It is known that a projective translation plane Π can be represented in a projective space of even order, following the papers of André [2] , Bruck and Bose [3] ) and Vincenti [1] .

A subplane of Π is affine and non-affine depending on whether it intersects the line at infinity in a subline or in one point.

An affine subplane of order $q$ is represented by every transversal plane to the spread. All that holds also in case Π is the Desarguesian plane $PG\left(2,q^r\right)$ when the spread is a regular spread (cf. [2] - [6] for $r=2$, [1] for $r=3$).

Denote $\Pi =PG\left(2,q^r\right)$, $\Sigma =PG\left(2r,q\right)$, ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ and $\mathcal{S}$ is a regular spread of $\left(r-1\right)$-subspaces.

There exist $q^{r-1}+q^{r-2}+\cdots +q^2+q+1$ affine subplanes of $\Pi =PG\left(2,q^r\right)$ of order $q$ having the same subline at infinity and through one fixed affine point, while $q^{2r-2}$ affine subplanes having no affine point in common partition the affine points of Π (cf. Proposition 3.6, Theorem 3.7)

A variety $V_2^{2r-1}$ of Σ is a ruled variety of $PG\left(2r,q\right)$ with the minimum order directrix a rational curve of order $r-1$ and a maximum order directrix a rational curve of order $r$, the two curves lying in two complementary spaces of dimension $r-1$ and $r$, respectively (cf. [7] , Capters 13, 8., 9.). The variety can be obtained by joining points of the two directrix curves corresponding via a projectivity.

In Propositions 4.3 - 4.6 and Theorem 4.7 some fundamental incidence properties of $V_2^{2r-1}$ are shown. Such properties allow to prove that $V_2^{2r-1}$ represents a non-affine subplane ${\Pi }_q$ of order $q$ of $PG\left(2,q^r\right)$ (cf. Theorem 4.8). The properties of ${\Pi }_q$ of being a plane, translate into further incidence properties of the affine points of $V_2^{2r-1}$ (cf. Corollary 4.9).

An example is then shown by choosing $q$ and $r$ such that $\gcd \left(q-1,r\right)=d\ne 0$ (cf. Paragraph 4.2).

In Theorem 5.2 a maximal bundle $ℬ$ of varieties $V_2^{2r-1}$ having in common only a curve of order $r$ is constructed.

To conclude, linear codes are associated with the projective systems related both to a variety $V_2^{2r-1}$ and to the bundle $ℬ$, then their basic parameters are calculated (cf. Proposition 5.1, Theorems 5.3 - 5.5).

Note that a part of Section 3 is necessarily common with previous articles, this representing a generalization as announced in the abstract.

Referring to the Section 2 of [1] , denote $F=GF\left(q\right)$ a finite field, $q=p^s$, $p$ an odd prime, $\bar{F}$ the algebraic closure of the field $F$, $F^{n+1}$ the $\left(n+1\right)$-dimensional vector space over $F$, $PG\left(n,q\right)=Pr{F}^{n+1}$ the n-dimensional projective space contraction of $F^{n+1}$ over $F$. It is considered a sub-geometry of $\overline{PG\left(n,q\right)}$, the projective geometry over $\bar{F}$. A subspace of $PG\left(n,q\right)$ of dimension $h$ is denoted h-space.

For the Definition of a variety $V_u^v$ of dimension $u$ and order $v$ of $PG\left(n,q\right)$ see [1] , Definition 2.1.

From [7] , p. 290, 7., follows the definition of a ruled variety $V_2^{n-1}$ of $PG\left(n,q\right)$ (cf. Lemma 2.2 of [1] ).

Let Σ be the projective space $PG\left(2r,q\right)$, ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ a hyperplane of Σ, $\mathcal{S}$ a spread of $\left(r-1\right)$-spaces of ${\Sigma }^{\prime }$ (for the definition of spread, regulus and regular spread cf. [3] and [1] , Definition 2.3 and the representation).

A transversal line $l$ to $\mathcal{S}$ is a line of ${\Sigma }^{\prime }$ such that $l\notin S$ for every $S\in \mathcal{S}$. As $\mathcal{S}$ is regular, then the line $l$ meets $q+1$ subspaces of $\mathcal{S}$ consisting of a regulus (cf. [1] , Definition 2.3).

For the following definitions and results, see [8] and [9] .

