unique minimum order 2 directrix.

2) The 3-space joining two generatrix lines cannot contain the plane ${\pi }_{\infty }$ .

3) The 3-space joining one generatrix line and the plane ${\pi }_{\infty }$ meets no other generatrix.

4) Three generatrices $g_1\mathrm{,}{g}_2\mathrm{,}{g}_3$ are joint by a hyperplane H that contains the plane ${\pi }_{\infty }$ , so that $H\cap V_2^5=\left\{g_1\mathrm{,}{g}_2\mathrm{,}{g}_3\right\}\cup {\mathcal{C}}^2$ .

5) A hyperplane contains neither a fixed directrix, nor a fixed generatrix.

Proof. Let $g\mathrm{,}{g}^{\prime }$ be two generatrix lines. Denote $G_{\infty }=g\cap {\pi }_{\infty }$ , ${G^{\prime }}_{\infty }=g^{\prime }\cap {\pi }_{\infty }$ , with $G_{\infty }\mathrm{,}{G^{\prime }}_{\infty }\in {\mathcal{C}}^2$ , $G=g\cap {\mathcal{C}}^3$ , $G^{\prime }=g^{\prime }\cap {\mathcal{C}}^3$ , $G\mathrm{,}{G}^{\prime }\in S_0$ . Let r be the line $G_{\infty }{G^{\prime }}_{\infty }$ , $r\subset {\pi }_{\infty }$ , $r^{\prime }$ the line $G{G}^{\prime }$ , $r^{\prime }\subset S_0$ . Assume $g\mathrm{,}{g}^{\prime }$ meet in a point. Hence they define a plane $\tau =\langle r\mathrm{,}{r}^{\prime }\rangle$ so that $r\cap r^{\prime }=P$ with $P\in {\pi }_{\infty }\cap S_0$ , a contradiction.

As ${\mathcal{C}}^3$ has no points in ${\pi }_0$ , then all the $q+1$ generatrices are affine and the points of them are affine except the points of ${\mathcal{C}}^2$ , therefore the affine points of $V_2^5$ are $q\left(q+1\right)=q^2+q$ .

1) Assume a hyperplane H meets $V_2^5$ in a directrix curve $\mathcal{C}\ne {\mathcal{C}}^2$ lying in a plane $\tau$ . Then $V_2^5$ is contained at most in the 5-space generated by $\tau$ and ${\pi }_{\infty }$ , a contradiction. The conic ${\mathcal{C}}^2$ 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 ${\pi }_{\infty }$ and two generatrix lines, $g_1\mathrm{,}{g}_2$ . Denote $G_i=g_i\cap {\mathcal{C}}^3$ , $i=\mathrm{1,2}$ and $G_1{G}_2\cap {\Sigma }^{\prime }=G$ . The line $G_1{G}_2$ belongs to $S_0$ and to S, so that the point G is a common point of ${\pi }_{\infty }$ and ${\pi }_0$ , a contradiction.

3) Let $S=\langle g\mathrm{,}{\pi }_{\infty }\rangle$ , with $g\in \mathcal{G}$ , be a 3-space. If $S\cap g^{\prime }\ne \varnothing$ with $g^{\prime }\ne g$ , $g^{\prime }\in \mathcal{G}$ , then $g^{\prime }\subset S$ , so that S contains two generatrix lines and the plane ${\pi }_{\infty }$ , a contradiction to (2).

4) Assume three generatrices $g_1\mathrm{,}{g}_2\mathrm{,}{g}_3$ are joint by a 4-space $S^{\prime }$ . As $S^{\prime }$ contains the three points $G_{1_{\infty }}\mathrm{,}{G}_{2_{\infty }}\mathrm{,}{G}_{3_{\infty }}\in {\mathcal{C}}^2$ , $G_{i\infty }=g_i\cap {\mathcal{C}}^2$ , $i=\mathrm{1,2,3}$ , then ${\pi }_{\infty }\subset S^{\prime }$ and ${\mathcal{C}}^2\subset S^{\prime }$ . As $S^{\prime }$ cannot contain $V_2^5$ , a hyperplane $H\supset S^{\prime }$ and through a further point $P\in V_2^5\backslash S^{\prime }$ 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. [6] , p. 288, 5.). Therefore a hyperplane H containing three generatrices $g_1\mathrm{,}{g}_2\mathrm{,}{g}_3$ , contains the non collinear points $G_{1_{\infty }}\mathrm{,}{G}_{2_{\infty }}\mathrm{,}{G}_{3_{\infty }}$ hence the whole plane ${\pi }_{\infty }$ and then the conic directrix ${\mathcal{C}}^2$ . Therefore $H\cap V_2^5=\left\{g_1\mathrm{,}{g}_2\mathrm{,}{g}_3\right\}\cup {\mathcal{C}}^2$ that is a curve of order 5 (and H contains no further point of $V_2^5$ ).

