Conics in Baer subplanes

TL;DR

This paper explores the geometric relationship between conics in tangent Baer subplanes of PG(2,q^2) and normal rational curves in PG(4,q), revealing a correspondence mediated by transversal lines of the regular spread.

Contribution

It establishes a bijective correspondence between conics in tangent Baer subplanes and certain normal rational curves in PG(4,q), linking two geometric frameworks.

Findings

Conics in tangent Baer subplanes correspond to normal rational curves meeting transversal lines.

Every such normal rational curve in PG(4,q) corresponds to a conic in a tangent Baer subplane.

The study clarifies the geometric structure of conics via the André/Bruck-Bose representation.

Abstract

This article studies conics and subconics of $PG(2,q^2)$ and their representation in the Andr\'e/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG(2,q^2)$ corresponds in $PG(4,q)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3 and 4-dimensional normal rational curve in $PG(4,q)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG(2,q^2)$.

Figures

Equations

$$\phi :\mathrm{PG}(2,q^2)\backslash {\ell }_{\infty }$$

$$\text{mbox}suchthatA=(x_0+x_1\tau ,y_0+y_1\tau ,z)$$

$$\bar{T}=(\delta,1,0)\ \longmapsto\ [T]=\big{\langle}\,(d_{0},d_{1},1,0,0),\ (t_{0}d_{1},d_{0}+t_{1}d_{1},0,1,0)\,\big{\rangle}.$$

$$g$$

$$g^q$$

$$\bar{T}=(\delta ,1,0)\in {\ell }_{\infty }⟷T=\delta A_0+A_1\in g,\delta \in {\mathbb{F}}_{q^2}\cup \{\infty \},$$

$$\bar{P}=(\alpha ,1,0)⟷P=\alpha A_0+A_1\in g^{\text{mbox}H},\alpha \in {\mathbb{F}}_{q^4}\cup \{\infty \}.$$

$$f_{\infty }=y_0^2+y_1^2{t}_0-x_{0}z\text{mbox}and{f}_0=y_1^2{t}_1+2y_0{y}_1-x_{1}z.$$

Full text

Keywords: Bruck-Bose representation, Baer subplanes, conics, subconics

AMS code: 51E20

1 Introduction

This article investigates the representation of conics and subconics of ${\rm PG}(2,q^{2})$ in the Bruck-Bose representation in ${\rm PG}(4,q)$. The Bruck-Bose representation of ${\rm PG}(2,q^{2})$ uses a regular spread $\mathcal{S}$ in the hyperplane at infinity of ${\rm PG}(4,q)$. The regular spread $\mathcal{S}$ has two unique transversal lines $g,g^{q}$ in the quadratic extension ${\rm PG}(4,q^{2})$. There are several known characterisations of objects of ${\rm PG}(4,q)$ in terms of their relationship with these transversal lines. Firstly, a conic ${\cal C}$ in ${\rm PG}(4,q)$ corresponds to a Baer subline of ${\rm PG}(2,q^{2})$ iff the extension of ${\cal C}$ to a conic of ${\rm PG}(4,q^{2})$ contains a point of $g$ and a point of $g^{q}$, see [13]. A ruled cubic surface ${\cal V}$ in ${\rm PG}(4,q)$ corresponds to a Baer subplane of ${\rm PG}(2,q^{2})$ iff the extension of ${\cal V}$ to ${\rm PG}(4,q^{2})$ contains $g$ and $g^{q}$, see [13]. Further, an orthogonal cone ${\cal U}$ corresponds to a classical unital of ${\rm PG}(2,q^{2})$ iff the extension of ${\cal U}$ to ${\rm PG}(4,q^{2})$ contains $g$ and $g^{q}$, see [16]. Hence the interaction of certain objects with the transversals of $\mathcal{S}$ is intrinsic to their characterisation in ${\rm PG}(2,q^{2})$. In this article we study conics and subconics of ${\rm PG}(2,q^{2})$, and determine their relationship with the transversals of $\mathcal{S}$ in the Bruck-Bose setting in ${\rm PG}(4,q)$. In particular, we characterise normal rational curves of ${\rm PG}(4,q)$ whose extension meets the transversals as subconics of ${\rm PG}(2,q^{2})$.

The article is set out as follows. Section 2 introduces the necessary background and proves some preliminary results. In particular, in order to study how objects of the Bruck-Bose representation relate to the transversals of the regular spread $\mathcal{S}$, we formally define in Section 2.6 the notion of special sets in ${\rm PG}(4,q)$. Further, in Section 2.11, we consider a Baer subplane ${\mathscr{B}}$ tangent to $\ell_{\infty}$, and give a geometric construction via ${\rm PG}(4,q)$ that partitions the affine points of ${\mathscr{B}}$ into $q$ conics, one of which is degenerate. In Section 3, we discuss how the notion of specialness relates to the known Bruck-Bose representation of Baer sublines and Baer subplanes.

In Section 4, we investigate non-degenerate conics of ${\rm PG}(2,q^{2})$ in the ${\rm PG}(4,q)$ Bruck-Bose representation. In particular, we investigate the corresponding structure in the quadratic extension to ${\rm PG}(4,q^{2})$. We show that in ${\rm PG}(4,q^{2})$, the (extended) structure corresponding to a non-degenerate conic $\mathcal{O}$ is the intersection of two quadrics which meet $g$ in the two points (possibly repeated or in an extension) corresponding to $\mathscr{O}\cap\ell_{\infty}$.

In Section 5 we characterise the Bruck-Bose representation of conics contained in Baer subplanes. In ${\rm PG}(2,q^{2})$, let ${\mathscr{B}}$ be a Baer subplane tangent to $\ell_{\infty}$, and ${\cal C}$ a non-degenerate conic contained in ${\mathscr{B}}$. We show that in ${\rm PG}(4,q)$, ${\cal C}$ corresponds to a normal rational curve that meets the transversals of the regular spread. Conversely, we characterise every normal rational curve in ${\rm PG}(4,q)$ that meets the transversals of the regular spread as corresponding to a non-degenerate conic in a Baer subplane of ${\rm PG}(2,q^{2})$.

While the proofs in Section 4 are largely coordinate based, the proofs in Section 5 use geometrical arguments.

2 Background and Preliminary Results

In this section we give the necessary background, introduce the notation we use in this article, and prove a number of preliminary results.

2.1 Conjugate points

For $q$ a prime power, we denote the unique finite field of order $q$ by $\mathbb{F}_{q}$. We use the phrase conjugate points in several different settings. Firstly, consider the automorphism $x\mapsto x^{q}$ for $x\in{{\mathbb{F}}_{q^{r}}}$, it induces an automorphic collineation of ${\rm PG}(n,q^{r})$ where a point $X=(x_{0},\ldots,x_{n})\mapsto X^{q}=(x_{0}^{q},\ldots,x_{n}^{q})$. The points $X,X^{q},\ldots,X^{q^{n}-1}$ are called conjugate. Secondly, let ${\mathscr{B}}$ be a Baer subplane of ${\rm PG}(2,q^{2})$, them there is a unique involutory collineation that fixes ${\mathscr{B}}$ pointwise, and we call this map conjugacy with respect to ${\mathscr{B}}$. Note that $P,Q\in\ell_{\infty}$ are conjugate with respect to the secant Baer subplane ${\mathscr{B}}$ if and only if $P,Q$ are conjugate with respect to the Baer subline ${\mathscr{B}}\cap\ell_{\infty}$.

2.2 Spreads in ${\rm PG}(3,q)$

The following construction of a regular spread of ${\rm PG}(3,q)$ will be needed, see [15] for more information on spreads. Embed ${\rm PG}(3,q)$ in ${\rm PG}(3,q^{2})$ and let $g$ be a line of ${\rm PG}(3,q^{2})$ disjoint from ${\rm PG}(3,q)$. The line $g$ has a conjugate line $g^{q}$ with respect to the map $x\mapsto x^{q}$, $x\in{{\mathbb{F}}_{q^{2}}}$, and $g^{q}$ is also disjoint from ${\rm PG}(3,q)$. Let $P_{i}$ be a point on $g$; then the line $\langle P_{i},P_{i}^{q}\rangle$ meets ${\rm PG}(3,q)$ in a line. As $P_{i}$ ranges over all the points of $g$, we obtain $q^{2}+1$ lines of ${\rm PG}(3,q)$ that partition ${\rm PG}(3,q)$. These lines form a regular spread $\mathcal{S}$ of ${\rm PG}(3,q)$. The lines $g$, $g^{q}$ are called the (conjugate skew) transversal lines of the regular spread $\mathcal{S}$. Conversely, given a regular spread $\mathcal{S}$ in ${\rm PG}(3,q)$, there is a unique pair of transversal lines in ${\rm PG}(3,q^{2})$ that generate $\mathcal{S}$ in this way.

