Rita Vincenti
Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy.
DOI: 10.4236/ojdm.2025.154005   PDF    HTML   XML   82 Downloads   433 Views   Citations

Abstract

This note investigates the hyperplane sections of a ruled variety $V_2^{2r-1}$ embedded in $PG\left(2r,q\right)$ , which yield caps associated with specific arcs in $PG\left(2,q\right)$ . We construct a partition of the affine points of $V_2^{2r-1}$ into caps, corresponding to a partition of the affine plane $PG\left(2,q\right)$ into conics.

Keywords

Finite Geometry, Translation Planes, Spreads, Varieties, Caps

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1. Introduction

In the André/Bruck and Bose representation of $PG\left(2,q^2\right)$ in $PG\left(4,q\right)$ (cf. [1] [2]) a non-affine Baer subplane $B$ corresponds to a ruled variety $V_2^3$ (cf. [3] [4]). In [5] is proved that a non degenerate conic in $B$ is a rational normal curve of $V_2^3$ .

Using that technique, in [6] is studied the representation of the projective plane $PG\left(2,q^r\right)$ in $PG\left(2r,q\right)$ and of a non-affine subplane $PG\left(2,q\right)$ in a variety $V_2^{2r-1}$ .

More precisely, if $\text{Π}=PG\left(2,q^r\right)$ with kernel $F=GF\left(q\right)$ , then it can be represented in a $2r$ -dimensional projective space $\text{Σ}=PG\left(2r,q\right)$ , by fixing a hyperplane ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ and a spread $\mathcal{S}$ of ${\Sigma }^{\prime }$ consisting of $\left(r-1\right)$ -dimensional subspaces, with $\left|\mathcal{S}\right|=q^r+1$ .

The affine points of $\text{Π}$ correspond to the points of $\Sigma \backslash {\Sigma }^{\prime }$ , the points at infinity correspond to the elements of $\mathcal{S}$ , and the affine lines are represented by the $r$ -subspaces $S_r$ of $\text{Σ}$ such that $S_r\cap {\Sigma }^{\prime }\in \mathcal{S}$ . The line at infinity is represented by the spread $\mathcal{S}$ itself. If $\text{Π}$ is Desarguesian, then the spread $\mathcal{S}$ is regular (cf. [1] [2], see also [4], and [3] for $r=2$ , [7] for $r=3$ , and [6] for the general case).

A subplane of $\text{Π}$ is affine or non-affine (also referred to as tangent) depending on whether it intersects the line at infinity in a subline or in one point, respectively.

An affine subplane of order $q$ is represented by a transversal plane to the spread; that is, a plane of $\text{Σ}$ intersecting $q+1$ elements of the spread.

A non-affine subplane $\pi$ of $\text{Π}=PG\left(2,q^r\right)$ of order $q$ is represented by a variety $V_2^{2r-1}$ , a ruled variety in $\text{Σ}=PG\left(2r,q\right)$ whose minimum order directrix a rational curve of order $r-1$ and with a maximum order directrix is a rational curve of order $r$ . These two curves lie in two complementary spaces of dimension $r-1$ and $r$ , respectively. The variety $V_2^{2r-1}$ can be obtained by joining corresponding points on the two directrix curves via a projectivity (cf. [8], Cap.13, 8., 9. and [6], Section 4).

Building on the results obtained in [5], Theorems 3.1 and 3.2 for $r=2$ , this note studies a generalization for $r>2$ . Several properties of the hyperplanes of $\text{Σ}$ and their intersections with the variety $V_2^{2r-1}$ are established, an essential step in demonstrating how to represent substructures of the plane $\pi$ in the space $\text{Σ}$ (cf. Subsection 3.1) and viceversa (cf. Subsection 3.2), specifically certain types of arcs and caps.

In Theorem 3.7 is shown how to construct in $V_2^{2r-1}$ a rational normal curve of order $r+1$ that represents a conic in $\pi$ . The paper concludes with Theorem 3.8 which presents a partition of the affine points of $V_2^{2r-1}$ into caps corresponding to a partition of $\pi$ into conics.

2. Preliminary Notes and Results

Denote $F=GF\left(q\right)$ a finite field, $q=p^s$ , $p$ an odd prime, $\overline{F}$ 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 $F$ . The geometry $PG\left(n,q\right)$ is considered a sub-geometry of $\overline{PG\left(n,q\right)}$ , the projective geometry over $\overline{F}$ . A subspace of $PG\left(n,q\right)$ of dimension $h$ (an $h$ -space) is denoted by $S_h$ (cf. [7], Section 2), possibly with an apex used when needed to distinguish between different subspaces.

Definition 2.1. A $k$ -arc $\mathcal{K}$ in $PG\left(n,q\right)$ is a set of $k\ge n+1$ points no $n+1$ of which are lie in a hyperplane.

A $k$ -cap $\mathcal{K}$ of $PG\left(n,q\right)$ , $n\ge 3$ is a set of $k$ points no three of which are collinear.

A tangent of $\mathcal{K}$ is a line which has exactly one point in common with $\mathcal{K}$ .

See Thas [9].

A curve of order $r$ is denoted ${\mathcal{C}}^r$ , possibly with a subscript used when needed to distinguish between different curves.

Definition 2.2. A rational normal curve ${\mathcal{C}}^n$ of $PG\left(n,q\right)$ consists of $q+1$ points ($q\ge n$ ) no $n+1$ of which in a hyperplane $S_{n-1}$ (that is, a hyperplane meets ${\mathcal{C}}^n$ in at most $n$ points).

See Hirschfeld [10] p. 229, Theorem 21.1.1, (iv).

Consequence—No set of $n$ points lie in an $S_{n-2}$ ; no set of $n-1$ points lies in an $S_{n-3}$ ; and so on, down to the fact that no three points lie on a line. That is, an $S_{n-2}$ meets the curve in at most $n-1$ points, an $S_{n-3}$ in $n-2$ points,..., a line in 2 points).

In $PG\left(3,q\right)$ , $q$ odd, a $\left(q+1\right)$ -arc is a twisted cubic, that is, a rational normal curve of degree 3 (cf. [10], pp.242-243, Theorem 21.2.3).

Definition 2.3. A variety $V_u^v$ of dimension $u$ and of order $v$ of $PG\left(n,q\right)$ is the set of the rational points of a projective variety ${\overline{V}}_u^v$ of $\overline{PG\left(n,q\right)}$ defined by a finite set of polynomials with coefficients in the field $F$ .

Definition 2.4. The ruled variety $V_2^{n-1}$ of $PG\left(n,q\right)$ , $n\ge 4$ and $n\ne 5$ , is generated by the $q+1$ lines joining the corresponding points of two birationally (projectively) equivalent curves of order $m$ and $n-1-m$ , respectively, lying in two complementary subspaces of the same dimensions, $m$ and $n-1-m$ respectively. As such directrix curves have no point in common, then the number of points of $V_2^{n-1}$ is ${\left(q+1\right)}^2$ and the order is the sum of the orders of the curves.