5) Let $g\mathrm{,}{g}^{\prime }$ two generatrices of $V_2^5$ . Denote S the 3-space containing them. Let H be a generic hyperplane with $H\supset S$ and assume H contains a fix directrix $\mathcal{C}$ . Let P be a point of $V_2^5$ , $P\in H\backslash \mathcal{C}$ . Denote $S_4=\langle S\mathrm{,}P\rangle$ . Then every hyperplane containing $S_4$ and $S_4$ itself, would contain the generatrix $g_P$ through P, so that $g\mathrm{,}{g}^{\prime }\mathrm{,}{g}_P\subset S_4$ , a contradiction to (4). An analogous contradiction is reached if we assume a generic hyperplane containing $g\mathrm{,}{g}^{\prime }$ contained a fix generatrix (cf. [6] , 6., pp. 289-290).

The following propositions are a rereading in the current case of [6] , pp. 287-290.

Proposition 3.2 A hyperplane H containing two generatrices, contains a residual cubic curve $\mathcal{C}$ lying in a 3-space $S\subset H$ , S skew to ${\pi }_{\infty }$ . $\mathcal{C}$ is irreducible and is a directrix.

Proof. In [6] , 3., p. 287, the 2nd paragraph, is affirmed that a hyperplane H meets $V_2^5$ in a rational normal curve of order 5 (as it lives in a 5-space) or in a curve of order $m<5$ met by all the generatrix lines and in $5-m$ generatrices. In our case it is $m=2$ or $m=3$ . Set $m=2$ . If a hyperplane H contains the unique conic directrix ${\mathcal{C}}^2$ (cf. 1), Proposition 3.1), then it must contain $5-m=3$ generatrix lines and viceversa (cf. 4), Proposition 3.1).

Set $m=3$ . If a hyperplane contains $5-m=2$ generatrix lines, then it meets $V_2^5$ in a residual cubic curve $\mathcal{C}$ and viceversa.

Assume the cubic $\mathcal{C}$ exists in a plane π. Let H be a hyperplane containing π and three further points $P\mathrm{,}Q\mathrm{,}R\in V_2^5\backslash \pi$ . Then H contains also the 3 generatrix lines $g_P\mathrm{,}{g}_Q\mathrm{,}{g}_R$ as all the generatrix meet $\mathcal{C}$ . In such a way $H\cap V_2^5\supset \left\{\mathcal{C}\mathrm{,}{g}_P\mathrm{,}{g}_Q\mathrm{,}{g}_R\right\}$ , that is a curve of order 6, a contradiction. Hence $\mathcal{C}$ lies in a 3-space S.

If such a space S met ${\pi }_{\infty }$ , then a hyperplane $H\supseteq \langle S\mathrm{,}{\pi }_{\infty }\rangle$ would contain $V_2^5$ , a contradiction, therefore $S\cap {\pi }_{\infty }=\varnothing$ .

Assume a cubic curve $\mathcal{C}$ is reducible. Of course, the unique possibility is that $\mathcal{C}$ consists of at most 3 generatrix lines. In such a case $\mathcal{C}$ would meet the conic ${\mathcal{C}}^2$ and then the plane ${\pi }_{\infty }$ , a contradiction with $S\cap {\pi }_{\infty }=\varnothing$ .

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 $4q+1$ points.

Proof. An irreducible cubic curve $\mathcal{C}\subset V_2^5$ is a rational normal curve that is, it lies in a 3-space S (cf. Proposition 3.2). If $\mathcal{C}\subset V_2^5$ is a cubic curve, for any two generatrix lines $g\mathrm{,}{g}^{\prime }\in \mathcal{G}$ is uniquely defined the hyperplane $H=\langle g\mathrm{,}{g}^{\prime }\mathrm{,}S\rangle$ .