2.3 The Bruck-Bose representation

We will use the linear representation of a finite translation plane of dimension at most two over its kernel, due independently to André [1] and Bruck and Bose [9, 10]. Let $\Sigma_{\infty}$ be a hyperplane of ${\rm PG}(4,q)$ and let $\mathcal{S}$ be a spread of $\Sigma_{\infty}$. We use the phrase a subspace of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$ to mean a subspace of ${\rm PG}(4,q)$ that is not contained in $\Sigma_{\infty}$. Consider the following incidence structure: the points of ${\cal A(\cal S)}$ are the points of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$; the lines of ${\cal A(\cal S)}$ are the planes of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$ that contain an element of $\mathcal{S}$; and incidence in ${\cal A(\cal S)}$ is induced by incidence in ${\rm PG}(4,q)$. Then the incidence structure ${\cal A(\cal S)}$ is an affine plane of order $q^{2}$. We can complete ${\cal A(\cal S)}$ to a projective plane ${\cal P(\cal S)}$; the points on the line at infinity $\ell_{\infty}$ have a natural correspondence to the elements of the spread $\mathcal{S}$. We call this the Bruck-Bose representation of $\mathcal{P}(\mathcal{S})$ in ${\rm PG}(4,q)$. The projective plane ${\cal P(\cal S)}$ is the Desarguesian plane ${\rm PG}(2,q^{2})$ if and only if $\mathcal{S}$ is a regular spread of $\Sigma_{\infty}\cong{\rm PG}(3,q)$ (see [6]). We use the following notation in the Bruck-Bose setting.

• •

$\mathcal{S}$ is a regular spread with transversal lines $g,g^{q}$.

• •

An affine point of ${\rm PG}(2,q^{2})\backslash\ell_{\infty}$ is denoted with a capital letter, $A$ say, and $[A]$ denotes the corresponding point of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$.

• •

A point on $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$ is denoted with an over-lined capital letter, $\bar{T}$ say, and the corresponding spread line is denoted $[T]$.

• •

The points of $\ell_{\infty}$ are in 1-1 correspondence with the points of $g$; for a point $\bar{T}\in\ell_{\infty}$, we denote the corresponding point of $g$ by $T$.

• •

A set of points $\mathcal{X}$ in ${\rm PG}(2,q^{2})$ corresponds to a set of points denoted $[\mathcal{X}]$ in ${\rm PG}(4,q)$.

We will work in the extension of ${\rm PG}(4,q)$ to ${\rm PG}(4,q^{2})$ and to ${\rm PG}(4,q^{4})$. Let $\mathcal{K}$ be a primal of ${\rm PG}(4,q)$, so $\mathcal{K}$ is the set of points of ${\rm PG}(4,q)$ satisfying a homogeneous equation $f(x_{0},\ldots,x_{4})=0$, with coefficients in $\mathbb{F}_{q}$. We define $\mathcal{K}{{}^{\mbox{\tiny\char 73}}}$ to be the (unique) primal of ${\rm PG}(4,q^{2})$ which is the set of points of ${\rm PG}(4,q^{2})$ satisfy the same homogeneous equation $f=0$. Note that if $\mathcal{K}=\Pi$ is an $r$-dimensional subspace of ${\rm PG}(4,q)$, then $\Pi{{}^{\mbox{\tiny\char 73}}}$ is the (unique) $r$-dimensional subspace of ${\rm PG}(4,q^{2})$ containing $\Pi$. Further, if ${\cal V}$ is a variety of ${\rm PG}(4,q)$, so ${\cal V}$ is the intersection of primals $\mathcal{K}_{1},\ldots,\mathcal{K}_{s}$, then we define ${\cal V}{{}^{\mbox{\tiny\char 73}}}={\mathcal{K}}_{1}^{\mbox{\tiny\char 73}}\cap\cdots\cap{\mathcal{K}}_{s}^{\mbox{\tiny\char 73}}$. Similarly, we can extend a primal $\mathcal{K}$ to ${\rm PG}(4,q^{4})$, and we denote the resulting set by ${\mathcal{K}^{\mbox{\tiny\char 72}}}$. The transversals $g,g^{q}$ of the regular spread $\mathcal{S}$ lie in ${\rm PG}(4,q^{2})$, and we denote their extensions to lines of ${\rm PG}(4,q^{4})$ by $g^{\mbox{\tiny\char 72}},{g^{q}}^{\mbox{\tiny\char 72}}$ respectively.

2.4 Ruled cubic surfaces in ${\rm PG}(4,q)$

A ruled cubic surface ${\cal V}$ of ${\rm PG}(4,q)$ consists of a line directrix $t$, a conic directrix ${\cal C}$ lying in a plane disjoint from $t$, and a set of $q+1$ pairwise disjoint generator lines joining the points of $t$ and ${\cal C}$ according to a projectivity $\omega\in{\rm PGL}(2,q)$. That is, let $\theta,\phi\in\mathbb{F}_{q}\cup\{\infty\}$ be the non-homogeneous coordinates of $t,{\cal C}$ respectively, and $\omega\colon(1,\theta)\mapsto(1,\phi)$, then the generators of ${\cal V}$ are the lines joining points of $t$ to the corresponding point of ${\cal C}$ under $\omega$. We will need the following result which shows how hyperplanes of ${\rm PG}(4,q)$ meet a ruled cubic surface.

Result 2.1

[17]** A hyperplane of ${\rm PG}(4,q)$ meets a ruled cubic surface in one of the following.

• •

The line directrix; $(q^{2}-q)/2$ hyperplanes do this.

• •

The line directrix and one generator line; $q+1$ hyperplanes do this.

• •

The line directrix and two generator lines; $(q^{2}+q)/2$ hyperplanes do this.

• •

A conic and a generator line; $q^{3}+q^{2}$ hyperplanes do this.

• •

A twisted cubic curve (which meets the line directrix in a unique point); $q^{4}-q^{2}$ hyperplanes do this.

Corollary 2.2

Let $\Pi$ be a hyperplane of ${\rm PG}(4,q)$ that meets a ruled cubic surface ${\cal V}$ in a twisted cubic ${\cal N}$. Then ${\cal N}$ meets each generator line of ${\cal V}$ in a unique point.

Proof   If ${\cal N}$ meets a generator line $\ell$ of ${\cal V}$ in two points, then the 3-space $\Pi$ containing ${\cal N}$ also contains $\ell$, which is not possible by Result 2.1. Hence ${\cal N}$ meets each generator line in at most one point. As ${\cal N}$ contains $q+1$ points, each generator of ${\cal V}$ contains a unique point of ${\cal N}$. $\square$