The $q+1$ lines are generatrices (or, generatrix lines).

See Bertini [8], Cap.9, n.1–3, Cap.13, n.1–8, p. 290, 7., Vincenti [7], Lemma 2.2, and [6].

From [8], p. 287, 3., follows

RESULT 1—In $PG\left(n,q\right)$ a hyperplane $S_{n-1}$ meets a ruled variety $V_2^{n-1}$ in one of the following ways:

1) in a rational normal curve of degree $n-1$ (provided $q\ge n-1$ )

or,

2) in a curve of degree $m<n-1$ met by all the generatrix lines and in $n-1-m$ generatrix lines and not composed of two or more distinct curves.

3) Every irreducible curve ${\mathcal{C}}^m$ of degree $m\le n-1$ contained in $V_2^{n-1}$ is a rational normal curve, that is, it lies in an $m$ -dimensional space $S_m$ .

Note that since throughout this paper we refer to rational normal curves ${\mathcal{C}}^h$ of $PG\left(h,q\right)$ for some $h\ge 2$ , we assume $q\ge h$ (see Definition 2.2).

Let $\text{Σ}$ be the projective space $PG\left(2r,q\right)$ , $r\ge 2$ , ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ a hyperplane of $\text{Σ}$ , $\mathcal{S}$ a regular spread of $\left(r-1\right)$ -spaces of ${\Sigma }^{\prime }$ . It is $\left|\mathcal{S}\right|=q^r+1$ . For the definition of spread, regulus and regular spread see [2] and [7], Definition 2.3 and the representation.

The Desarguesian plane $PG\left(2,q^r\right)$ is represented by $\text{Σ}$ and by the spread $\mathcal{S}$ of ${\Sigma }^{\prime }$ according the André/Bruck and Bose method (cf. [1] [2]).

Let $r=2$ . In such a case the projective plane is $PG\left(2,q^2\right)$ , $\text{Σ}=PG\left(4,q\right)$ , ${\Sigma }^{\prime }=PG\left(3,q\right)$ , $\mathcal{S}$ is a regular spread of $q^2+1$ lines of ${\Sigma }^{\prime }$ . If $B=PG\left(2,q\right)$ denotes a non-affine Baer subplane of $PG\left(2,q^2\right)$ , and $P_{\infty }\in B$ its the unique point on the line at infinity, then $B$ is represented by a variety $V_2^3$ , which has as its linear directrix a line $r_{\infty }\in \mathcal{S}$ and as a conic directrix a conic $\mathcal{C}$ lying in a plane that meets ${\Sigma }^{\prime }$ in a line $l_{\infty }\in \mathcal{S}$ , $l_{\infty }\ne r_{\infty }$ with $\mathcal{C}\cap l_{\infty }=\varnothing$ (cf. [3]-[5]).

RESULT 2—A non degenerate conic $\mathcal{C}$ of a non-affine Baer subplane $B$ through $P_{\infty }$ is represented on the variety $V_2^3$ by either a twisted cubic curve (and viceversa), or by a normal rational curve of order 4 depending on whether $P_{\infty }$ belongs to $\mathcal{C}$ or not.

See [5], Theorem 3.1 and Theorem 3.2.

Let $r\ge 2$ , $\text{Π}=PG\left(2,q^r\right)$ . It is $\text{Σ}=PG\left(2r,q\right)$ , ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ , $\mathcal{S}$ is a regular spread of $\left(r-1\right)$ -subspaces, $\left|\mathcal{S}\right|=q^r+1$ .

RESULT 3—A non-affine subplane $\pi =PG\left(2,q\right)$ of $\Pi$ having only one point $P_{\infty }$ at infinity is represented in $\Sigma$ by a ruled variety $V_2^{2r-1}$ . Such a variety is the locus of the lines connecting corresponding points (via a projectivity) of a curve ${\mathcal{C}}_{\infty }^{r-1}$ of a subspace $S_{r-1}^{\infty }\in \mathcal{S}$ and of a curve ${\mathcal{C}}_0^r$ of a subspace $S_r^0\subset \Sigma$ such that $S_r^0\cap {\Sigma }^{\prime }=S_{r-1}^0\ne S_{r-1}^{\infty }$ and ${\mathcal{C}}_0^r\cap S_{r-1}^0=\varnothing$ . Such lines are the generatrix lines, the curves ${\mathcal{C}}^r=S_r\cap V_2^{2r-1}$ with $S_r\cap {\Sigma }^{\prime }\in \mathcal{S}$ are directrices of $V_2^{2r-1}$ . The $q+1$ lines of $\pi$ through $P_{\infty }$ are represented by the generatrix lines, the other lines by the $q^2$ directrix curves of $V_2^{2r-1}$ .

See [6], Theorems 4.7, 4.8.

RESULT 4—A subspace $S_i$ of $S_r=PG\left(r,q\right)$ with $i\ge r-k$ , meets a variety $V_k^n$ in a variety $V_{i+k-r}^n$ .

See [8], p. 191, 3., comma 2.

3. Main Results

3.1. From Σ to Π

Represent $\text{Π}=PG\left(2,q^r\right)$ in $\text{Σ}=PG\left(2r,q\right)$ with $r\ge 2$ , $q\ge 2r-1$ .

Let $\mathcal{S}$ be a regular spread of $\left(r-1\right)$ -spaces of a hyperplane ${\Sigma }^{\prime }=PG\left(2r-1,q\right)$ , $\left|\mathcal{S}\right|=q^r+1$ . The elements of $\mathcal{S}$ are the points at infinity of $\text{Π}$ , the $r$ -spaces $S_r$ such that $S_r\cap {\Sigma }^{\prime }\in \mathcal{S}$ are the affine lines of $\text{Π}$ , $\mathcal{S}$ is the line at infinity $l_{\infty }$ of $\text{Π}$ .

Note that a transversal $r$ -space, that is, an $S_r$ with $S_r\cap {\Sigma }^{\prime }\notin \mathcal{S}$ can represent a Baer subplane $\beta$ of $\text{Π}$ only if $r$ is even. This is because $\beta$ would have order $q^{\frac{r}{2}}$ , and hence the $\left(r-1\right)$ -space $S_r\cap {\Sigma }^{\prime }$ must intersect exactly $q^{\frac{r}{2}}+1$ elements of $\mathcal{S}$ .