Let H be a hyperplane. If $H\cap V_2^5$ were an irreducible curve of order 5, then $\left|H\cap V_2^5\right|=q+1$ . If $H\cap V_2^5=\left\{g_1\mathrm{,}{g}_2\mathrm{,}{\mathcal{C}}^3\right\}$ , $g_1\mathrm{,}{g}_2\in \mathcal{G}$ , then $\left|H\cap V_2^5\right|=3q+1$ . If $H\cap V_2^5=\left\{g_1\mathrm{,}{g}_2\mathrm{,}{g}_3\mathrm{,}{\mathcal{C}}^2\right\}$ , $g_1\mathrm{,}{g}_2\mathrm{,}{g}_3\in \mathcal{G}$ , then $\left|H\cap V_2^5\right|=4q+1$ .

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 $\mathcal{C}\mathrm{,}{\mathcal{C}}^{\prime }\subset V_2^5$ belong to a same 3-space S. Then any hyperplane $H\supset S$ meets $V_2^5$ in a curve of order at least 6, a contradiction.

2) Let $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$ be two cubic curves with $\mathcal{C}\subset S$ , ${\mathcal{C}}^{\prime }\subset S^{\prime }$ , where $S,S^{\prime }$ are 3-spaces, $S\ne S^{\prime }$ from (1). Assume the curves have at least 2 points in common, $P\mathrm{,}Q\in \mathcal{C}\cap {\mathcal{C}}^{\prime }$ , $P\ne Q$ . Then $S\cap S^{\prime }\supset PQ$ so that the hyperplane $H=\langle S\mathrm{,}{S}^{\prime }\rangle$ meets $V_2^5$ in a curve of order 6, a contradiction.

3.2. Bundles of Cubics on $V_2^5$ and a Non-Affine Subplane

Denote $F=GF\left(q\right)$ , $\Sigma =PG\left(\mathrm{6,}q\right)$ . Let ${\Sigma }^{\prime }=PG\left(\mathrm{5,}q\right)$ be a hyperplane of Σ, $\mathcal{S}$ a regular spread of ${\Sigma }^{\prime }$ .

Choose a coordinate system $\left(x\mathrm{,}y\mathrm{,}t\right)=\left(x_1\mathrm{,}{x}_2\mathrm{,}{x}_3\mathrm{,}{y}_1\mathrm{,}{y}_2\mathrm{,}{y}_3\mathrm{,}t\right)$ in Σ so that $t=0$ represents ${\Sigma }^{\prime }$ , $\left(x\mathrm{,}y\right)$ are internal coordinates for ${\Sigma }^{\prime }$ and for a point $P\in \Sigma \backslash {\Sigma }^{\prime }$ , $P\approx \left(x\mathrm{,}y\mathrm{,}t\right)=\left(x_1\mathrm{,}{x}_2\mathrm{,}{x}_3\mathrm{,}{y}_1\mathrm{,}{y}_2\mathrm{,}{y}_3\mathrm{,}t\right)=F^{\mathrm{<mo>\; *\; </mo>}}\left(x\mathrm{,}y\mathrm{,}t\right)$ , $F^{\mathrm{<mo>\; *\; </mo>}}=F\backslash \left\{0\right\}$ .

The spread $\mathcal{S}$ can be represented as follows

$\mathcal{S}=\left\{{\pi }_{\infty }=\left(\mathrm{0,}y\right)\mathrm{<mo>\; |\; </mo>}y\in F^3\right\}\cup \left\{{\pi }_m=\left(x\mathrm{,}xm\right)\mathrm{<mo>\; |\; </mo>}x\mathrm{,}m\in F^3\right\}$

where $y=xm$ is the multiplication in the field $F^3$ . Such a multiplication can be represented also by $y=xM$ with M a 3 × 3 matrix over F so that $xM=xm$ . The set $\mathcal{M}=\left\{M\mathrm{<mo>\; |\; </mo>}xM=xm\right\}$ is a field isomorphic to ${\left(F^3\right)}^2$ , strictly transitive over $F^3$ .

Denote $\mathcal{R}=\left\{{\pi }_{\infty }\mathrm{,}{\pi }_k\mathrm{<mo>\; |\; </mo>}k\in F\right\}$ the regulus of $\mathcal{S}$ represented by the scalar matrices $kI$ .

From now on we choose $q\equiv 1\left(\mathrm{mod}3\right)$ , so that in $F=GF\left(q\right)$ $\frac{q-1}{3}$ elements are cubes, while the remaining ones are non cubes.