Definition 2.1 A linear ${\left[n,k\right]}_q$-code $C$ of length $n$ is a k-dimensional subspace of the vector space $F^n$. The dual code of $C$ is the $\left(n-k\right)$-dimensional subspace $C^{\perp }$ of $F^n$ and it is an ${\left[n,n-k\right]}_q$-code.

For $t\ge 1$ the t-th higher weight of $C$ is defined by

$d_t=d_t\left(C\right)=\text{min}\left\{\Vert D\Vert \text{for}\text{all}D<C,\dim D=t\right\},$

where $\Vert D\Vert$ is the number of indices $i$ such that there exists $v\in D$ with $v_i\ne 0$.

Note that $d_1=d_1\left(C\right)$ is the classical minimum distance of $C$, the Hamming distance.

An ${\left[n,k\right]}_q$-code $C$ of minimum distance $d$ is also denoted ${\left[n,k,d\right]}_q$-code.

Definition 2.2 An ${\left[n,k\right]}_q$-projective system $\mathcal{X}$ of the projective space $PG\left(k-1,q\right)$ is a collection of $n$ not necessarily distinct points. It is called non-degenerate if these $n$ points are not contained in any hyperplane.

Assume that $\mathcal{X}$ consists of $n$ distinct points having rank $k$.

For each point of $X$ choose a generating vector. Denote by $M$ the matrix having as rows such $n$ vectors and let $C_{\mathcal{X}}$ be the linear code having $M^t$ as a generator matrix. The code $C_{\mathcal{X}}$ is the k-dimensional subspace of $F^n$ which is the image of the mapping from the dual k-dimensional space ${\left(F^k\right)}^{\text{*}}$ onto $F^n$ that calculates every linear form over the points of $X$.

Hence the length $n$ of codeword of $C_{\mathcal{X}}$ is the cardinality of $X$, the dimension of $C_{\mathcal{X}}$ being just $k$.

There exists a natural [1-1] correspondence between the equivalence classes of a non-degenerate ${\left[n,k\right]}_q$-projective system $X$ and a non-degenerate ${\left[n,k\right]}_q$-code $C_{\mathcal{X}}$ such that if $X$ is an ${\left[n,k\right]}_q$-projective system and $C_{\mathcal{X}}$ is the corresponding code, then the non-zero codewords of $C_{\mathcal{X}}$ correspond to hyperplanes of $PG\left(k-1,q\right)$, up to a non-zero factor, the correspondence preserving the parameters $n,k,d_t$.

Generally, subcodes $D$ of $C_{\mathcal{X}}$ of dimension $r$ correspond to (projective) subspaces of codimension $r$ of $PG\left(k-1,q\right)$, therefore $d_1=d_1\left(C_{\mathcal{X}}\right)=n-\text{max}\left\{\left|X\cap H\right|:H<PG\left(k-1,q\right),\text{codim}H=1\right\}$.

If $d$ is the minimum weight of a linear code $C_{\mathcal{X}}$ then $C_{\mathcal{X}}$ is an s-error-correcting code for all $s\le \lfloor \frac{d-1}{2}\rfloor$. We call $\lfloor \frac{d-1}{2}\rfloor$ the error-correcting capability of $C_{\mathcal{X}}$.

From now on denote $\Sigma =PG\left(2r,q\right)$ the 2r-dimensional geometry over the field $F=GF\left(q\right)$, ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ a hyperplane of Σ, $\mathcal{S}$ a regular spread of $\left(r-1\right)$-spaces of ${\Sigma }^{\prime }$. Clearly $\left|\mathcal{S}\right|=q^r+1$.

Let $\Pi =PG\left(2,q^r\right)$ be the Desarguesian plane over the field $GF\left(q^r\right)$. Denote $l_{\infty }$ the line at infinity of Π. Represent Π in $\Sigma =PG\left(2r,q\right)$ by the spread $\mathcal{S}$.

Define the following incidence structure (points, lines, incidence, respectively) where

,

$ℒ=\left\{{ℒ}_0=\left\{S_r\subset \Sigma \backslash {\Sigma }^{\prime }|S_r\cap {\Sigma }^{\prime }\in \mathcal{S}\right\}\right\}\cup \left\{l_{\infty }=\mathcal{S}\right\}$,

$ℐ$ is defined as follows

If $P\in \Sigma \backslash {\Sigma }^{\prime },l\in {ℒ}_0$ then $Pℐl\iff P\in l$, no point of $\Sigma \backslash {\Sigma }^{\prime }$ is incident $l_{\infty }$, $S_{r-1}ℐl_{\infty }$ for all $S_{r-1}\in \mathcal{S}$, $S_{r-1}ℐl$ where $l\in {ℒ}_0\iff l\cap {\Sigma }^{\prime }=S_{r-1}$.