There are two ways to extend the ruled cubic surface to ${\rm PG}(4,q^{2})$, we show that they are equivalent. The ruled cubic surface ${\cal V}$ is a variety whose points are the exact intersection of three quadrics, ${\cal V}=\mathscr{Q}_{0}\cap\mathscr{Q}_{1}\cap\mathscr{Q}_{2}$ (see for example [3]). So extending this variety to ${\rm PG}(4,q^{2})$ yields ${{\cal V}^{\mbox{\tiny\char 73}}}={\mathscr{Q}}^{\mbox{\tiny\char 73}}_{0}\cap{\mathscr{Q}}^{\mbox{\tiny\char 73}}_{1}\cap{\mathscr{Q}}^{\mbox{\tiny\char 73}}_{2}$. Alternatively, we can extend ${\cal V}$ to ${\rm PG}(4,q^{2})$ as in [13]: namely extending the line directrix $t$ and conic directrix ${\cal C}$ to ${\rm PG}(4,q^{2})$ by taking $\theta,\phi\in\mathbb{F}_{q^{2}}\cup\{\infty\}$, and extending the projectivity $\omega$ to act over $\mathbb{F}_{q^{2}}$. We denote this extension by ${\cal V}^{\prime}$, thus ${\cal V}^{\prime}$ is the ruled cubic surface with line directrix $t{{}^{\mbox{\tiny\char 73}}}$, conic directrix ${\cal C}{{}^{\mbox{\tiny\char 73}}}$, and ruled using the (extended) projectivity $\omega$. We show that these two extensions ${{\cal V}^{\mbox{\tiny\char 73}}}$, ${\cal V}^{\prime}$ are the same. The surface ${\cal V}$ contains exactly $q^{2}$ conics ${\cal C}_{1},\ldots,{\cal C}_{q^{2}}$, and these conics cover each point of ${\cal V}\backslash t$ $q$ times (see [2] for more details). Hence both sets ${{\cal V}^{\mbox{\tiny\char 73}}},{\cal V}^{\prime}$ contain the extended conics ${{\cal C}}_{1}^{\mbox{\tiny\char 73}},\ldots,{{\cal C}}_{q^{2}}^{\mbox{\tiny\char 73}}$. Moreover, these conics together with $t{{}^{\mbox{\tiny\char 73}}}$ cover all the points of ${\cal V}^{\prime}$, and so ${{\cal V}^{\mbox{\tiny\char 73}}}$ contains ${\cal V}^{\prime}$. However, ${{\cal V}^{\mbox{\tiny\char 73}}}$ is the intersection of three quadrics over $\mathbb{F}_{q^{2}}$, whose intersection over $\mathbb{F}_{q}$ is a ruled cubic surface. By [5], all ruled cubic surfaces are projectively equivalent, hence ${{\cal V}^{\mbox{\tiny\char 73}}}$ and ${\cal V}^{\prime}$ are the same ruled cubic surface of ${\rm PG}(4,q^{2})$.

2.5 Coordinates in Bruck-Bose

We now show how the coordinates of points in ${\rm PG}(2,q^{2})$ relate to the coordinates of the corresponding points in the Bruck-Bose representation in ${\rm PG}(4,q)$. See [2, Section 3.4] for more details. Let $\tau$ be a primitive element in ${{\mathbb{F}}_{q^{2}}}$ with primitive polynomial $x^{2}-t_{1}x-t_{0}$ over ${{\mathbb{F}}_{q}}$. Then every element $\alpha\in{{\mathbb{F}}_{q^{2}}}$ can be uniquely written as $\alpha=a_{0}+a_{1}\tau$ with $a_{0},a_{1}\in{{\mathbb{F}}_{q}}$. That is, ${{\mathbb{F}}_{q^{2}}}=\{x_{0}+x_{1}\tau\mid x_{0},x_{1}\in{{\mathbb{F}}_{q}}\}.$ It is useful to keep in mind the relationships: $\tau\tau^{q}=-t_{0}$, $\tau+\tau^{q}=t_{1}$, $t_{0}/\tau=-\tau^{q}=\tau-t_{1}$ and $\tau^{q^{2}}=1$. Points in ${\rm PG}(2,q^{2})$ have homogeneous coordinates $(x,y,z)$ with $x,y,z\in{{\mathbb{F}}_{q^{2}}}$, not all zero. We let the line at infinity $\ell_{\infty}$ have equation $z=0$, so affine points of ${\rm PG}(2,q^{2})$ have coordinates $(x,y,1)$, with $x,y\in{{\mathbb{F}}_{q^{2}}}$. Points in ${\rm PG}(4,q)$ have homogeneous coordinates $(x_{0},x_{1},y_{0},y_{1},z)$ with $x_{0},x_{1},y_{0},y_{1},z\in{{\mathbb{F}}_{q}}$, not all zero. We let the hyperplane at infinity $\Sigma_{\infty}$ have equation $z=0$, so the affine points of ${\rm PG}(4,q)$ have coordinates $(x_{0},x_{1},y_{0},y_{1},1)$, with $x_{0},x_{1},y_{0},y_{1}\in{{\mathbb{F}}_{q}}$. Let $A$ be an affine point in ${\rm PG}(2,q^{2})$ with coordinates $A=(x_{0}+x_{1}\tau,y_{0}+y_{1}\tau,z)$, where $x_{0},x_{1},y_{0},y_{1},z\in{{\mathbb{F}}_{q}}$, $z\neq 0$. The map

[TABLE]

is a bijection from the affine points of ${\rm PG}(2,q^{2})$ to the affine points of ${\rm PG}(4,q)$, called the Bruck-Bose map. We can extend this to a projective map; for a point $\bar{T}=(\delta,1,0)\in\ell_{\infty}$, write $\delta=d_{0}+d_{1}\tau\in{{\mathbb{F}}_{q^{2}}}$, $d_{0},d_{1}\in{{\mathbb{F}}_{q}}$, then

[TABLE]

The transversal lines $g,g^{q}$ of $\mathcal{S}$ have coordinates given by:

[TABLE]

Recall that each line of the regular spread $\mathcal{S}$ meets the transversal $g$ of $\mathcal{S}$ in a point. The 1-1 correspondence between points of $\ell_{\infty}$ and points of $g$ is:

[TABLE]

that is, ${T}={[T]^{\mbox{\tiny\char 73}}}\cap g$ and ${[T]^{\mbox{\tiny\char 73}}}={T}T^{q}$.

2.5.1 Coordinates and the quartic extension ${\rm PG}(4,q^{4})$

We will be interested in non-degenerate conics of ${\rm PG}(2,q^{2})$, and one of the cases to consider is when a conic $\mathscr{C}$ is exterior to $\ell_{\infty}$, and so meets $\ell_{\infty}$ in two points which lie in the quadratic extension of ${\rm PG}(2,q^{2})$ to ${\rm PG}(2,q^{4})$. That is, $\mathscr{C}$ meets $\ell_{\infty}$ in two points $\bar{P},\,\bar{Q}$ over ${{\mathbb{F}}_{q^{4}}}$. Note that $\bar{P},\,\bar{Q}$ are conjugate with respect to the map $x\mapsto x^{q^{2}}$, $x\in{{\mathbb{F}}_{q^{4}}}$, that is $\bar{Q}=\bar{P}^{q^{2}}$. There is no direct representation for the point $\bar{P}$ in the Bruck-Bose representation in ${\rm PG}(4,q)$. However, there is a related point in the quartic extension ${\rm PG}(4,q^{4})$. We can extend the 1-1 correspondence between points $\ell_{\infty}$ and points of $g$ to a 1-1 correspondence between points of the quadratic extension of $\ell_{\infty}$ and points of the extended transversal $g^{\mbox{\tiny\char 72}}$ in ${\rm PG}(4,q^{4})$, so

[TABLE]

If $\bar{P}=(\alpha,1,0)$ for some $\alpha\in{{\mathbb{F}}_{q^{4}}}\backslash{{\mathbb{F}}_{q^{2}}}$, that is $\bar{P}\in{\rm PG}(2,q^{4})\backslash{\rm PG}(2,q^{2})$, then in ${\rm PG}(4,q^{4})$, the corresponding point ${P}$ lies in $g^{\mbox{\tiny\char 72}}\backslash g$, and the conjugate point $P^{q}=\alpha^{q}A_{0}^{q}+A_{1}^{q}$ lies on ${g^{q}}^{\mbox{\tiny\char 72}}\backslash g^{q}$. As $\bar{P}\notin{\rm PG}(2,q^{2})$, the line $PP^{q}$ is not a line of the spread $\mathcal{S}$; $PP^{q}$ is a line of ${\rm PG}(4,q^{4})$ that does not meet $\Sigma_{\infty}$.