Let ${\mathcal{C}}_{\infty }$ be an irreducible conic of ${\pi }_{\infty }$ and ${\mathcal{C}}_0$ a skew cubic curve in the 3-space $S_0$ of $\Sigma \backslash {\Sigma }^{\prime }$ through ${\pi }_0$ so that ${\mathcal{C}}_0\cap {\pi }_0=\varnothing$ (cf. [7] , p. 234, Corollary 4, ${\mathcal{N}}_5$ ):

${\mathcal{C}}_{\infty }=\left\{\left(\mathrm{0,0,0,1,}\lambda \mathrm{,}{\lambda }^2\mathrm{,0}\right)\mathrm{<mo>\; |\; </mo>}\lambda \in GF\left(q\right)\right\}\cup \left\{O^{\prime }=\left(\mathrm{0,0,0,0,0,1,0}\right)\right\}$

${\mathcal{C}}_0=\left\{\left(1,\lambda ,{\lambda }^2,0,0,0,s-{\lambda }^3\right)|\lambda \in GF\left(q\right)\right\}\cup \left\{O=\left(0,0,0,0,0,0,1\right)\right\}$ ,

where $s\in GF\left(q\right)$ is a non cube.

The two curves are referred through a projectivity $\Lambda \mathrm{<mo>\; :\; </mo>}{\mathcal{C}}_0\to {\mathcal{C}}_{\infty }$ represented by having inserted the same parameter $\lambda$ for which it is agreed that the points are considered corresponding to each other, plus $\Lambda \left(O\right)=O^{\prime }$ .

Denote $\mathcal{V}$ the ruled variety $V_2^5$ defined by ${\mathcal{C}}_{\infty }$ and ${\mathcal{C}}_0$ . The curves ${\mathcal{C}}_{\infty }$ and ${\mathcal{C}}_0$ are directrix curves of $\mathcal{V}$ , the set $\mathcal{G}$ of the lines connecting corresponding points are the generatrix lines of $\mathcal{V}$ .

Let us consider the affinity $\phi$ of Σ represented by the following 6 × 6 matrix in 3 × 3 blocks

$M_{\phi }=\left(\begin{array}{cc}I& kI\\ 0& I\end{array}\right)$

so that the extended projectivity $\bar{\phi }$ is represented by the 7 × 7 matrix $M_{\bar{\phi }}$ obtained from $M_{\phi }$ by adding the vector $\left(\mathrm{0,0,0,0,0,0,1}\right)$ as the 7th column and the 7th row.

Theorem 3.5 1) For every point $P\in O{O}^{\prime }\backslash \left\{O^{\prime }\right\}$ there exists a bundle $C_P$ of q cubic curves on the variety $\mathcal{V}=V_2^5$ having the point P in common, each curve lying in one 3-space intersecting a plane of $\mathcal{R}\backslash {\pi }_{\infty }$ . Each bundle cover the $q^2$ points of $\mathcal{V}\backslash O{O}^{\prime }$ .

2) Such cubic curves are $q^2$ .

Proof. 1) For each point $A=\left(0,a,0\right)=\left(0,0,0,a_1,a_2,a_3,0\right)\in {\pi }_{\infty }$ it is $\bar{\phi }\left(A\right):=\left(A\right)M_{\bar{\phi }}=A$ , that is, ${\pi }_{\infty }$ is pointwise fixed. For a point $B=\left(b,0,0\right)=\left(b_1,b_2,b_3,0,0,0,0\right)\in {\pi }_0$ it is $\bar{\phi }\left(B\right):=\left(B\right)M_{\bar{\phi }}=\left(b,kb,0\right)$ , that is, $\bar{\phi }\left({\pi }_0\right)={\pi }_k$ , and $\bar{\phi }\left(O\right)=O$ . Hence $\bar{\phi }\left(S_0\right)$ is a 3-space $S_k$ through O with $S_k\cap {\Sigma }^{\prime }={\pi }_k$ . The cubic ${\mathcal{C}}_0\subset S_0$ is mapped onto a cubic ${\mathcal{C}}_k\subset S_k$ with $O\in {\mathcal{C}}_k$ and ${\mathcal{C}}_k\cap {\pi }_k=\varnothing$ . Therefore there exists a bundle C₀ of q cubic curves through O collecting the $q^2$ points of $\mathcal{V}\backslash O{O}^{\prime }$ .