Lemma 3.1 ${\Pi }^{\prime }\cong \Pi$.

Proof. See [3] .

From [1] and [3] , Definitions 2.7, 2.8, Propositions 2.9, referred to the current dimension, follows that the affine points of Π are represented by the $q^{2r}$ affine points of $\Sigma \backslash {\Sigma }^{\prime }$, the points at infinity by the $q^r+1$ subspaces of $\mathcal{S}$. The affine lines of Π are represented by the r-spaces $S_r$ of $\Sigma \backslash {\Sigma }^{\prime }$ such that the subspaces $S_{r-1}=S_r\cap {\Sigma }^{\prime }$ belong to $\mathcal{S}$, the line at infinity $l_{\infty }$ by the spread $\mathcal{S}$.

Definition 3.2 A subplane ${\pi }^{\prime }=\left(P^{\prime },L^{\prime },I^{\prime }\right)$ of a plane $\pi =\left(P,L,I\right)$ is a subgeometry of $\pi$, that is, an incidence substructure for which $P^{\prime }\subset P$, for each line $l^{\prime }\in L^{\prime }$ there exists a line $l\in L$ such that $l^{\prime }\subset L$ and $I^{\prime }=I$.

Definition 3.3 A subplane of $\Pi =PG\left(2,q^r\right)$ 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 $t$ be any transversal line to $\mathcal{S}$, that is a line meeting $q+1$ $\left(r-1\right)$-spaces of $\mathcal{S}$. As $\mathcal{S}$ is regular, these $q+1$ elements form a regulus $ℛ\subset \mathcal{S}$ (cf. [3] , Lemma 12.2). Choose and fix a plane $\alpha \subset \Sigma \backslash {\Sigma }^{\prime }$ through the line $t$, that is, a transversal plane.

As in Proposition 2.9 of [1] , one can easily prove

Proposition 3.4 The plane $\alpha$ is isomorphic to a subplane $\pi \cong PG\left(2,q\right)$ of Π whose points at infinity are represented by the $q+1$ $\left(r-1\right)$-spaces of $ℛ$, the lines of $\pi$ being represented by the sublines intersections of $\alpha$ with the r-spaces of $\Sigma$ through the $\left(r-1\right)$-spaces of $ℛ$. As the line at infinity of $\pi$ is a subline of the infinity line of Π, then $\pi$ is an affine subplane.

For the construction of transversal lines to $ℛ$ in ${\Sigma }^{\prime }$ the procedure is similar to the one used for the dimension 5 (cf. [1] , Proposition 2.10).

Proposition 3.5 The set of the transversal lines to $ℛ$ has cardinality $q^{r-1}+q^{r-2}+\cdots +q^2+q+1$, that is, they are as many as the points of an $\left(r-1\right)$-subspace of $ℛ$.

Proof. Denote $S_1,S_2,S_3$ three $\left(r-1\right)$-subspaces of the regulus $ℛ$. Fix a point $P\in S_1$ and denote $S=\langle P,S_2\rangle$ the r-space of ${\Sigma }^{\prime }$ direct sum of $P$ and $S_2$ and $S^{\prime }=\langle P,S_3\rangle$ the r-space of ${\Sigma }^{\prime }$ direct sum of $P$ and $S_3$. Lying in a $\left(2r-1\right)$-dimensional subspace, then $S\cap S^{\prime }=t$ is a line. As a line of $S$, $t$ meets $S_2$ in a point, as a line of $S^{\prime }$, $t$ meets $S_3$ in a point. Therefore $t$ is a transversal line to the subspaces $S_1,S_2,S_3$. As $S_1,S_2,S_3$ belong to the regulus $ℛ$, the line $t$ 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 $S_1$, that is, $q^{r-1}+q^{r-2}+\cdots +q^2+q+1$.

Proposition 3.6 The cardinality of the affine subplanes of Π isomorphic to $PG\left(2,q\right)$ having the same subline of $q+1$ points at infinity and containing one affine point is $q^{r-1}+q^{r-2}+\cdots +q^2+q+1$.