2.6 $g$-special sets

When studying curves of ${\rm PG}(2,q^{2})$ in the ${\rm PG}(4,q)$ Bruck-Bose setting, the transversals $g,g^{q}$ of the regular spread $\mathcal{S}$ play an important role in characterisations. Let ${\cal V}$ be a variety or rational curve in ${\rm PG}(4,q)$, we are interested in how ${{\cal V}^{\mbox{\tiny\char 73}}}$ meets $g,g^{q}$ in the extension to ${\rm PG}(4,q^{2})$. Note that if ${{\cal V}^{\mbox{\tiny\char 73}}}$ meets $g$ in a point $P$, then as ${\cal V}$ is defined over ${{\mathbb{F}}_{q}}$, ${{\cal V}^{\mbox{\tiny\char 73}}}$ also meets $g^{q}$ in the point $P^{q}$. A non-degenerate conic ${\cal C}$ in ${\rm PG}(4,q)$ is called a $g$-special conic if in ${\rm PG}(4,q^{2})$, ${\cal C}{{}^{\mbox{\tiny\char 73}}}$ contains one point of $g$, and one point of $g^{q}$. A twisted cubic ${\cal N}$ in ${\rm PG}(4,q)$ is called a $g$-special twisted cubic if in ${\rm PG}(4,q^{2})$, ${{\cal N}^{\mbox{\tiny\char 73}}}$ contains one point of $g$, and one point of $g^{q}$. A 4-dimensional normal rational curve ${\cal N}$ in ${\rm PG}(4,q)$ is called a $g$-special normal rational curve if in ${\rm PG}(4,q^{2})$, ${{\cal N}^{\mbox{\tiny\char 73}}}$ contains two points of $g$ (possibly repeated) and two points of $g^{q}$. Further, ${\cal N}$ is called $g^{\mbox{\tiny\char 72}}$-special if in the quartic extension ${\rm PG}(4,q^{4})$, ${{\cal N}^{\mbox{\tiny\char 72}}}$ contains two points of the extended transversal $g^{\mbox{\tiny\char 72}}\backslash g$. A ruled cubic surface ${\cal V}$ in ${\rm PG}(4,q)$ is called a $g$-special ruled cubic surface if in ${\rm PG}(4,q^{2})$, ${{\cal V}^{\mbox{\tiny\char 73}}}$ contains $g$ and $g^{q}$.

2.7 Representations of Baer sublines and subplanes

We use the following representations of Baer sublines and subplanes of ${\rm PG}(2,q^{2})$ in ${\rm PG}(4,q)$, see [2] for more details.

Result 2.3

Let $\mathcal{S}$ be a regular spread in a $3$-space $\Sigma_{\infty}$ in ${\rm PG}(4,q)$ and consider the representation of the Desarguesian plane $\mathcal{P}(\mathcal{S})={\rm PG}(2,q^{2})$ defined by the Bruck-Bose construction.

A Baer subline of $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$ corresponds to a regulus of $\mathcal{S}$. 2. 2.

A Baer subline of ${\rm PG}(2,q^{2})$ that meets $\ell_{\infty}$ in a point corresponds to a line of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$. 3. 3.

A Baer subplane of ${\rm PG}(2,q^{2})$ secant to $\ell_{\infty}$ corresponds to a plane of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$ not containing a spread line. 4. 4.

A Baer subline of ${\rm PG}(2,q^{2})$ that is disjoint from $\ell_{\infty}$ corresponds in ${\rm PG}(4,q)$ to a $g$-special conic. 5. 5.

A Baer subplane tangent to $\ell_{\infty}$ at a point $\bar{T}$ corresponds in ${\rm PG}(4,q)$ to a $g$-special ruled cubic surface containing the corresponding spread line $[T]$.

Moreover, the converse of each of these correspondences holds.

Remark 2.4

The correspondences in parts 2 and 3 are not exact at infinity. The exact at infinity representation of a Baer subline that meets $\ell_{\infty}$ in a point $T$ is an affine line that meets the spread line $[T]$ union with the spread line $[T]$. Similarly, the exact at infinity representation of a secant Baer subplane is a plane $\alpha$ not containing a spread line, union the lines of $\mathcal{S}$ that $\alpha$ meets.

2.8 Representations of subconics

The representation of non-degenerate conics contained in a Baer subplane was considered in [17].

Result 2.5

[17]** Let ${\cal C}$ be a non-degenerate conic contained in a Baer subplane ${\mathscr{B}}$ of ${\rm PG}(2,q^{2})$.

Suppose ${\mathscr{B}}$ is secant to $\ell_{\infty}$, then ${\cal C}$ corresponds to a non-degenerate conic in the plane $[{\mathscr{B}}]$ of ${\rm PG}(4,q)$. 2. 2.

Suppose ${\mathscr{B}}$ is tangent to $\ell_{\infty}$, ${\mathscr{B}}\cap\ell_{\infty}\in{\cal C}$, and $q\geq 3$. Then ${\cal C}$ corresponds to a twisted cubic on the ruled cubic surface $[{\mathscr{B}}]$ of ${\rm PG}(4,q)$. 3. 3.

Suppose ${\mathscr{B}}$ is tangent to $\ell_{\infty}$, ${\mathscr{B}}\cap\ell_{\infty}\notin{\cal C}$, and $q\geq 4$. Then ${\cal C}$ corresponds to a 4-dimensional normal rational curve on the ruled cubic surface $[{\mathscr{B}}]$ of ${\rm PG}(4,q)$.

In Section 5, we show that the 3- and 4-dimensional normal rational curves of Result 2.5 are $g$-special. Conversely, we show that every $g$-special normal rational curve in ${\rm PG}(4,q)$ corresponds to a non-degenerate conic contained in a tangent Baer subplane.

Remark 2.6

The correspondence in Result 2.5 parts 1 and 2 is not exact at infinity (compare with Remark 2.4). For example, in part 2, the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$ is in $[{\cal C}]$, and the twisted cubic $[{\cal C}]$ meets $\Sigma_{\infty}$ in a point of $[T]$. The exact-at-infinty representation is: the set $[{\cal C}]$ is a twisted cubic union the spread line $[T]$. We use the simpler, not exact-at-infinity correspondence given in Result 2.5 as it does not lead to any confusion. **

2.9 The circle geometry $CG(2,q)$

Circle geometries $CG(d,q)$, $d\geq 2$ were introduced in [7, 8], and we summarise the results we need here. Note that $CG(2,q)$ is an inversive plane. We can construct $CG(2,q)$ from the line ${\rm PG}(1,q^{2})$, in this case the circles are the Baer sublines of ${\rm PG}(1,q^{2})$. Equivalently, we can construct $CG(2,q)$ from the lines of a regular spread $\mathcal{S}$ of ${\rm PG}(3,q)$, in this case the circles are the reguli contained in $\mathcal{S}$. Using the representation of $CG(2,q)$ as $\ell_{\infty}\cong{\rm PG}(1,q^{2})$, we can use properties of the circle geometry to deduce several properties of the projective plane ${\rm PG}(2,q^{2})$. If $\bar{P},\,\bar{Q}$ are two distinct points on $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$, then there is a unique partition of $\ell_{\infty}$ into $\bar{P},\,\bar{Q}$ and $q-1$ Baer sublines $\ell_{1},\ldots,\ell_{q-1}$, where the points $\bar{P},\,\bar{Q}$ are conjugate with respect to each Baer subline $\ell_{i}$. Further, if ${\mathscr{B}}$ is a Baer subplane secant to $\ell_{\infty}$, such that $\bar{P},\,\bar{Q}$ are conjugate with respect to ${\mathscr{B}}$, then ${\mathscr{B}}$ meets $\ell_{\infty}$ in one of the Baer sublines $\ell_{i}$. Of particular interest is an application to conics.

Result 2.7

Let $\mathscr{O}$ be a non-degenerate conic of ${\rm PG}(2,q^{2})$ that meets $\ell_{\infty}$ in $\{\bar{P},\,\bar{Q}\}$. Then there is a unique partition of the $q^{2}-1$ affine points of $\mathscr{O}$ into $q-1$ subconics ${\cal C}_{1},\ldots,{\cal C}_{q-1}$, lying in Baer subplanes ${\mathscr{B}}_{1},\ldots,{\mathscr{B}}_{q-1}$ which are secant to $\ell_{\infty}$. Further, the Baer sublines ${\mathscr{B}}_{i}\cap\ell_{\infty}$ are either equal or disjoint

The properties of the circle geometry also lead to properties of a regular spread $\mathcal{S}$ in ${\rm PG}(3,q)$. Let $g,g^{q}$ be the transversals of $\mathcal{S}$, so $g,g^{q}$ lie in ${\rm PG}(3,q^{2})$. Consider the set of lines of ${\rm PG}(3,q^{2})$ that meet both $g$ and $g^{q}$. This set is called the hyperbolic congruence of $g,g^{q}$ in [14]. Note that if two distinct lines in the hyperbolic congruence meet, then they meet on $g$ or $g^{q}$. The hyperbolic congruence contains the extended spread lines ${[P]^{\mbox{\tiny\char 73}}}=PP^{q}$ for $P\in g$ and the lines $PQ^{q}$ for distinct $P,Q\in g$. The lines $PQ^{q}$ have an interesting relationship with the regular spread $\mathcal{S}$.