Let $P=\left(\mathrm{0,0,0,0,0,}h\mathrm{,1}\right)$ be a point of $O{O}^{\prime }\backslash \left\{O\mathrm{,}{O}^{\prime }\right\}$ and denote ${\tau }_h$ the associated translation. Therefore ${\tau }_h\left(O\right)=P$ and ${\tau }_h\left({\text{C}}_0\right)={\text{C}}_P$ .

2) The cardinality of $\left\{{\text{C}}_P\mathrm{<mo>\; |\; </mo>}P\in O{O}^{\prime }\backslash \left\{O^{\prime }\right\}\right\}$ is $q^2$ as for each point $P\in O{O}^{\prime }\backslash \left\{O^{\prime }\right\}$ it is $\left|{\text{C}}_P\right|=q$ and the points of $O{O}^{\prime }\backslash \left\{O^{\prime }\right\}$ are q.

Note that, chosen ${\pi }_{\infty }$ and $S_0$ , the variety $V_2^5$ selections in the spread $\mathcal{S}$ the regulus $\mathcal{R}$ to which ${\pi }_{\infty }$ and ${\pi }_0$ belong.

Denote Π the projective plane $PG\left(\mathrm{2,}{q}^3\right)$ . Represent Π in Σ, $\Pi =\left(\mathcal{P}\mathrm{,}\mathcal{L}\mathrm{,}\mathcal{I}\right)$ as in Lemma 2.6.

Denote ${\mathcal{V}}^{\prime }$ the set of the $q^2$ affine points of $\mathcal{V}=V_2^5$ .

Let ${\Pi }_q=\left({\mathcal{P}}^{\prime }\mathrm{,}{\mathcal{L}}^{\prime }\mathrm{,}{\mathcal{I}}^{\prime }\right)$ be the incidence substructure of Π defined as follows:

${\mathcal{P}}^{\prime }=\left\{P\in {\mathcal{V}}^{\prime }\right\}\cup \left\{{\pi }_{\infty }\right\}$ ,

${\mathcal{L}}^{\prime }=\left\{\mathcal{C}\in C_P\mathrm{<mo>\; |\; </mo>}P\in O{O}^{\prime }\backslash \left\{O^{\prime }\right\}\right\}\cup \mathcal{G}$ ,

${\mathcal{I}}^{\prime }$ is defined as follows

${\mathcal{I}}^{\prime }=\mathcal{I}$ restricted to the affine points and lines, ${\pi }_{\infty }{\mathcal{I}}^{\prime }g$ for all $g\in \mathcal{G}$ .

Theorem 3.6 Π_q is a non-affine subplane of Π of order q.

Proof. It is known from [8] [9] pp. 160-161 and [4] pp. 40-41, that if in an incidence structure the following four properties hold

$\left(\begin{array}{ccc}1& 3& -\\ 2& -& 6^2\end{array}\right)$

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 $\mathcal{V}$ are the affine points of the $q+1$ generatrix lines of $\mathcal{G}$ , that is, they are $q\left(q+1\right)=q^2+q$ to which the point at infinity ${\pi }_{\infty }$ has to be added. Hence $\left|{\mathcal{P}}^{\prime }\right|=q^2+q+1$ , that is, 1 - holds.

From Theorem 3.5 follows $\left|\left\{\mathcal{C}\in C_P\mathrm{<mo>\; |\; </mo>}P\in O{O}^{\prime }\backslash \left\{O^{\prime }\right\}\right\}\right|=q^2$ . As $\left|\mathcal{G}\right|=q+1$ then $\left|{\mathcal{L}}^{\prime }\right|=q^2+q+1$ , that is, 2 - holds.

Each cubic curve of C_P has as many points as ${\mathcal{C}}_0$ has, that is $q+1$ . Each generatrix line $g\in \mathcal{G}$ has q affine points and the point ad infinity ${\pi }_{\infty }$ , 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 ${\pi }_{\infty }$ . Hence 6² holds.

To end proving that Π_q is a subplane of Π it needs to verify that Π_q 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 $g\in \mathcal{G}$ is contained in a unique 3-space $S=\langle g\mathrm{,}{\pi }_{\infty }\rangle$ which meets no other generatrix (cf. Proposition 3.1, 3)) and every cubic of C_P lies in a unique 3-space (cf. Proposition 3.4, 1)) meeting ${\Sigma }^{\prime }$ in a plane of $\mathcal{R}$ (cf. Theorem 3.5, 1)).

From Theorem 3.6 follows.