Proof. Let $r_0$ be a transversal line to $ℛ$, ${\alpha }_0\supset r_0$ a transversal plane, $O$ an affine point of ${\alpha }_0$. Denote $\left\{t_i|i=0,\cdots ,q^{r-1}+q^{r-2}+\cdots +q^2+q\right\}$ the transversal lines to the regulus $ℛ$. Each of the $q^{r-1}+q^{r-2}+\cdots +q^2+q+1$ planes ${\alpha }_i=\langle O,r_i\rangle$ represents an affine subplane ${\pi }_i$ of Π, ${\pi }_i\cong PG\left(2,q\right)$ (cf. Proposition 3.4).

Choose and fix a transversal line $t$. Consider the bundle $\left(t\right)$ of the planes of $\Sigma \backslash {\Sigma }^{\prime }$ having the line $t$ as axis. Each plane $\alpha \in \left(t\right)$ is isomorphic to $PG\left(2,q\right)$ (cf. Proposition 3.4) and it is an affine subplane of Π having a same subline of $q+1$ points on the line at infinity.

Theorem 3.7 The planes of $\left(t\right)$ are $q^{2r-2}$ and partition the $q^{2r}$ affine points of Π.

Proof. The planes of $\left(t\right)$ are parallel to each other, therefore they have no affine point in common otherwise they would coincide. Each such a plane contains $q^2$ affine points.

Let $h=\left|\left(t\right)\right|$. As a line and an independent point define a plane, fixed the line $t$, there are $q^{2r}$ choices for a point in $\Sigma \backslash {\Sigma }^{\prime }$ to get the plane $\langle t,P\rangle \subset \Sigma \backslash {\Sigma }^{\prime }$, this number to be divided by $q^2$, which equals the choices of an affine point on a same plane, hence $h=q^{2r-2}$.

In $\text{Σ}=PG\left(2r,q\right)$ consider two normal rational curves ${\mathcal{C}}^m$ and ${\mathcal{C}}^{2r-m-1}$ of order $m$ and $2r-m-1$, respectively in two complementary subspaces $S_m$ and $S_{2r-m-1}$ of Σ. Each of them consists of $q+1$ points (cf. [10] , Theorem 21.1.1). They are projectively equivalent. From Lemma 1 follows that a ruled rational surface $V_2^{2r-1}$ of order $2r-1$ of $PG\left(2r,q\right)$ is generated by connecting corresponding points of the two directrices ${\mathcal{C}}^m$ and ${\mathcal{C}}^{2r-m-1}$ of $V_2^{2r-1}$, (cf. [7] , p. 290, 7.). The variety consists of $q+1$ skew lines and ${\left(q+1\right)}^2$ points.

Choose and fix $m=r-1$ so that $2r-m-1=r$.

For our purpose to choose appropriately a directrix ${\mathcal{C}}^r$ in an r-dimensional subspace of Σ, some considerations have to be made.

It is well known that a rational normal curve ${\mathcal{C}}^r$ of order $r$ of an r-dimensional geometry $PG\left(r,q\right)$ can be defined by $r+1$ independent binary forms of order $r$, $g_i\left(s_0,s_1\right)\in F\left[s_0,s_1\right]$, $i=0,1,\cdots ,r$, or by $r+1$ functions $f_i\left(s\right):=g_i\left(1,s\right)$ where at least one of $f_i$ has degree $r$. Moreover it must be $q\ge r$ (cf. [10] , p. 229).

A hyperplane of the geometry $PG\left(r,q\right)$ meets ${\mathcal{C}}^r$ in at most $r$ points, corresponding to the solutions of an equation of degree $r$ over $F=GF\left(q\right)$.

The orbits of the hyperplanes under the action of the group of the projectivities of $PG\left(r,q\right)$ fixing ${\mathcal{C}}^r$, correspond just to such possibilities (for $r=3$ cf. [10] , pp. 229-230, and p. 234, Corollary 4, ${\mathcal{N}}_5$).

For our construction, we need an r-curve having no point in the hyperplane of $PG\left(r,q\right)$ chosen as hyperplane at infinity. Therefore it needs to find irreducible polynomials over $F$ of degree $r$. Two ways are indicated in [11] and in [12] .

However we show an example of what is written above.

Let us introduce coordinates $\left(x_0,x_1,\cdots ,x_r\right)$ in $PG\left(r,q\right)$ so that a curve ${\mathcal{C}}^r$ of order $r$ can be expressed as follows ${\mathcal{C}}^r=\left\{\left(1,s,s^2,\cdots ,s^{\left(r-1\right)},f_r\left(s\right)\right)|s\in F^{+}\right\}$, where $f_r\left(s\right)$ is an irreducible polynomial of degree $r$ (for the symbology see [10] , p. 229).