Result 2.8

[8]** Let $[P],[Q]$ be two lines of a regular spread $\mathcal{S}$ in ${\rm PG}(3,q)$, and denote their intersection with transversal $g$ of $\mathcal{S}$ by $P,Q$ respectively. Then there is a unique partition of $\mathcal{S}$ into $[P],[Q]$ and $q-1$ reguli $\mathcal{R}_{1},\ldots,\mathcal{R}_{q-1}$. Denote the opposite regulus of $\mathcal{R}_{i}$ by $\mathcal{R}_{i}^{\prime}$. Then the set $\{[P],[Q],\mathcal{R}_{1}^{\prime},\ldots,\mathcal{R}_{q-1}^{\prime}\}$ is a regular spread with transversals $PQ^{q}$, $P^{q}Q$.

We will show that the lines in the hyperbolic congruence of $g,g^{q}$ are related to the Bruck-Bose representation of non-degenerate conics of ${\rm PG}(2,q^{2})$ in ${\rm PG}(4,q)$.

2.10 Normal rational curves contained in quadrics

Next, we show that if a normal rational curve is contained in a quadric in ${\rm PG}(4,q)$, then the containment also holds in the quadratic extension ${\rm PG}(4,q^{2})$, provided $q$ is not small.

Lemma 2.9

In ${\rm PG}(4,q)$, $q>7$, let ${\cal N}$ be a 4-dimensional normal rational curve and $\mathscr{Q}$ a quadric, with ${\cal N}\subset\mathscr{Q}$. Then in the quadratic extension ${\rm PG}(4,q^{2})$, ${{\cal N}^{\mbox{\tiny\char 73}}}\subset{\mathscr{Q}^{\mbox{\tiny\char 73}}}$.

Proof   Without loss of generality, let ${\cal N}=\{P_{\theta}=(1,\theta,\theta^{2},\theta^{3},\theta^{4})\,|\,\theta\in{{\mathbb{F}}_{q}}\cup\{\infty\}\}$. Let $\mathscr{Q}$ have equation $g(x_{0},x_{1},x_{2},x_{3},x_{4})=0$. Consider $g(P_{\theta})=g(1,\theta,\theta^{2},\theta^{3},\theta^{4})=f(\theta)$. As $\mathscr{Q}$ is a quadric, $f(\theta)$ is a polynomial in $\theta$ of degree at most 8. Now as ${\cal N}\subset\mathscr{Q}$, $f(P_{\theta})=0$ for all $\theta\in{{\mathbb{F}}_{q}}\cup\{\infty\}$. So if $q+1>8$, $f$ is identically 0, and so $f(P_{\theta})=0$ for all $\theta\in{{\mathbb{F}}_{q^{2}}}$. Using $\theta=\infty$, this implies that the coefficient of $\theta^{8}$ is zero, thus the degree of $f$ is at most $7$. As $f(\theta)=0$ for the $q$ values of $\theta\in{{\mathbb{F}}_{q}}$, it follows that $f$ has $q$ roots, so if $q>7$ then $f$ is the zero polynomial, thus $f(\theta)=0$ for all $\theta$ in any extension of ${{\mathbb{F}}_{q}}$, and so $g(P_{\theta})=0$ for all $\theta$ in any extension of ${{\mathbb{F}}_{q}}$. So if $q>7$, the point $P_{\theta}$, $\theta\in{{\mathbb{F}}_{q^{2}}}\cup\{\infty\}$, lies on ${\mathscr{Q}^{\mbox{\tiny\char 73}}}$, and so ${{\cal N}^{\mbox{\tiny\char 73}}}\subset{\mathscr{Q}^{\mbox{\tiny\char 73}}}$. $\square$

The bound on $q$ in Lemma 2.9 is tight as shown by the following example. In ${\rm PG}(4,7)$, let ${\cal N}$ be the normal rational curve ${\cal N}=\{P_{\theta}=(1,\theta,\theta^{2},\theta^{3},\theta^{4})\,|\,\theta\in{\rm GF}(7)\cup\{\infty\}\}$ and let $\mathscr{Q}$ be the quadric with equation $f(x_{0},x_{1},x_{2},x_{3},x_{4})=-x_{0}x_{1}-x_{3}^{2}+x_{2}x_{4}+x_{3}x_{4}$. First note that $f(P_{\theta})=\theta^{7}-\theta=0$ for all $\theta\in{\rm GF}(7)$. Further, $P_{\infty}=(0,0,0,0,1)$, so $f(P_{\infty})=0$. Hence ${\cal N}\subset\mathscr{Q}$ in ${\rm PG}(4,7)$. Now extend ${\rm GF}(7)$ to ${\rm GF}(7^{2})$ using a primitive element $\tau$. The point $P_{\tau}=(1,\tau,\tau^{2},\tau^{3},\tau^{4})$ lies in the extended curve ${{\cal N}^{\mbox{\tiny\char 73}}}$. However $f(P_{\tau})=\tau^{7}-\tau\neq 0$ as $\tau\not\in{\rm GF}(7)$, and so $P_{\tau}$ does not lie on the extended quadric ${\mathscr{Q}^{\mbox{\tiny\char 73}}}$, that is ${{\cal N}^{\mbox{\tiny\char 73}}}\not\subset{\mathscr{Q}^{\mbox{\tiny\char 73}}}$.

2.11 Baer pencils and partitions of Baer subplanes

In this section we investigate the representation in ${\rm PG}(2,q^{2})$ of a $3$-space of ${\rm PG}(4,q)$. We use this to partition tangent Baer subplanes into conics.

Definition 2.10

A Baer pencil $\mathcal{K}$ in ${\rm PG}(2,q^{2})$ is the cone of $q+1$ lines joining a vertex point $P$ to a Baer subline base $b$. An $\ell_{\infty}$-Baer pencil $\mathcal{K}$ is a Baer pencil with vertex in $\ell_{\infty}$, and base $b$ meeting $\ell_{\infty}$ in a point.

Let $\mathcal{K}$ be a Baer pencil, then every line of ${\rm PG}(2,q^{2})$ not through the vertex of $\mathcal{K}$ meets $\mathcal{K}$ in a Baer subline. Also note that an $\ell_{\infty}$-Baer pencil $\mathcal{K}$ contains $\ell_{\infty}$, and a further $q^{3}$ affine points. It is straightforward to characterise the $\ell_{\infty}$-Baer pencils of ${\rm PG}(2,q^{2})$ in ${\rm PG}(4,q)$.

Lemma 2.11

Let $\Pi$ be a $3$-space in ${\rm PG}(4,q)$ distinct from $\Sigma_{\infty}$. Then $\Pi$ corresponds in ${\rm PG}(2,q^{2})$ to an $\ell_{\infty}$-Baer pencil with vertex corresponding to the unique spread line in $\Pi$. Conversely, any $\ell_{\infty}$-Baer pencil in ${\rm PG}(2,q^{2})$ corresponds to a $3$-space of ${\rm PG}(4,q)$.

We look at how $\ell_{\infty}$-Baer pencils meet a tangent Baer subplane.

Theorem 2.12

Let ${\mathscr{B}}$ be a Baer subplane in ${\rm PG}(2,q^{2})$ tangent to $\ell_{\infty}$ at the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$. An $\ell_{\infty}$-Baer pencil with vertex $\bar{P}\neq\bar{T}$ meets ${\mathscr{B}}$ in either a non-degenerate conic through $\bar{T}$; or in two lines, namely the unique line of ${\mathscr{B}}$ whose extension contains $\bar{P}$, and one line through $\bar{T}$. Of the $\ell_{\infty}$-Baer pencils with vertex $\bar{P}$, there are $q^{2}-1$ of the first type, and $q+1$ of the second type (each containing one of the $q+1$ lines of ${\mathscr{B}}$ through $\bar{T}$).