Fix an $\left(r-1\right)$ -space $S_{r-1}^{\infty }$ of $\mathcal{S}$ , and choose a curve ${\mathcal{C}}_{\infty }^{r-1}\subset S_{r-1}^{\infty }$ of order $r-1$ . Let $S_r^0$ be an $r$ -space such that $S_r^0\cap {\Sigma }^{\prime }=S_{r-1}^0\ne S_{r-1}^{\infty }$ , and let curve ${\mathcal{C}}_0^r\subset S_r^0$ be a curve satisfying ${\mathcal{C}}_0^r\cap S_{r-1}^0=\varnothing$ .

Let $\pi =PG\left(2,q\right)$ be a non-affine subplane of $\text{Π}$ of order $q$ with $P_{\infty }\in l_{\infty }$ as its unique point at infinity, corresponding to the $\left(r-1\right)$ -space $S_{r-1}^{\infty }$ . Then $\pi$ is represented by the ruled variety $\mathcal{V}=V_2^{2r-1}$ , obtained by connecting corresponding points of ${\mathcal{C}}_{\infty }^{r-1}$ and of ${\mathcal{C}}_0^r$ (see Result 3).

The point $P_{\infty }$ is the center of a bundle of $q+1$ lines in $\pi$ . The remaining $q^2$ lines of $\pi$ each determine a unique point on the line $l_{\infty }$ . Therefore, there exists a subset $\overline{\mathcal{S}}\subset \mathcal{S}$ , with $\left|\overline{\mathcal{S}}\right|=q^2$ corresponding to these points.

NOTE 1—For each element $S_{r-1}\in \overline{\mathcal{S}}$ , there exists a unique $r$ -space $S_r$ that meets the ruled variety $\mathcal{V}$ in a directrix curve ${\mathcal{C}}^r$ , which corresponds to a line of $\pi$ . This is the only line of $\pi$ whose point at infinity is $S_{r-1}$ . Similarly, for $S_{r-1}^0$ , there exists a unique $r$ -space $S_r^0$ that intersects $\mathcal{V}$ in the directrix curve ${\mathcal{C}}_0^r\subset S_r^0$ .

Since ${\mathcal{C}}_{\infty }^{r-1}$ is a rational normal curve, it consists of $q+1$ points, no $r$ of which in a hyperplane $S_{r-2}$ . This holds under the assumption $q\ge r-1$ , which is satisfied by the hypothesis $q\ge 2r-1$ (cf. Definition 2.2).

Choose a subset consisting of $r-1$ independent points. Let denote the $\left(r-2\right)$ -space of $S_{r-1}^{\infty }$ generated by the points of . For $r=2$ the set is a singleton.

Let be the set of the $r-1$ generatrix lines joining the points of and the corresponding $r-1$ points of ${\mathcal{C}}_0^r$ . Let ${\mathcal{G}}^{\prime }=\left\{{g^{\prime }}_r,\cdots ,{g^{\prime }}_{q+1}\right\}$ be the set of the remaining $q+2-r$ generatrix lines.

The hyperplane intersects the ruled variety $\mathcal{V}$ in the union of the curve ${\mathcal{C}}_0^r$ and the set of generatrix lines (cf. Result 1, 2)).

Consider the subspace , the direct sum of the two subspaces.

The bundle ${ℬ}_{2r-2}$ of hyperplanes with axes $S_{2r-2}$ contains $q+1$ hyperplanes. Among them are the hyperplane ${\Sigma }^{\prime }$ and the hyperplane .

Each hyperplane $H^{\prime }\in {ℬ}_{2r-2}\backslash {\Sigma }^{\prime }$ intersects all $q+1$ generatrix lines . If $H^{\prime }\ne H$ and contains no generatrix line, then the $r-1$ lines of meet $H^{\prime }$ in the points of . The remaining $q+2-r$ lines in ${\mathcal{G}}^{\prime }$ meet $H^{\prime }$ in affine points. Denote this set of affine points by $\mathcal{Q}=\left\{Q_r,\cdots ,Q_{q+1}\right\}$ such a set.

Lemma 3.1. If $r>2$ , then the set ${ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ contains both hyperplanes that include one generatrix line from and hyperplanes that contain no generatrix line from .

If $r=2$ , there exists exactly one hyperplane in ${ℬ}_{2r-2}\backslash {\Sigma }^{\prime }$ that contains a generatrix line.

Proof. From Result 1, it follows that a hyperplane $H^{\prime }\in {ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ intersects the ruled variety $\mathcal{V}$ either in a rational normal curve ${\mathcal{C}}^{2r-1}$ of degree $2r-1$ , or in a curve of order $m<n-1$ , which is met by all the generatrix lines and in $n-1-m$ generatrix lines.

Let $r>2$ . Assume that $H^{\prime }\cap \mathcal{V}$ contains at least two generatrix lines, . Since all generatrix lines intersect each directrix curve, denote $A_i$ and $A_j$ the points on $g_i$ and $g_j$ , respectively, that lie on the directrix curve ${\mathcal{C}}_0^r\subset S_r^0$ . Note that $A_i,A_j\notin \mathcal{Q}$ . Because $S_{r-1}^0\subset H^{\prime }$ and the line $A_i{A}_j$ meets $S_{r-1}^0$ , it follows that $H^{\prime }$ contains the entire $S_0^r$ and $H^{\prime }=H$ , which contradicts the assumption.

Hence $H^{\prime }\cap \mathcal{V}$ contains at most one generatrix line.

Now, assume that each of the $q-1$ hyperplanes of ${ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ contains exactly one generatrix line from . Let $H^{\prime }$ and $H^{″}$ be two distinct hyperplanes from , and let $g_1\subset H^{\prime }$ , $g_2\subset H^{″}$ be two generatrix lines such that .

Two distinct hyperplanes in ${ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ can intersect only in the space $S_{2r-2}$ . Therefore such generatrix lines they contain must be distinct. Since $q\ge 2r-1>r$ , we have , which implies that there more generatrix lines than elements in , contradicting the definition of . This contradiction completes the argument.

Hence in ${ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ there are both hyperplanes containing a generatrix of and hyperplanes that contain none.

If $r=2$ , , then there is only one generatrix line $g$ through . Hence, there exists a unique hyperplane in ${ℬ}_{2r-2}\backslash {\Sigma }^{\prime }$ that contains $g$ .

The following cases should be considered.

Theorem 3.2. i) If $r\ge 2$ and $H^{\prime }\cap \mathcal{V}$ contains no generatrix line, then $H^{\prime }\cap \mathcal{V}$ is a rational normal curve ${\mathcal{C}}^{2r-1}$ .

ii₁) If $r>2$ and $H^{\prime }\cap \mathcal{V}$ contains a generatrix line , then $H^{\prime }\cap \mathcal{V}=g\cup {\mathcal{C}}^{2r-2}$ .