Example 4.1 The curve ${\mathcal{C}}^r=\left\{\left(1,s,s^2,\cdots ,s^{\left(r-1\right)},f_r\left(s\right)\right)|s\in F^{+}\right\}$ and the hyperplane $x_r=0$ have no point in common.

Another way to find irreducible polynomials of a given degree, is obtained by considering the problem of searching in $F^{\text{*}}=F\backslash \left\{0\right\}$ the elements that are not r-th powers.

Given $q=p^n$, $p$ a prime, a field $F=GF\left(q\right)$ and a positive integer $r$, denote $d=\gcd \left(r,q-1\right)$, the great common divisor.

Lemma 4.2 The subset $N_r\subset F^{*}$ of the non-r-th powers has cardinality $\frac{\left(q-1\right)\left(d-1\right)}{d}$, so that if $d\ne 1$ each polynomial $x^r-h\in F\left[x\right]$ with $h\in N_r$ is irreducible over $F$.

Proof. Denote $\phi :F^{\text{*}}\to F^{\text{*}}$ the mapping $\phi :x\mapsto x^r$. Set $\ker \phi =\left\{x\in F^{*}|x^r=1\right\}$, the subset of the r-th roots of unity. Then

$\left|\ker \phi \right|=\gcd \left(r,q-1\right)=d$. Hence $\phi \left(F^{*}\right)\cong \frac{F^{*}}{\ker \phi }$ so that in $F^{\text{*}}$ there are $\frac{\left|F^{*}\right|}{\left|\ker \phi \right|}=\frac{q-1}{d}$ elements that are r-th powers. If $d\ne 1$, then the complementary set $N_r=F^{\text{*}}\backslash \phi \left(F^{\text{*}}\right)$ of the elements that are not r-th powers has cardinality $\frac{\left(q-1\right)\left(d-1\right)}{d}$, hence every polynomial $x^r-h$ with $h\in N_r$ is irreducible over $F$.

NOTE 1—A rational normal curve ${\mathcal{C}}^r$ of an r-space consists of $q+1$ points ( $q\ge r$) no $r+1$ of which in a hyperplane $S_{r-1}$ (that is, a hyperplane meets ${\mathcal{C}}^r$ in at most $r$ points, cf. [10] , p. 229, Theorem 21.1.1, (iv)). Hence $r$ points lie in no $S_{r-2}$, $r-1$ points in no $S_{r-3}$.

Choose and fix an $\left(r-1\right)$-space $S_{r-1}^{\infty }\in \mathcal{S}$ and a rational normal curve ${\mathcal{C}}_{\infty }^{r-1}\subset S_{r-1}^{\infty }$ of order $r-1$. Let $S_r^0$ be an r-dimensional subspace of $\Sigma \backslash {\Sigma }^{\prime }$ such that $S_r^0\cap {\Sigma }^{\prime }=S_{r-1}^0\in \mathcal{S}\backslash S_{r-1}^{\infty }$ and ${\mathcal{C}}_0^r\subset S_r^0$ a rational normal curve of it of order $r$ with ${\mathcal{C}}_0^r\cap S_{r-1}^0=\varnothing$.

Let $\Lambda :{\mathcal{C}}_{\infty }^{r-1}\to {\mathcal{C}}_0^r$ be a projectivity. Represent ${\mathcal{C}}_{\infty }^{r-1}=\left\{G_{i_{\infty }}|i=1,\cdots ,q+1\right\}$, ${\mathcal{C}}_0^r=\left\{G_i=\Lambda G_{i_{\infty }}|i=1,\cdots ,q+1\right\}$. Denote $V_2^{2r-1}$ the variety arising by connecting corresponding points of ${\mathcal{C}}_{\infty }^{r-1}$ and ${\mathcal{C}}_0^r$ via Λ (cf. [7] , p. 291). The curves ${\mathcal{C}}_{\infty }^{r-1}$ and ${\mathcal{C}}_0^r$ are directrix curves of $V_2^{2r-1}$, 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 $V_2^{2r-1}$. The set $\mathcal{G}$ partitions the variety.

Let $H$ be any hyperplane. In a suitable complexification of Σ, $H\cap V_2^{2r-1}$ is a curve of order $2r-1$ (cf. [7] , p. 288, 5.).