Proof   In ${\rm PG}(4,q)$, let $X$ be a point on the spread line $[T]$, and let $\alpha=\langle X,[P]\rangle$. Label the $3$-spaces of ${\rm PG}(4,q)$ (not equal to $\Sigma_{\infty})$ that contain the plane $\alpha$ by ${\mathscr{L}}=\{\Pi_{1},\ldots,\Pi_{q}\}$. By Lemma 2.11, each $3$-space in ${\mathscr{L}}$ corresponds to an $\ell_{\infty}$-Baer pencil of ${\rm PG}(2,q^{2})$ with vertex $P$. Result 2.1 describes how a $3$-space meets the ruled cubic surface $[{\mathscr{B}}]$. As each 3-space in ${\mathscr{L}}$ meets $[T]$ in one point, and the $3$-spaces in ${\mathscr{L}}$ partition the affine points, we deduce that one of the 3-spaces in ${\mathscr{L}}$, $\Pi_{1}$ say, meets $[{\mathscr{B}}]$ in a conic and the generator line of $[{\mathscr{B}}]$ through $X$, and the remaining 3-spaces in ${\mathscr{L}}$ meet $[{\mathscr{B}}]$ in a twisted cubic ${\cal N}_{i}=\Pi_{i}\cap[{\mathscr{B}}]$, $i=2,\ldots,q$. By Result 2.5, the twisted cubics ${\cal N}_{i}$ each correspond in ${\rm PG}(2,q^{2})$ to non-degenerate conics in ${\mathscr{B}}$ that contains $\bar{T}$. Note that there is a unique plane of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$ that contains the spread line $[P]$ and meets $[{\mathscr{B}}]$ in a conic; namely the plane that corresponds in ${\rm PG}(2,q^{2})$ to the unique line m_{\mbox{\raisebox{-0.25pt}{\scalebox{0.55}{{\boldmath{!P}}}}}} through $\bar{P}$ that meets ${\mathscr{B}}$ in a Baer subline. Hence $\Pi_{1}\cap[{\mathscr{B}}]$ contains the generator line $[m]$ of $[{\mathscr{B}}]$ through the point $X$, and a conic in the plane [m_{\mbox{\raisebox{-0.25pt}{\scalebox{0.55}{{\boldmath{!P}}}}}}]. This corresponds in ${\rm PG}(2,q^{2})$ to an $\ell_{\infty}$-Baer pencil with vertex $\bar{P}$ that meets $[{\mathscr{B}}]$ in the two Baer sublines m_{\mbox{\raisebox{-0.25pt}{\scalebox{0.55}{{\boldmath{!P}}}}}}\cap{\mathscr{B}} and $m$.

As there are $q+1$ choices for the point $X$ on $[T]$, there are $(q+1)(q-1)$ 3-spaces about $[P]$ that meets $[{\mathscr{B}}]$ in a twisted cubic, and $q+1$ that meet $[{\mathscr{B}}]$ in a line and a conic, giving the required number of Baer pencils. $\square$

The next result shows that a non-degenerate conic in ${\mathscr{B}}$ lies in a unique $\ell_{\infty}$-Baer pencil, and describes the relationship between the conic and the vertex of the pencil.

Theorem 2.13

Let ${\mathscr{B}}$ be a Baer subplane in ${\rm PG}(2,q^{2})$ tangent to $\ell_{\infty}$ at the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$ and let ${\cal C}$ be a non-degenerate conic in ${\mathscr{B}}$ with $\bar{T}\in{\cal C}$. Then ${\cal C}$ lies in a unique $\ell_{\infty}$-Baer pencil $\mathcal{K}$. Moreover, the vertex of $\mathcal{K}$ lies in the extension of ${\cal C}$ to ${\rm PG}(2,q^{2})$.

Proof   Let ${\cal C}$ be a non-degenerate conic in ${\mathscr{B}}$, with $\bar{T}={\mathscr{B}}\cap\ell_{\infty}\in{\cal C}$. As $\ell_{\infty}$ is not a line of ${\mathscr{B}}$, it is not the tangent line of ${\cal C}$ at the point $\bar{T}$. Let ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ be the extension of ${\cal C}$ to ${\rm PG}(2,q^{2})$, then $\ell_{\infty}$ is a secant to ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$, so ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}=\{\bar{T},\bar{L}\}$. We will show that ${\cal C}$ lies in a unique $\ell_{\infty}$-Baer pencil $\mathcal{K}$ which has vertex $\bar{L}$.

We first show that any point $X\in{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ projects ${\cal C}$ onto a Baer subline. Without loss of generality, let ${\cal C}=\{P_{\theta}=(1,\theta,\theta^{2})\,|\,\theta\in{\mathbb{F}_{q^{2}}}\cup\{\infty\}\}$, so ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}=\{(1,\theta,\theta^{2})\,|\,\theta\in{{\mathbb{F}}_{q}}\cup\{\infty\}\}$. Let $\omega\in{{\mathbb{F}}_{q^{2}}}\backslash{{\mathbb{F}}_{q}}$, so the point $X=(1,\omega,\omega^{2})$ lies in ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\backslash{\cal C}$. The projection of the point $P_{\theta}$, $\theta\in{{\mathbb{F}}_{q}}\cup\{\infty\}$ from $X$ onto the line $\ell$ with equation $x=0$ is $P_{\theta}^{\prime}=(0,1,\theta+\omega)$. That is, the projection of ${\cal C}$ from $X$ onto $\ell$ is the set $\{(0,1,\omega)+\theta(0,0,1)\,|\,\theta\in{{\mathbb{F}}_{q}}\cup\{\infty\}\}$, which is a Baer subline.

We next show that ${\cal C}$ lies in a unique $\ell_{\infty}$-Baer pencil. By Result 2.5, in ${\rm PG}(4,q)$, $[{\cal C}]$ is a twisted cubic meeting the spread line $[T]$ in one point, and $[{\cal C}]$ lies in a 3-space $\Pi$ that meets $[T]$ in exactly one point. Hence $\Pi$ contains a unique spread line $[P]$, with $\bar{P}\neq\bar{T}$. By Lemma 2.11, $\Pi$ corresponds to an $\ell_{\infty}$-Baer pencil $\mathcal{K}$ with vertex $\bar{P}$, so ${\cal C}$ lies in the pencil $\mathcal{K}$. If ${\cal C}$ were in two $\ell_{\infty}$-Baer pencils $\mathcal{K},\mathcal{K}^{\prime}$, then $[{\cal C}]$ would lie in two 3-spaces $\Pi_{\mathcal{K}}$, $\Pi_{\mathcal{K}^{\prime}}$, which is not possible.

Hence ${\cal C}$ lies in a unique $\ell_{\infty}$-Baer pencil $\mathcal{K}$ with some vertex $\bar{P}\in\ell_{\infty}$. Further, as argued above, the point $\bar{L}\in{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}$ projects ${\cal C}$ onto a Baer subline, and so ${\cal C}$ lies in an $\ell_{\infty}$-Baer pencil with vertex $\bar{L}$. Thus $\bar{P}=\bar{L}$ as required. $\square$

The $\ell_{\infty}$-Baer pencils give rise to partitions of the affine points of a tangent Baer subplane into $q$ conics: one degenerate and $q-1$ non-degenerate.

Corollary 2.14

Let ${\mathscr{B}}$ be a Baer subplane in ${\rm PG}(2,q^{2})$ tangent to $\ell_{\infty}$ at the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$. For each line $m$ of ${\mathscr{B}}$ through $\bar{T}$ and point $\bar{P}\in\ell_{\infty}$, $\bar{P}\neq\bar{T}$, there is a set of $q$ $\ell_{\infty}$-Baer pencils with vertex $\bar{P}$ that partition the affine points of ${\rm PG}(2,q^{2})$; and partition the affine points of ${\mathscr{B}}$ into $q$ conics through $\bar{T}$, one being degenerate. Moreover, the extension of each of these conics to ${\rm PG}(2,q^{2})$ contains the point $\bar{P}$ (see Figure 1).