In both cases, the curve consists of the $r-1$ points of and of $q+2-r$ affine points, each lying on one of the $q+2-r$ generatrix lines of ${\mathcal{G}}^{\prime }$ with no two of them lying on the same directrix.

ii₂) If $r=2$ and $H^{\prime }\cap \mathcal{V}$ contains a generatrix line $g$ , then $H^{\prime }\cap \mathcal{V}=g\cup \mathcal{C}$ where $\mathcal{C}$ is a conic directrix.

Proof. i) Assume that $H^{\prime }$ contains no generatrix line from .

The hyperplane $H^{\prime }$ intersects all $q+1$ generatrix lines of ${\mathcal{G}}_{}\cup {\mathcal{G}}^{\prime }$ . The $r-1$ lines of are intersected at the points of , while the remaining $q+2-r$ lines of ${\mathcal{G}}^{\prime }$ are intersected at affine points. Denote $\mathcal{Q}=\left\{Q_r,\cdots ,Q_{q+1}\right\}$ the set of these affine points.

Then, from Result 1, 1), it follows that $H\cap \mathcal{V}={\mathcal{C}}^{2r-1}$ , where is a rational normal curve. The condition $q\ge 2r-1$ , although assumed as hypothesis, can be proved although it was taken as a hypothesis, can be proven (cf. [10], Theorem 21.1.1, (i)).

Assume that a point $Q\in \mathcal{Q}$ lies on a generatrix line . Then the line $g$ , containing two points in $H^{\prime }$ , must lie entirely in $H^{\prime }$ , which contradicts the assumption. A similar contradiction arises if either a point $Q$ and a point , or two points $Q_j,Q_h\in \mathcal{Q}$ , lie on the same generatrix $g$ .

Hence, the $q+2-r$ affine points of $\mathcal{Q}$ are distributed one per line over the $q+1-\left(r-1\right)=q+2-r$ generatrix lines $\left\{{g^{\prime }}_r,\cdots ,{g^{\prime }}_{q+1}\right\}$ .

Assume that $H^{\prime }$ contains two points $A,B$ on a directrix curve. Since $H^{\prime }\supset S_{2r-2}\supset S_{r-1}^0$ this directrix must be ${\mathcal{C}}_0^r$ , because the line $AB$ intersects $S_{r-1}^0$ . Therefore $H^{\prime }$ would contain the entire $S_0^r$ , implying that $H^{\prime }=H$ , which is a contradiction.

ii₁) If $H^{\prime }$ contains one generatrix , then from Result 1, 2), it follows that $2r-1-m=1$ so that $m=2r-2$ . Therefore $H^{\prime }\cap \mathcal{V}=g\cup {\mathcal{C}}^{2r-2}$ . From Result 3, since ${\mathcal{C}}^{2r-2}$ is irreducible, it is a rational normal curve and hence lies in a subspace ${S^{\prime }}_{2r-2}$ of $H^{\prime }$ . Note that ${\mathcal{C}}^{2r-2}$ , in addition to $\mathcal{Q}$ , contains the entire set , including the point , otherwise it would have only $q$ points. Moreover, it meets all the generatrix lines.

There are no affine point of ${\mathcal{C}}^{2r-2}$ on $g$ ; otherwise, ${\mathcal{C}}^{2r-2}$ would have $q+2$ points. Therefore, the line $g$ intersects ${\mathcal{C}}^{2r-2}$ in exactly one point and does not lie in $S_{2r-2}$ . The proof of the first property of ${\mathcal{C}}^{2r-2}$ is analogous to the proof in i), where contradictions arise from the possibility that $H^{\prime }$ contains more than one generatrix. The proof of the second one is similar.

ii₂) Let $r=2$ . Then and $H^{\prime }=S_3$ . The line $g\subset H^{\prime }$ is the unique generatrix line, so the residual curve of $H^{\prime }\cap \mathcal{V}$ is a conic $\mathcal{C}$ . Let $\alpha$ denote the plane of $\mathcal{C}$ ; clearly, $\alpha \subset H^{\prime }$ .

Assume that the line $l=\alpha \cap {\Sigma }^{\prime }$ contains the point $P$ . If $l$ coincides with the line ${\mathcal{C}}_{\infty }^1$ , then $\alpha$ would contain $q$ skew generatrix lines of $V_2^3$ , which is a contradiction. If $l$ is a line of ${\Sigma }^{\prime }$ that meets $q+1$ lines of the spread, ${\mathcal{C}}_{\infty }^1$ included, then $\alpha$ is a transversal plane and thus represents an affine Baer subplane of $\text{Π}$ . Since it would share $q$ points with the non-affine Baer subplane represented by $V_2^3$ , the two would have to coincide, again leading to a contradiction. Therefore $l$ is a line of $\mathcal{S}\backslash {\mathcal{C}}_{\infty }^1$ , and we conclude that $H^{\prime }\cap V_2^3=g\cup \mathcal{C}$ , where $\mathcal{C}$ is a directrix curve, that, as required, intersects all the generatrix lines.

Let $H^{\prime }$ be a hyperplane in the bundle ${ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ that contains no generatrix line. Denote by $\mathcal{Q}=\left\{H^{\prime }\cap {g^{\prime }}_i=Q_i|i=r,\cdots ,q+1\right\}$ the set of the $q+2-r$ affine points where $H^{\prime }$ intersects the generatrix lines ${g^{\prime }}_i$ .

Let ${\mathcal{Q}}^{\prime }=\left\{{Q^{\prime }}_r,\cdots ,{Q^{\prime }}_{q+1}\right\}$ be the subset of points of the projective plane $\pi \subset \text{Π}$ , represented in $\text{Σ}$ by the points of $\mathcal{Q}$ , to which we add the point $P_{\infty }$ , represented by $S_{r-1}^{\infty }$ . Denote by $\mathcal{K}={\mathcal{Q}}^{\prime }\cup \left\{P_{\infty }\right\}$ this subset of points of $\pi$ .

Proposition 3.3. The set $\mathcal{K}$ is a $\left(q+3-r\right)$ -arc of $\pi$ . It is maximal, that is, a $\left(q+1\right)$ -arc, when $r=2$ . In this case, if $q$ is odd, $\mathcal{K}$ is a conic.

Proof. First, note that $\mathcal{K}$ has cardinality $q+1-\left(r-1\right)+1=q+3-r$ .

If three affine points of $\mathcal{K}$ were collinear, then the corresponding points in $\mathcal{Q}$ would lie on a directrix curve, contradicting Theorem 3.2. Similarly, if the points ${Q^{\prime }}_i,{Q^{\prime }}_j,P_{\infty }$ were collinear, their line would pass through $P_{\infty }$ , implying that the two affine points $Q_i$ and $Q_j$ of ${\mathcal{C}}^{2r-1}$ lie on a generatrix line of $\mathcal{V}$ , again contradicting both our assumption and Theorem 3.2.