Proposition 4.3 The variety $V_2^{2r-1}$ consists of $q+1$ mutually skew affine generatrix lines and of $q^2+q$ affine points.

a) A directrix curve $\mathcal{C}\ne {\mathcal{C}}_{\infty }^{r-1}$ cut by a hyperplane on $V_2^{2r-1}$ cannot lie in an $\left(r-1\right)$-space. The curve ${\mathcal{C}}_{\infty }^{r-1}$ is the unique minimum order $\left(r-1\right)$ directrix.

If a space $S$ contains $r$ points of ${\mathcal{C}}_{\infty }^{r-1}$, then $S\supset {\mathcal{C}}_{\infty }^{r-1}$.

Moreover $k\le r$ generatrix lines are independent and belong to a $\left(2k-1\right)$-space.

b) An r-space containing $S_{r-1}^{\infty }$ contains at most one generatrix line.

c) The r-space joining one generatrix line and the $\left(r-1\right)$-space $S_{r-1}^{\infty }$ meets no other generatrix in an affine point.

d) $r$ generatrices $\left\{g_i|i=1,\cdots ,r\right\}$ are joint by a hyperplane $H$ that contains the $\left(r-1\right)$-space $S_{r-1}^{\infty }$, so that $H\cap V_2^{2r-1}=\left\{g_i|i=1,\cdots ,r\right\}\cup {\mathcal{C}}_{\infty }^{r-1}$.

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

Proof. The proof of the first statement is analogous to that of Proposition 3.1 of [1] .

a) Assume a hyperplane $H$ meets $V_2^{2r-1}$ in a directrix curve ${\mathcal{C}}^{r-1}\ne {\mathcal{C}}_{\infty }^{r-1}$ lying in a $\left(r-1\right)$-space $S$. Then $V_2^{2r-1}$ is contained at most in the $\left(2r-1\right)$-space generated by $S$ and $S_{r-1}^{\infty }$ and the variety generated by the two curves would have order at most $2r-2$, a contradiction. Hence the curve ${\mathcal{C}}_{\infty }^{r-1}$ is the unique minimum order $r-1$ directrix.

For the proof of the last two statements see [7] , 5., 6., pp. 288-289.

b) Assume $S$ is an r-space containing $S_{r-1}^{\infty }$ and two generatrix lines $g_1,g_2$. Denote $G_i=g_i\cap {\mathcal{C}}_0^r$, $i=1,2$ then the line $G_1{G}_2$ belongs to both $S_r^0$ and $S$ so that the point $G=G_1{G}_2\cap {\Sigma }^{\prime }$ is a common point of $S_{r-1}^{\infty }$ and $S_{r-1}^0$, a contradiction.

c) Denote $S=\langle g,S_{r-1}^{\infty }\rangle$ with $g\in \mathcal{G}$, an r-space. Assume that for $g^{\prime }\in \mathcal{G}\backslash \left\{g\right\}$ with $g^{\prime }\cap {\mathcal{C}}_{\infty }^{r-1}={G^{\prime }}_{\infty }$ is $S\cap g^{\prime }\backslash {G^{\prime }}_{\infty }\ne \varnothing$. Then $g^{\prime }\subset S$, so that $S$ contains two generatrix lines and the $\left(r-1\right)$-space $S_{r-1}^{\infty }$, a contradiction to b).

d) Assume $r$ generatrices $\left\{g_i|i=1,\cdots ,r\right\}$ are joint by a $\left(2r-2\right)$-space $S^{\prime }$. As $S^{\prime }$ contains the $r$ independent points $G_{i\infty }=g_i\cap {\mathcal{C}}_{\infty }^{r-1}$, $i=1,\cdots ,r$, $G_{i_{\infty }}\in S_{r-1}^{\infty }$, then $S_{r-1}^{\infty }\subset S^{\prime }$ and ${\mathcal{C}}_{\infty }^{r-1}\subset S^{\prime }$. As $S^{\prime }$ cannot contain $V_2^{2r-1}$, a hyperplane $H\supset S^{\prime }$ and through a further point $P\in V_2^{2r-1}\backslash S^{\prime }$ should contain also the generatrix $g_P$ through $P$. Hence $H$ would meet $V_2^{2r-1}$ in $\left(r+1\right)$ generatrix lines and in a curve of order $r-1$, that is, in a curve of order $2r$, a contradiction (cf. [7] , p. 288, 5.). Hence $H\cap V_2^{2r-1}=\left\{g_i|i=1,\cdots ,r\right\}\cup {\mathcal{C}}_{\infty }^{r-1}$, that is, a curve of order $2r-1$ (and $H$ contains no further point of $V_2^{2r-1}$).