Proof   The proof of Theorem 2.12 gives a construction for these partitions. The line $m$ corresponds in ${\rm PG}(4,q)$ to a line $[m]$ that meets the spread line $[T]$ in a point $X$. Let ${\mathscr{L}}$ be the set of $q$ $3$-spaces of ${\rm PG}(4,q)\backslash\Sigma_{\infty}$ containing the plane $\alpha=\langle X,[P]\rangle$. These 3-spaces partition the affine points of ${\rm PG}(4,q)$ and hence partition the affine points of $[{\mathscr{B}}]$. As argued in the proof of Theorem 2.12, one of the 3-spaces in ${\mathscr{L}}$ gives rise in ${\rm PG}(2,q^{2})$ to two lines in ${\mathscr{B}}$, and the remaining $q-1$ give rise to non-degenerate conics of ${\mathscr{B}}$ containing $\bar{T}$. By Theorem 2.13, the extension of these conics to ${\rm PG}(2,q^{2})$ contains the point $\bar{P}$. $\square$

3 Specialness and Baer sublines and subplanes

Parts 4 and 5 of Result 2.3 illustrate that the concept of $g$-specialness is important in the Bruck-Bose representation of Baer substructures. In this section we discuss how parts 1 and 3 of Result 2.3 relate to the notion of specialness.

Let $b$ be a Baer subline of $\ell_{\infty}$, then by Result 2.3(1), in ${\rm PG}(4,q)$, $[b]$ is a regulus contained in the regular spread $\mathcal{S}$. Hence in ${\rm PG}(4,q^{2})$, the transversals $g,g^{q}$ of $\mathcal{S}$ are lines of the regulus opposite to ${[b]^{\mbox{\tiny\char 73}}}$. That is, the regulus $[b]$ is closely related to the transversals of $\mathcal{S}$. There is another way to express this relationship.

Theorem 3.1

Let $b$ be a Baer subline of $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$. Then in the Bruck-Bose representation in ${\rm PG}(4,q)$, each non-degenerate conic contained in the regulus $[b]$ is a $g$-special conic. 2. 2.

Conversely, every $g$-special conic in $\Sigma_{\infty}$ lies in a unique regulus of $\mathcal{S}$, and so corresponds to a Baer subline of $\ell_{\infty}$.

Proof   Let $b$ be a Baer subline of $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$. By Result 2.3(1), in ${\rm PG}(4,q)$, $[b]$ is a regulus contained in the regular spread $\mathcal{S}$. There are $q^{3}-q$ planes of $\Sigma_{\infty}$ that meet the regulus $[b]$ is a non-degenerate conic, namely the planes that do not contain a line of $[b]$. Let $\alpha$ be such a plane, so $\alpha$ contains a unique spread line $[L]$, and ${\cal C}=[b]\cap\alpha$ is a non-degenerate conic. In ${\rm PG}(4,q^{2})$, the transversal $g$ meets each extended spread line, and so $g$ meets at least three lines of the extended regulus ${[b]^{\mbox{\tiny\char 73}}}$, hence $g$ is a line of the opposite regulus. In particular, each point of $g$ lies on one line of ${[b]^{\mbox{\tiny\char 73}}}$. Now ${\cal C}{{}^{\mbox{\tiny\char 73}}}$ is the exact intersection ${[b]^{\mbox{\tiny\char 73}}}\cap{\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}}$, and ${\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}}$ meets $g$ in one point, hence ${\cal C}{{}^{\mbox{\tiny\char 73}}}$ contains the points $g\cap{\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}}$, $g^{q}\cap{\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}}$, and so ${\cal C}$ is a $g$-special conic.

Conversely, let ${\cal C}$ be a $g$-special conic in $\Sigma_{\infty}$. So ${\cal C}$ lies in a plane $\alpha$, moreover, $\alpha$ contains a spread line $[L]$, and in ${\rm PG}(4,q^{2})$, ${\cal C}{{}^{\mbox{\tiny\char 73}}}$ contains the points $X=g\cap{[L]^{\mbox{\tiny\char 73}}}$, $X^{q}=g^{q}\cap{[L]^{\mbox{\tiny\char 73}}}$. Let $\mathcal{K}$ be the set of lines of $\mathcal{S}$ that meet ${\cal C}$, we need to show that $\mathcal{K}$ is a regulus. Let $[P_{1}],[P_{2}],[P_{3}]$ be three lines of $\mathcal{K}$ and let $\mathcal{R}$ be the unique regulus containing the three lines. By the argument above, ${\cal D}=\mathcal{R}\cap\alpha$ is a $g$-special conic, and ${\cal D}{{}^{\mbox{\tiny\char 73}}}$ contains the points $X,X^{q}$. So ${\cal C}{{}^{\mbox{\tiny\char 73}}}$, ${\cal D}{{}^{\mbox{\tiny\char 73}}}$ have five points in common, namely $X,X^{q},[P_{i}]\cap\alpha$, $i=1,2,3$. Hence ${\cal C}{{}^{\mbox{\tiny\char 73}}}={\cal D}{{}^{\mbox{\tiny\char 73}}}$ and so $\mathcal{K}=\mathcal{R}$. That is, the points of ${\cal C}$ lie on lines of a regulus of $\mathcal{S}$, which by Result 2.3 corresponds to a Baer subline of $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$. $\square$

Furthermore, the regulus $[b]$ has a relationship to the lines in the hyperbolic congruence of $g,g^{q}$.

Theorem 3.2

Let $b$ be a Baer subline of $\ell_{\infty}$, and let $\bar{P},\,\bar{Q}\in\ell_{\infty}$ be conjugate with respect to $b$. Then in ${\rm PG}(4,q^{2})$, the lines $PQ^{q}$, $P^{q}Q$ are lines of the regulus ${[b]^{\mbox{\tiny\char 73}}}$.

Proof   Let $\bar{P},\,\bar{Q}$ be two points of $\ell_{\infty}$ that are conjugate with respect to a Baer subline $b\subset\ell_{\infty}$. By Result 2.3, in ${\rm PG}(4,q)$, $[b]$ is a regulus of $\mathcal{S}$. By Result 2.8, the unique partition of $\mathcal{S}\backslash\{[P],[Q]\}$ into reguli contains the regulus $[b]$; and in ${\rm PG}(4,q^{2})$, the lines $PQ^{q},P^{q}Q$ meet each line of the regulus opposite to ${[b]^{\mbox{\tiny\char 73}}}$. Hence the lines $PQ^{q},P^{q}Q$ are lines of the regulus ${[b]^{\mbox{\tiny\char 73}}}$. $\square$

Remark 3.3

Given a Baer subline $b$ of $\ell_{\infty}$, the points of $\ell_{\infty}\backslash\{b\}$ can be partitioned into pairs of points $\{\bar{P}_{i},\bar{Q}_{i}\}$ which are conjugate with respect to $b$. Hence the $q^{2}-q$ lines $P_{i}Q_{i}^{q},(P_{i}Q_{i}^{q})^{q}$ are exactly the lines of ${\rm PG}(4,q^{2})$ in the regulus ${[b]^{\mbox{\tiny\char 73}}}$ that are not lines of ${\rm PG}(4,q)$. **

We now consider a Baer subplane ${\mathscr{B}}$ of ${\rm PG}(2,q^{2})$ secant to $\ell_{\infty}$. By Result 2.3, $[{\mathscr{B}}]$ is a plane of ${\rm PG}(4,q)$, and the line $[{\mathscr{B}}]\cap\Sigma_{\infty}$ meets $q+1$ lines of $\mathcal{S}$ which form a regulus denoted by $\mathcal{R}$. As noted above, in ${\rm PG}(4,q^{2})$, the transversals $g,g^{q}$ are lines of the regulus opposite to $\mathcal{R}$. Moreover, by Theorem 3.2 the extended regulus $\mathcal{R}{{}^{\mbox{\tiny\char 73}}}$ contains the line $PQ^{q}$ where the corresponding points $\bar{P},\bar{Q}\in\ell_{\infty}$ are conjugate with respect to ${\mathscr{B}}$.