The arc $\mathcal{K}$ is maximal when $q+3-r=q+1$ , that is, when $r=2$ , In this case, $\text{Π}=PG\left(2,q^2\right)$ and $\pi$ is a Baer subplane. If $q$ is odd, $\mathcal{K}$ is a conic.

Denote by $t$ the tangent line of $\mathcal{K}$ at the point $P_{\infty }$ , and let $g_t$ be the corresponding line of the variety $\mathcal{V}$ in $\text{Σ}$ .

Corollary 3.4. The line $t$ is represented in $\Sigma$ by a generatrix line $g_t$ such that $g_t\notin H^{\prime }$ .

Proof. The tangent line $t$ to $\mathcal{K}$ at the point $P_{\infty }$ is a line through $P_{\infty }$ and, clearly, it lies in the subplane $\pi$ . Since all lines through $P_{\infty }$ in $\pi$ are represented in $\text{Σ}$ by the generatrix lines of $\mathcal{V}$ , it follows that $g_t$ is a generatrix line of $\mathcal{V}$ . By hypothesis, $H^{\prime }$ contains no generatrix lines; therefore $g_t\notin H^{\prime }$ .

3.2. From Π to Σ

Let $\mathcal{C}\subset \pi$ be a non degenerate conic containing the unique point at infinity $P_{\infty }$ of $\pi$ . Denote by $\left\{{P^{\prime }}_1,\cdots ,{P^{\prime }}_q\right\}$ the $q$ affine points of $\mathcal{C}$ , and by $\left\{{g^{\prime }}_1,\cdots ,{g^{\prime }}_q\right\}$ the $q$ affine lines of $\pi$ joining $P_{\infty }$ with the points $\left\{{P^{\prime }}_1,\cdots ,{P^{\prime }}_q\right\}$ . Let $t$ be the tangent line to $\mathcal{C}$ at $P_{\infty }$ .

Let ${\mathcal{K}}_{\mathcal{C}}$ be the subset of $\mathcal{V}$ corresponding in $\text{Σ}$ to the points of $\mathcal{C}$ , $\mathcal{K}=\left\{P_1,\cdots ,P_q\right\}$ is the set of the $q$ affine points of ${\mathcal{K}}_{\mathcal{C}}$ representing $\left\{{P^{\prime }}_1,\cdots ,{P^{\prime }}_q\right\}$ , $\mathcal{G}=\left\{g_1,\cdots ,g_q,g_t\right\}$ the set of the $q+1$ generatrix lines of $\mathcal{V}$ corresponding to $\left\{{g^{\prime }}_1,\cdots ,{g^{\prime }}_q,g_t\right\}$ where $g_t$ is the generatrix representing $t$ , $T$ the point $g_t\cap {\mathcal{C}}_{\infty }^{r-1}$ .

Theorem 3.5. i) The set ${\mathcal{K}}_{\mathcal{C}}$ consists of $q+1$ points of $\mathcal{V}$ . Among them, the $q$ affine points of $\mathcal{K}$ each lie on a distinct generatrix in the set $\left\{g_1,\cdots ,g_q\right\}$ . The point $T$ , which lies on the curve ${\mathcal{C}}_{\infty }^{r-1}$ , is the unique point at infinity of ${\mathcal{K}}_{\mathcal{C}}$ .

ii) The set ${\mathcal{K}}_{\mathcal{C}}$ forms a $\left(q+1\right)$ -cap, with the line $g_t$ being the tangent to ${\mathcal{K}}_{\mathcal{C}}$ at the point $T$ .

Proof. i) Obviously no point of $\mathcal{K}$ belongs to the generatrix $g_t$ as no affine point of the conic $\mathcal{C}$ belongs to the tangent $t$ .

Assume two points of $\mathcal{K}$ lie on the same generatrix $g\in \left\{g_1,\cdots ,g_q\right\}$ . That would imply that the corresponding two points of the conic $\mathcal{C}$ are collinear with $P_{\infty }$ , contradicting the fact that $\mathcal{C}$ is non-degenerate. Hence, the $q$ affine points of $\mathcal{K}$ lie each on a distinct generatrix among $g_1,\cdots ,g_q$ , all different from the tangent line $g_t$ ; that is, $P_i\in g_i$ for $i=1,\cdots ,q$ .

Assume that ${\mathcal{K}}_{\mathcal{C}}$ contains a point $T^{\prime }\in {\mathcal{C}}_{\infty }^{r-1}$ , with $T^{\prime }\ne T$ . Then the generatrix containing $T^{\prime }$ must be a line $g_i\in \mathcal{G}$ for some $i=1,\cdots ,q$ . Since $P_i\in g_i$ , the entire generatrix $g_i=\overline{T^{\prime }{P}_i}$ would lie in the configuration. This would introduce to $\mathcal{K}$ the remaining $q-1$ affine points of the generatrix $g_i$ , whose corresponding points in $\pi$ would lie on a line through $P_{\infty }$ . That line would then be added to the conic $\mathcal{C}$ , contradicting its non-degeneracy. Therefore $T^{\prime }=T$ , and $T$ is the unique point at infinity of ${\mathcal{K}}_{\mathcal{C}}$ .

ii) First, note that no two points of $\mathcal{K}$ lie on the same generatrix, since no two affine points of the conic $\mathcal{C}$ corresponding to them are collinear with $P_{\infty }$ .

Assume that $P_i,P_j,P_h\in \mathcal{K}$ are collinear, and let $l$ be the line passing through them. Let ${P^{\prime }}_i,{P^{\prime }}_j,{P^{\prime }}_h$ be the corresponding points on the conic $\mathcal{C}$ . The line $l$ is not a generatrix line, and therefore it determines, via its intersection point $l\cap {\Sigma }^{\prime }$ an element ${S^{\prime }}_{r-1}\in \mathcal{S}$ , and subsequently a space ${S^{\prime }}_r$ such that ${S^{\prime }}_r\cap {\Sigma }^{\prime }={S^{\prime }}_{r-1}$ . This space ${S^{\prime }}_r$ contains a directrix curve ${{\mathcal{C}}^{\prime }}^r$ .

The line in $\pi$ through $P_i$ and $P_j$ is represented in $\text{Σ}$ by the curve ${{\mathcal{C}}^{\prime }}^r$ , that is the same line in $\pi$ through $P_i,P_h$ and $P_j,P_h$ . Hence, the three points ${P^{\prime }}_i,{P^{\prime }}_j,{P^{\prime }}_h\in \mathcal{C}$ must lie on a common line in $\pi$ , a contradiction to the fact that $\mathcal{C}$ is a non-degenerate conic.