e) Let ${\mathcal{G}}_{r-1}=\left\{g_i|i=1,\cdots ,r-1\right\}$ be a subset of $r-1$ generatrices of $V_2^{2r-1}$. Denote $S=S_{2\left(r-1\right)-1}=S_{2r-3}$ the subspace containing ${\mathcal{G}}_{r-1}$ (cf. [7] , 6., p. 289). Let $H$ be a hyperplane with $H\supset S$ and assume $H$ contains a residual and fix directrix curve $\mathcal{C}$ of order $r$. Let $P$ be a point of $V_2^{2r-1}$, $P\in H\backslash \mathcal{C}$. Denote $S^{″}=\langle S,P\rangle$. Then every hyperplane containing $S^{\prime }$ and $S^{\prime }$ itself, would contain the generatrix $g_P$ through $P$, so that ${\mathcal{G}}_{r-1}\cup \left\{g_P\right\}\subset S^{\prime }$, a contradiction to d).

An analogous contradiction is reached if we assumed a generic hyperplane $H$ with $g,g^{\prime }\subset H$ contained a fix generatrix (cf. [7] , 6., pp. 289-290).

From [7] , pp. 287-290 follows

Proposition 4.4 A hyperplane $H$ containing $r-1$ generatrices contains a residual curve $\mathcal{C}$ of order $r$ of an r-space $S\subset H$. Moreover $S$ is skew to $S_{r-1}^{\infty }$, $\mathcal{C}$ is irreducible and is a directrix.

Proof. A hyperplane $H$ meets $V_2^{2r-1}$ in a rational normal curve of order $2r-1$ or in a curve of order $m<2r-1$ met by all generatrices and in $2r-1-m$ generatrices. In the current case $m=r$ or $m=r-1$ can happen.

If a hyperplane meets $V_2^{2r-1}$ in ${\mathcal{C}}_{\infty }^{r-1}$, the unique directrix curve of order $r-1$ (see a), Proposition 4.3), then it contains $2r-1-\left(r-1\right)=r$ generatrix lines and viceversa (see d), Proposition 4.3). If a hyperplane contains $2r-1-r=r-1$ generatrices, then it meets $V_2^{2r-1}$ in a residual curve $\mathcal{C}$ of order $r$.

Assume $\mathcal{C}$ irreducible and contained in an $\mu$-space $S_{\mu }$, with $\mu <r$. Let $H$ be the hyperplane containing $S_{\mu }$ and $2r-1-\mu$ points $P_i$ of $V_2^{2r-1}$ and then also the $2r-1-\mu$ generatrix lines $g_{P_i}$. In such a case $H$ would meet $V_2^{2r-1}$ in a curve of order $r+2r-1-\mu >2r-1$, a contradiction. Hence each curve $\mathcal{C}$ irreducible of order $r$, lives in an r-space $S$ and it is a directrix curve, that is, meets each generatrix in one point (cf. [7] , 3. p. 287). If such an r-space $S$ met $S_{r-1}^{\infty }$, then a hyperplane $H\supseteq \langle S,S_{r-1}^{\infty }\rangle$ would contain $V_2^{2r-1}$, a contradiction. Hence $S\cap S_{r-1}^{\infty }=\varnothing$.

Assume $\mathcal{C}$ is reducible. The unique possibility is that it consists of $r$ generatrix lines. Let $H$ and $H^{\prime }$ be two hyperplanes. Assume $H$ has in common with $V_2^{2r-1}$ $r$ generatrices and $H^{\prime }$ has in common with $V_2^{2r-1}$ other $r$ generatrices. Denote $S_{2r-2}=H\cap H^{\prime }$. By varying the hyperplanes in the bundle $\left(S_{2r-2}\right)$ of hyperplanes, both each hyperplane and the space $S_{2r-2}$ itself would have in common with $V_2^{2r-1}$ the locus of all these points. Such a locus would be a directrix contained in all the hyperplanes of the bundle. Therefore such a directrix curve should exist in all the hyperplanes of the bundle, a contradiction to Proposition 4.3, e) (cf. [7] , 6. p. 290).