Corollary 3.7 Let P, Q be two points of ${\mathcal{V}}^{\prime }$ . If PQ is not a generatrix line then P, Q belong to one directrix cubic, if $P\in {\mathcal{V}}^{\prime }$ and $Q={\pi }_{\infty }$ then the line PQ of Π_q is the generatrix $g_P$ .

The properties of Π_q 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 $V_2^5\backslash {\mathcal{C}}_{\infty }$ . Then P, Q are joined by one generatrix line or by one directrix cubic $\mathcal{C}\subset S$ , where S is a 3-space with $S\cap {\Sigma }^{\prime }={\pi }_k\in \mathcal{R}\backslash {\pi }_{\infty }$ . Every two directrix cubic curves of $V_2^5\backslash {\mathcal{C}}_{\infty }$ meet in one point.

In Theorem 3.5, 1), is shown that the variety $\mathcal{V}$ selects a regulus $\mathcal{R}$ to which both ${\pi }_{\infty }\mathrm{,}{\pi }_0$ belong. Denote ${\pi }_{\infty }\mathrm{<mo>\; :\; </mo>}={\pi }_{1_{\infty }}$ , ${\mathcal{C}}_{\infty }\mathrm{<mo>\; :\; </mo>}={\mathcal{C}}_{1_{\infty }}$ . $\mathcal{R}\mathrm{<mo>\; :\; </mo>}={\mathcal{R}}_1$ . Fix the directrix cubic curve ${\mathcal{C}}_0\subset S_0$ .

Theorem 3.9 There exists a bundle $\mathcal{B}$ of varieties $V_2^5$ with the cubic ${\mathcal{C}}_0$ as directrix, any to varieties having ${\mathcal{C}}_0$ in common, $\left|\mathcal{B}\right|=q^2$ .

Proof. The construction is done step by step, by choosing at each step a plane of the spread $\mathcal{S}$ out the regulus identified by the variety of the previous step, and a directrix conic in it.

Step 1 - Construct the variety ${\mathcal{V}}_1=V_2^5$ starting from the conic ${\mathcal{C}}_{1_{\infty }}$ and the cubic ${\mathcal{C}}_0\subset S_0$ . In $\mathcal{S}\backslash {\mathcal{R}}_1$ are $q^3-q$ possible choices for the next step.

Step 2 - Choose a plane ${\pi }_{2_{\infty }}\in \mathcal{S}\backslash {\mathcal{R}}_1$ . Fix a conic ${\mathcal{C}}_{2_{\infty }}$ in it and construct the variety ${\mathcal{V}}_2=V_2^5$ starting from the conic ${\mathcal{C}}_{2_{\infty }}$ and the cubic ${\mathcal{C}}_0\subset S_0$ . Let ${\mathcal{R}}_2$ be the regulus of $\mathcal{S}$ to which ${\pi }_{2_{\infty }}$ and ${\pi }_0$ belong. In $\mathcal{S}\backslash \left\{{\mathcal{R}}_1\mathrm{,}{\mathcal{R}}_2\right\}$ are $q^3-2q$ possible choices for the next step.

Step 3 - Choose a plane ${\pi }_{3_{\infty }}\in \mathcal{S}\backslash \left\{{\mathcal{R}}_1\mathrm{,}{\mathcal{R}}_2\right\}$ . Fix a conic ${\mathcal{C}}_{3_{\infty }}$ in it and construct the variety ${\mathcal{V}}_3=V_2^5$ starting from the conic ${\mathcal{C}}_{3_{\infty }}$ and ${\mathcal{C}}_0\subset S_0$ . Let ${\mathcal{R}}_3$ be the regulus of $\mathcal{S}$ to which ${\pi }_{3_{\infty }}$ and ${\pi }_0$ belong. In $\mathcal{S}\backslash \left\{{\mathcal{R}}_1\mathrm{,}{\mathcal{R}}_2\mathrm{,}{\mathcal{R}}_3\right\}$ are $q^3-3q$ possible choices for the next step. And so on.

The procedure ends evidently at the q²-th step. Therefore $\mathcal{B}=\left\{{\mathcal{V}}_i\mathrm{<mo>\; |\; </mo>}i=\mathrm{1,2,}\cdots \mathrm{,}{q}^2\right\}$ and $\left|\mathcal{B}\right|=q^2$ .

Conjecture - A variety $V_2^{2r-1}$ of $PG\left(2r\mathrm{,}q\right)$ represents a non-affine subplane of order q of $PG\left(\mathrm{2,}{q}^r\right)$ 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.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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