The line $g_t\in \mathcal{V}$ , corresponding of the tangent $t$ to the conic $\mathcal{C}$ at the point $P_{\infty }$ , has only the point $T$ in common with ${\mathcal{K}}_{\mathcal{C}}$ . Therefore, $g_t$ is the tangent line to ${\mathcal{K}}_{\mathcal{C}}$ at $T$ .

To complete the characterization of ${\mathcal{K}}_{\mathcal{C}}\subset \mathcal{V}$ , it remains to understand the nature of ${\mathcal{K}}_{\mathcal{C}}$ in the case where it is contained in the intersection of $\mathcal{V}$ with a hyperplane.

Let $S^{\prime }=S_{2r-1}$ be a hyperplane such that ${\mathcal{K}}_{\mathcal{C}}\subset S^{\prime }$ . Necessarily $S^{\prime }\ne {\Sigma }^{\prime }$ .

The subspace ${S^{\prime }}_{2r-2}=S^{\prime }\cap {\Sigma }^{\prime }$ may either intersect each element of the spread $\mathcal{S}$ in subspaces of the same dimension, or contain one element of $\mathcal{S}$ entirely.

Note that the number of points in ${S^{\prime }}_{2r-2}$ is $q^{2r-2}+q^{2r-3}+\cdots +q+1$ .

Since this number is not divisible by $q^r+1=\left|\mathcal{S}\right|$ , the first case, where ${S^{\prime }}_{2r-2}$ intersects each element of the spread $\mathcal{S}$ in subspaces of equal dimension, cannot occur. Hence ${S^{\prime }}_{2r-2}$ must contain one element of the spread.

1) Assume that ${S^{\prime }}_{2r-2}\supset S_{r-1}^{\infty }$ , so that $S^{\prime }\supset S_{r-1}^{\infty }$ .

Then $S^{\prime }$ contains the curve ${\mathcal{C}}_{\infty }^{r-1}$ as well as the $q$ generatrix lines of $\mathcal{G}$ (excluding $g_t$ ) that connect the $q$ affine points of ${\mathcal{K}}_{\mathcal{C}}$ to the corresponding $q$ points of ${\mathcal{C}}_{\infty }^{r-1}\backslash \left\{T\right\}$ . Let $g$ denote the set of the affine points on the line $g_t$ . Then $S^{\prime }\cap \mathcal{V}=\mathcal{V}\backslash \left\{g\right\}$ , which consists of the $q$ generatrix lines together with ${\mathcal{C}}_{\infty }^{r-1}$ , that is, a reducible curve of order $q+r+1>2r-1$ , contradicting Result 1.

2) Assume ${S^{\prime }}_{2r-2}$ contains an element of ${S^{\prime }}_{r-1}\in \mathcal{S}\backslash \left\{S_{r-1}^{\infty }\right\}$ , so that $S^{\prime }\supset {S^{\prime }}_{r-1}$ .

Since $S^{\prime }\ne {\Sigma }^{\prime }$ , it follows that $S^{\prime }\cap S_{r-1}^{\infty }$ is a subspace ${\overline{S}}_{r-2}$ , which intersects the curve ${\mathcal{C}}_{\infty }^{r-1}$ in $r-1$ points, including the point $T$ (cf. Definition 2.2).

Define ${S^{\prime }}_{2r-2}={S^{\prime }}_{r-1}+{\overline{S}}_{r-2}$ .

There are $q+1$ hyperplanes containing ${S^{\prime }}_{2r-2}$ , one of which one is ${\Sigma }^{\prime }$ (which contains $S_{r-1}^{\infty }$ ). Another hyperplane contains an $r$ -dimensional subspace ${S^{\prime }}_r$ such that ${S^{\prime }}_r\cap {\Sigma }^{\prime }={S^{\prime }}_{r-1}$ . This subspace ${S^{\prime }}_r$ represents a line $d$ of $\text{Π}$ , and its intersection with $\mathcal{V}$ is a directrix curve ${\mathcal{C}}_d^r$ , which corresponds to the unique line of $\pi$ through the point at infinity represented by .

Since $S^{\prime }$ contains the $r-1$ generatrix lines, say $g_1,\cdots ,g_{r-1}$ , through the points of ${\overline{S}}_{r-2}$ , as well as the corresponding points of the directrix curve ${\mathcal{C}}_d^r$ , it follows that

$S^{\prime }\cap \mathcal{V}={\mathcal{C}}_d^r\cup \left\{g_1,\cdots ,g_{r-1}\right\},$

in accordance with Result 1.

Let $A,B\in \mathcal{K}$ be any two affine points of ${\mathcal{K}}_{\mathcal{C}}$ , and let $A_{\pi },B_{\pi }$ be the corresponding points in $\pi$ represented by $A$ and $B$ , respectively; note that $A_{\pi },B_{\pi }\in \mathcal{C}$ . By hypothesis $A$ and $B$ lie in the hyperplane $S^{\prime }$ , since $S^{\prime }\supset {\mathcal{K}}_{\mathcal{C}}$ . Then the point at infinity $R_{\infty }$ of the line $r=AB$ lies either in ${\overline{S}}_{r-2}$ or in ${S^{\prime }}_{r-1}$ ; that is, $R_{\infty }\in S_{r-1}^{\infty }\cup {S^{\prime }}_{r-1}$ .

The line $r_{\pi }=A_{\pi }{B}_{\pi }$ is a secant of the conic $\mathcal{C}$ , and therefore it cannot pass through $P_{\infty }$ . This implies that the point at infinity $R_{\infty }$ of the line $r=AB$ cannot lie in $S_{r-1}^{\infty }$ .

If, on the other hand, $R_{\infty }\in {S^{\prime }}_{r-1}$ , then $A,B\in {\mathcal{C}}_d^r$ . Since this would hold for any pair of affine points of $A,B\in \mathcal{K}$ , it would follow that $\mathcal{K}={\mathcal{C}}_d^r$ . However, in $\pi$ , the points of ${\mathcal{C}}_d^r$ are represented by the line $d$ , and thus $\mathcal{C}$ would have to coincide with the line $d$ , contradicting the assumption that $\mathcal{C}$ is a non-degenerate conic.

Therefore, ${\mathcal{K}}_{\mathcal{C}}$ cannot be contained in such a hyperplane $S^{\prime }$ representing a line of $\pi$ .

Based on cases of 1) and 2), we now make the following choices. Fix a subspapce ${S^{\prime }}_{r-1}\in \overline{\mathcal{S}}=\mathcal{S}\backslash S_{r-1}^{\infty }$ , and choose a subspace ${\overline{S}}_{r-2}\subset S_{r-1}^{\infty }$ such that ${\overline{S}}_{r-2}$ intersects the curve ${\mathcal{C}}_{\infty }^{r-1}$ in exactly $r-1$ points. Define

${S^{\prime }}_{2r-2}={S^{\prime }}_{r-1}+{\overline{S}}_{r-2}.$

In the bundle ${ℬ}_{2r-2}$ of hyperplanes with axes ${S^{\prime }}_{2r-2}$ , there are $q+1$ hyperplanes: one is $\text{Σ}\text{'}$ , and another is $H={S^{\prime }}_r+{\overline{S}}_{r-2}$ , where ${S^{\prime }}_r$ represents a line $d$ of $\pi$ . Thus,

$\left|{ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}\right|=q-1.$

Choose a hyperplane $S^{\prime }\in {ℬ}_{2r-2}\backslash \left\{{\Sigma }^{\prime },H\right\}$ .