Proposition 4.5 a) Each directrix curve of order $r$ is obtained by cutting $V_2^{2r-1}$ with the hyperplanes through any $r-1$ generatrices.

b) The cardinalities of the intersections of hyperplanes $H$ with $V_2^{2r-1}$ are $q+1,rq+1,\left(r+1\right)q+1$. It is $\max \left\{\left|H\cap V_2^{2r-1}\right|:H\text{hyperplane}\right\}=\left(r+1\right)q+1$.

c) The cardinalities of the intersections of hyperplanes $H$ with $V_2^{2r-1}\backslash {\mathcal{C}}_{\infty }^{r-1}$ are $\le q+1,r\left(q-1\right)+2,rq$. It is $\max \left\{\left|H\cap V_2^{2r-1}\backslash {\mathcal{C}}_{\infty }^{r-1}\right|:H\text{hyperplane}\right\}=rq$.

Proof. a) An irreducible curve $\mathcal{C}$ of order $r$ is a rational normal curve, that is, it lies in an $r$-space (cf. Proposition 4.4).

Let $\mathcal{C}\subset V_2^{2r-1}$ be a directrix curve of order $r$ and $H\supset \mathcal{C}$ a hyperplane. As $H$ cannot contain ${\mathcal{C}}_{\infty }^{r-1}$ otherwise $H\supset V_2^{2r-1}$, then $H$ must contain $r-1$ generatrix lines.

b) Let $H$ be a hyperplane. If $H\cap V_2^{2r-1}$ is an irreducible curve of order $2r-1$, then $\left|H\cap V_2^{2r-1}\right|=q+1$.

If $H\cap V_2^{2r-1}=\left\{g_1,\cdots ,g_{r-1},{\mathcal{C}}^r\right\}$, $g_i\in \mathcal{G}$ (cf. a)), then $\left|H\cap V_2^{2r-1}\right|=rq+1$.

If $H\cap V_2^{2r-1}=\left\{g_1,\cdots ,g_r,{\mathcal{C}}_{\infty }^{r-1}\right\}$, $g_i\in \mathcal{G}$ (cf. d), Proposition 4.3), then $\left|H\cap V_2^{2r-1}\right|=\left(r+1\right)q+1$.

That is we get the following possibilities: $q+1,rq+1,\left(r+1\right)q+1$.

It is easy to prove that $\left(r+1\right)q+1=\max \left\{\left|H\cap V_2^{2r-1}\right|:H\text{hyperplane}\right\}$.

c) Let $H$ be a hyperplane. If $H\cap V_2^{2r-1}$ is an irreducible curve of order $2r-1$, then $\left|H\cap V_2^{2r-1}\backslash {\mathcal{C}}_{\infty }^{r-1}\right|\le q+1$, depending on whether it has or does have not points on ${\Sigma }^{\prime }$.

If $H\cap V_2^{2r-1}=\left\{g_1,\cdots ,g_{r-1},{\mathcal{C}}^r\right\}$, $g_i\in \mathcal{G}$ (cf. a)), then $\left|H\cap V_2^{2r-1}\backslash {\mathcal{C}}_{\infty }^{r-1}\right|=\left(q-1\right)\left(r-1\right)+q+1=r\left(q-1\right)+2$.

If $H\cap V_2^{2r-1}=\left\{g_1,\cdots ,g_r,{\mathcal{C}}_{\infty }^{r-1}\right\}$, $g_i\in \mathcal{G}$ (cf. d), Proposition 4.3), then $\left|H\cap V_2^{2r-1}\backslash {\mathcal{C}}_{\infty }^{r-1}\right|=rq$.

That is, we get the following possibilities: $\le q+1,r\left(q-1\right)+2,rq$.

It is easy to prove that $rq=\max \left\{\left|H\cap V_2^{2r-1}\backslash {\mathcal{C}}_{\infty }^{r-1}\right|:H\text{hyperplane}\right\}$.

Proposition 4.6 a) No two directrix curves $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$ of order $r$ belong to a same r-space.

b) Two directrix curves of order $r$ meet in one point.

Proof. a) Assume $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$ belong to a same r-space $S$. Then a hyperplane $H\supset S$ would meet $V_2^{2r-1}$ in a curve of order at least $2r$, a contradiction.

b) Let $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$ contained in two different r-spaces, $S$ and $S^{\prime }$, respectively. Assume $\mathcal{C}\cap {\mathcal{C}}^{\prime }$ contains two different points, $P$ and $Q$. Then $S\cap S^{\prime }\supset PQ$ so that the hyperplane $H=\langle S,S^{\prime }\rangle$ meets $V_2^{2r-1}$ in a curve of order $2r$, a contradiction. If $\mathcal{C}\cap {\mathcal{C}}^{\prime }=\varnothing$, then by connecting corresponding points, $V_2^{2r-1}$ would contain a variety of order $2r$, a contradiction.