NOTE 2—From NOTE 1, it follows that for each choice of ${S^{\prime }}_{r-1}\in \overline{\mathcal{S}}$ , there are $q^2$ possibilities, and for each of these, there are $q-1$ hyperplanes like $S^{\prime }$ . Moreover, one must count the possible choices of $r-1$ independent points on ${\mathcal{C}}_{\infty }^{r-1}$ , each determining a distinct subspace ${\overline{S}}_{r-2}$ , and hence giving rise to different hyperplanes.

Assume that $S^{\prime }$ contains ${\mathcal{K}}_{\mathcal{C}}$ . In this case, the point $T=g_t\cap {\mathcal{C}}_{\infty }^{r-1}$ is one of the $r-1$ points of the intersection ${\overline{S}}_{r-2}\cap {\mathcal{C}}_{\infty }^{r-1}$ , and the line $g_t$ cannot contain any affine points. Denote by $\left\{g_1,\cdots ,g_{r-2},g_t\right\}$ the $r-1$ generatrix lines passing through the points of ${\overline{S}}_{r-2}\cap {\mathcal{C}}_{\infty }^{r-1}$ .

Theorem 3.6. i) $r>2$ . Then $S^{\prime }\cap \mathcal{V}$ consists of $r-2$ generatrix lines and a residual curve ${\mathcal{C}}^{r+1}$ representing ${\mathcal{K}}_{\mathcal{C}}$ . The curve ${\mathcal{C}}^{r+1}$ meets all the generatrix lines; it is a rational normal curve lying in a subspace $S_{r+1}\subset S^{\prime }$ , with ${\mathcal{C}}^{r+1}\cap {\overline{S}}_{r-2}=\left\{T\right\}$ .

ii) $r=2$ . Then ${\mathcal{K}}_{\mathcal{C}}=S^{\prime }\cap \mathcal{V}$ . If $q$ is odd, ${\mathcal{K}}_{\mathcal{C}}$ is a twisted cubic containing in $S^{\prime }$ .

In no case does $g_t$ , tangent to ${\mathcal{K}}_{\mathcal{C}}$ at the point at infinity, belong to $S^{\prime }$ .

Proof. i) $r>2$ . The intersection $S^{\prime }\cap \mathcal{V}$ must either be a curve of order $2r-1$ consisting of a total of $q+1$ points, or consist of $2r-1-m$ generatrix lines and of a residual curve of order $m<2r-1$ (cf. Result 1). Since $S^{\prime }$ contains ${\mathcal{K}}_{\mathcal{C}}$ , which includes $q$ affine points, it follows that $S^{\prime }\cap \mathcal{V}$ must consist of the $r-2=2r-1-m$ generatrix lines $\left\{g_1,\cdots ,g_{r-2}\right\}$ , and of a residual curve of order $m$ satisfying $2r-1-m=r-2$ , hence $m=r+1<2r-1$ . Such a curve ${\mathcal{C}}^{r+1}$ must intersect all the generatrix lines (cf. Result 1).

More precisely, the lines in $\left\{g_1,\cdots ,g_{r-2}\right\}$ intersect ${\mathcal{C}}^{r+1}$ at the corresponding points of ${\overline{S}}_{r-2}$ . The generatrix $g_t$ , which represents the tangent $t$ to the conic $\mathcal{C}$ at its infinite point, does not belong to $S^{\prime }$ (by hypothesis). Since $g_t$ contains no affine points of ${\mathcal{K}}_{\mathcal{C}}$ , it intersects ${\mathcal{C}}^{r+1}$ only at the point $T$ .

Therefore ${\mathcal{K}}_{\mathcal{C}}$ is represented by ${\mathcal{C}}^{r+1}$ , a rational normal curve of $\mathcal{V}$ , lying in a subspace $S_{r+1}$ of $S^{\prime }$ (cf. Result 1, 3)). Indeed, since $dim \left(S_{r+1}\cap {\overline{S}}_{r-2}\right)=0$ , which is equivalent to ${\mathcal{C}}^{r+1}\cap {\mathcal{C}}_{\infty }^{r-1}=\left\{T\right\}$ , the remaining $q$ affine points are distributed one on each of the remaining $q$ generatrix lines.

ii) For $r=2$ , case i) can still be applied since $2r-1=r+1$ . Specifically, note that $\mathcal{V}=V_2^3$ is embedded in $PG\left(4,q\right)$ , where the hyperplane $S^{\prime }$ is a 3-dimensional subspace. The curve at infinity, ${\mathcal{C}}_{\infty }^1={\mathcal{C}}_{\infty }^{2-1}$ , is a line $l_{\infty }$ . The hyperplane $S^{\prime }$ meets $l_{\infty }$ in exactly one point, $T$ , and intersects each generatrix line in exactly one of the $q$ affine point of ${\mathcal{K}}_{\mathcal{C}}$ .

From Theorem 3.5, it follows that no three points of ${\mathcal{K}}_{\mathcal{C}}$ are collinear. Therefore, in this case and for $q$ odd, ${\mathcal{K}}_{\mathcal{C}}$ is a twisted cubic curve of $S^{\prime }$ , that is, a normal rational curve of $S^{\prime }$ (cf. [10], Theorem 21.2.3, [5], Theorem 3.1).

In neither case i) nor ii), by construction, does $g_t$ belong to $S^{\prime }$ .

This result shows that for $r>2$ (that is, for $2r-1>r+1$ ), no hyperplane strictly defines a substructure of the variety $\mathcal{V}$ capable of representing a cap of $q+1$ points, unless one considers a subspace of dimension $r+1$ within a hyperplane $S^{\prime }$ , which can be constructed as follows.

First, remind that the curve $C_{\infty }^{r-1}$ of $S_{r-1}^{\infty }$ consists of $q+1$ points, no $r$ of which lie in a hyperplane of $S_{r-1}^{\infty }$ , with $q\ge r-1$ (since by hypothesis $q\ge 2r-1$ ; cf. Definition 2.2).

Choose a subset of $r-1$ points from $C_{\infty }^{r-1}$ . Let be the subspace generated by . Choose a subspace $S_{r-1}\in \mathcal{S}\backslash S_{r-1}^{\infty }$ . Denote by $\mathcal{G}=\left\{g_1,\cdots ,g_{r-2},g_t\right\}$ the set of the generatrix lines through the points of with .

Theorem 3.7. There exists a hyperplane $H^{\prime }$ containing a subspace $S_{r+1}$ , in which lies a rational normal curve $C^{r+1}$ . This curve $C^{r+1}$ is a cap of $\mathcal{V}$ , consisting of $q+1$ points, exactly one of which is at infinity. The curve ${\mathcal{C}}^{r+1}$ corresponds in the plane $\pi$ , to a conic passing through the point at infinity .

Proof. Set $r>2$ . Consider the set $\mathcal{G}\backslash \left\{g_t\right\}$ consisting of the $r-2$ generatrices $\left\{g_1,\cdots ,g_{r-2}\right\}$ , and denote by $S_{r-1}^{\text{*}}$ the subspace they generate. Let $H^{\prime }=S_{r-1}^{\text{*}}+S_{r-1}$ be the hyperplane defined as the span of $S_{r-1}^{\text{*}}$ and a chosen element $S_{r-1}\in \mathcal{S}\backslash S_{r-1}^{\infty }$ . Since $H^{\prime }$ contains $2r-1-m=r-2$ generatrix lines, it follows by Result 1 that $H^{\prime }\cap \mathcal{V}$ includes a residual curve ${\mathcal{C}}^{r+1}$ of order $m=r+1$ , which meets all the generatrix lines.

Since $H^{\prime }\cap S_{r-1}^{\infty }$ is a subspace of dimension $r-2$ , it follows that $H^{\prime }$ contains the entire subspace , and in particular, the point $T$ . Therefore, $H^{\prime }$ is a hyperplane of the bundle ${ℬ}_{2r-2}$ of hyperplanes.

The curve ${\mathcal{C}}^{r+1}$ is a rational normal curve, so it lies in a subspace $S_{r+1}\subset H^{\prime }$ that intersects ${S^{\prime }}_{r-2}$ in a single point. If ${\mathcal{C}}^{r+1}\subset S_{r+1}$ met each generatrix in an affine point, it would be a directrix. However, the maximum possible order of a directrix is $r<r+1$ , which leads to a contradiction. Therefore, ${\mathcal{C}}^{r+1}$ must intersect one of the generatrices in $\mathcal{G}$ at the point $P=S_{r+1}\cap {S^{\prime }}_{r-2}$ which lies in .

Assume $P\ne T$ . Then ${\mathcal{C}}^{r+1}$ meets $g_t$ in an affine point, implying that $g_t$ must lie in $H^{\prime }$ . This leads to a contradiction, as including $g_t$ would increase the dimension to $S_{r-1}^{\text{*}}$ . Therefore $P=T$ .

The remaining $q$ generatrices are each intersected by ${\mathcal{C}}^{r+1}$ at exactly one affine point, so that ${\mathcal{C}}^{r+1}$ contains $q$ affine points and one point at infinity. This configuration does not contradict the presence of the lines $\left\{g_1,\cdots ,g_{r-2}\right\}$ in $H^{\prime }$ , as they are already contained within in. Since ${\mathcal{C}}^{r+1}$ is a normal rational curve with $q+1$ points, it clearly forms a $\left(q+1\right)$ -cap.

Moreover, $H^{\prime }\ne {\Sigma }^{\prime }$ since ${\Sigma }^{\prime }$ contains no generatrices, and $H^{\prime }\ne H$ , as the hyperplane $H$ contains $r-1$ generatrices and the unique directrix curve of order $r$ .

If $r=2$ , then and $\mathcal{G}=\left\{g_t\right\}$ . Although the previous procedure can still be applied, to clarify this case, let $H^{\prime }=S_3$ be one of the $q-1$ hyperplanes around the plane $S_{r-1}+T=S_1+T$ distinct from both ${\Sigma }^{\prime }$ and $H$ , so that $H^{\prime }$ contains no conic directrix.

If $g_t\in H^{\prime }$ , then $2r-1-m=3-m=1$ , so that the residual curve $C$ would have order $m=2r-2=2$ . That is, $C$ would be a conic meeting all the generatrix lines (cf. Result 1). Therefore $C$ would be a directrix and $H^{\prime }=H$ , which is a contradiction.

Hence $H^{\prime }$ does not contain $g_t$ ; it meets the line $C_{\infty }^{r-1}=C^1$ in single point, namely, $T$ . Consequently, $H^{\prime }\cap \mathcal{V}$ is an irreducible rational normal curve of order $2r-1=3$ (cf. [5], Lemma 2.1) having $T$ as its unique point at infinity.

By construction, in both cases, it is immediate to verify that ${\mathcal{C}}^{r+1}$ represents a conic of $\pi$ passing through the point $P_{\infty }$ .

Theorem 3.8. There exists a partition of the affine points of the variety $\mathcal{V}=V_2^{2r-1}$ consisting of $q$ rational normal curves of order $r+1$ , together one generatrix line.

Proof. In the non-affine subplane $\pi$ , let $ℱ$ be the bundle of hyperosculating conics at the point $P_{\infty }$ , all sharing the common tangent line $t$ through $P_{\infty }$ . It is straightforward to verify that $\left|ℱ\right|=q$ . Consequently, the affine points of $ℱ\cup t$ provide a partition of the affine points of $\pi$ .

Denote by ${ℒ}_{ℱ}$ the set of $q$ curves of order $r+1$ in the variety $\mathcal{V}$ corresponding to the conics of the bundle $ℱ$ . Let $g_t$ be the generatrix line representing the tangent line $t$ at $P_{\infty }$ (cf. Theorem 3.5).

Since two distinct conics in $ℱ$ meet only at the point $P_{\infty }$ , the corresponding curves in ${ℒ}_{ℱ}$ have no affine points in common. The total number of affine points of $\mathcal{V}$ are $q^2+q$ , each curve in ${ℒ}_{ℱ}$ contains exactly $q$ affine points, so the union of all these curves accounts for $q\cdot q=q^2$ . Adding the $q$ affine points of the generatrix line $g_t$ , we cover all $q^2+q$ affine points of $\mathcal{V}$ .

4. Conclusion

The representation of a non-affine subplane $PG\left(2,q\right)$ of the projective plane $PG\left(2,q^r\right)$ in the variety $V_2^{2r-1}$ of $PG\left(2r,q\right)$ is a generalization introduced in a earlier work. In this paper, we studied the connection between the caps obtained from certain hyperplane sections of $V_2^{2r-1}$ and specific arcs in $PG\left(2,q\right)$ . This connection enabled us to establish a partition of the affine points of $V_2^{2r-1}$ into caps, corresponding to a partition of the affine points $PG\left(2,q\right)$ into conics.

Conflicts of Interest

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

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