Corollary 3.4

Let ${\mathscr{B}}$ be a Baer subplane of ${\rm PG}(2,q^{2})$ that is secant to $\ell_{\infty}$, and let $\bar{P},\bar{Q}\in\ell_{\infty}$ be conjugate with respect to ${\mathscr{B}}$. Then in ${\rm PG}(4,q^{2})$, the lines $PQ^{q}$, $P^{q}Q$ meet the plane ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$.

4 Conics of ${\rm PG}(2,q^{2})$

In [4], it is shown that a non-degenerate conic $\mathscr{O}$ in ${\rm PG}(2,q^{2})$ corresponds in ${\rm PG}(4,q)$ to the intersection of two quadrics. Moreover, this correspondence is exact-at-infinity: that is, an affine point $A\in{\rm PG}(2,q^{2})\backslash\ell_{\infty}$ lies in $\mathscr{O}$ if and only if the affine point $[A]\in{\rm PG}(4,q)\backslash\Sigma_{\infty}$ lies in $[\mathscr{O}]=\mathscr{Q}_{1}\cap\mathscr{Q}_{2}$; and a point $\bar{T}\in\ell_{\infty}$ lies in $\mathscr{O}$ if and only if the spread line $[T]$ is contained in $[\mathscr{O}]=\mathscr{Q}_{1}\cap\mathscr{Q}_{2}$. So the set $[\mathscr{O}]=\mathscr{Q}_{1}\cap\mathscr{Q}_{2}$ meets $\Sigma_{\infty}$ either in the empty set, or in 1 or 2 spread lines. We determine the relationship of $[\mathscr{O}]$ with the transversals $g,g^{q}$ of the regular spread $\mathcal{S}$.

The arguments used are coordinate based. A conic $\mathscr{O}$ has equation $f(x,y,z)=0$ where $f$ is a homogeneous equation of degree two over ${{\mathbb{F}}_{q^{2}}}$. Using the Bruck-Bose map, this can be written as $f_{\infty}(x_{0},x_{1},y_{0},y_{1},z)+\tau f_{0}(x_{0},x_{1},y_{0},y_{1},z)=0$ where $f_{\infty}=0$, $f_{0}=0$ are homogeneous quadratic equations over ${{\mathbb{F}}_{q}}$, and so they are equations of quadrics $\mathscr{Q}_{\infty}$, $\mathscr{Q}_{0}$ in ${\rm PG}(4,q)$, hence $[\mathscr{O}]=\mathscr{Q}_{\infty}\cap\mathscr{Q}_{0}$. Moreover, $[\mathscr{O}]$ is contained in the pencil of quadrics $\{\mathscr{Q}_{t}=t\mathscr{Q}_{\infty}+\mathscr{Q}_{0},t\in{{\mathbb{F}}_{q}}\cup\{\infty\}\}$ where $\mathscr{Q}_{t}$ has equation $f_{t}=tf_{\infty}+f_{0}=0$. There is a natural extension to ${\rm PG}(4,q^{2})$ and to ${\rm PG}(4,q^{4})$, namely ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}={\mathscr{Q}}_{\infty}^{\mbox{\tiny\char 73}}\cap{\mathscr{Q}}_{0}^{\mbox{\tiny\char 73}}$ and ${[{\mathscr{O}}]^{\mbox{\tiny\char 72}}}={\mathscr{Q}}_{\infty}^{\mbox{\tiny\char 72}}\cap{\mathscr{Q}}_{0}^{\mbox{\tiny\char 72}}$. In order to study subconics in Baer subplanes, we will need a full analysis of how these sets meet the hyperplane at infinity, which we give in this section. We first show that none of the quadrics ${\mathscr{Q}}_{t}^{\mbox{\tiny\char 73}}$, $t\in{{\mathbb{F}}_{q}}\cup\{\infty\}$, contain $g$, and so ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}$ does not contain $g$.

Theorem 4.1

Let $\mathscr{O}$ be a non-degenerate conic in ${\rm PG}(2,q^{2})$, so $[\mathscr{O}]=\mathscr{Q}_{\infty}\cap\mathscr{Q}_{0}$. In ${\rm PG}(4,q^{2})$, the quadric ${\mathscr{Q}}_{t}^{\mbox{\tiny\char 73}}=t{\mathscr{Q}}_{\infty}^{\mbox{\tiny\char 73}}+{\mathscr{Q}}_{0}^{\mbox{\tiny\char 73}}$, $t\in{{\mathbb{F}}_{q}}\cup\{\infty\}$, meets $g$ in [math], $1$ or $2$ points, according to whether $\mathscr{O}$ meets $\ell_{\infty}$ in [math], $1$ or $2$ points respectively.

Proof   Consider first the case when $\mathscr{O}$ is tangent to $\ell_{\infty}$. The group ${\rm PGL}(3,q^{2})$ is transitive on non-degenerate conics, and the subgroup fixing a non-degenerate conic $\mathscr{O}$ is transitive on the tangent lines of $\mathscr{O}$. Hence we can without loss of generality, prove the result for the conic $\mathscr{O}$ of equation $y^{2}=xz$ in ${\rm PG}(2,q^{2})$, which meets $\ell_{\infty}$ in one (repeated) point $\bar{T}=(1,0,0)$. The affine point $(x,y,1)=(x_{0}+x_{1}\tau,y_{0}+y_{1}\tau,1)$ is on $\mathscr{O}$ if $(y_{0}+y_{1}\tau)^{2}=x_{0}+x_{1}\tau$, that is $(y_{0}^{2}+y_{1}^{2}t_{0}-x_{0})+(y_{1}^{2}t_{1}+2y_{0}y_{1}-x_{1})\tau=0$. The solutions $(x_{0},x_{1},y_{0},y_{1},1)\in{\rm PG}(4,q)$ to this are the affine points in $[\mathscr{O}]$. That is, $[\mathscr{O}]$ is the intersection of the two quadrics $\mathscr{Q}_{\infty},\mathscr{Q}_{0}$ with homogeneous equations $f_{\infty}=0$, $f_{0}=0$ respectively, where

[TABLE]

Note that the intersection $[\mathscr{O}]=\mathscr{Q}_{\infty}\cap\mathscr{Q}_{0}$ is exact on $\Sigma_{\infty}$: both $\mathscr{Q}_{\infty}$ and $\mathscr{Q}_{0}$ contain the spread line $[T]=\{(a,b,0,0,0)\,|\,a,b\in{{\mathbb{F}}_{q}}\}$, and these are the only points of $\Sigma_{\infty}$ contained in both $\mathscr{Q}_{\infty}$ and $\mathscr{Q}_{0}$. Also note that in ${\rm PG}(4,q^{2})$, ${\mathscr{Q}}_{\infty}^{\mbox{\tiny\char 73}}$ and ${\mathscr{Q}}_{0}^{\mbox{\tiny\char 73}}$ both contain the extended spread line ${[T]^{\mbox{\tiny\char 73}}}$, and so both contain at least one point of $g$, namely ${[T]^{\mbox{\tiny\char 73}}}\cap g=A_{0}$. Also, $[\mathscr{O}]$ lies in the pencil of quadrics $\{\mathscr{Q}_{t}=t\mathscr{Q}_{\infty}+\mathscr{Q}_{0}\,|\,t\in{{\mathbb{F}}_{q}}\cup\{\infty\}\}$ where $\mathscr{Q}_{t}$ has equation $f_{t}=tf_{\infty}+f_{0}=0$. Recall that the transversal $g$ of $\mathcal{S}$ consists of the points G_{\beta}=\beta A_{0}+A_{1}=\big{(}\beta\tau^{q},\ -\beta,\ \tau^{q},\ -1,\ 0\big{)} for $\beta\in{{\mathbb{F}}_{q^{2}}}\cup\{\infty\}$. For $\beta\in{{\mathbb{F}}_{q^{2}}}$, we have $f_{\infty}(G_{\beta})=\tau^{q}(\tau^{q}-\tau)$ and $f_{0}(G_{\beta})=\tau-\tau^{q}$. Let $f_{t}=tf_{\infty}+f_{0}$, then $f_{t}(G_{\beta})=(\tau^{q}-\tau)(t\tau^{q}-1)$ which is never zero when $t\in{{\mathbb{F}}_{q}}$. Hence $G_{\infty}=A_{0}$ is the only point of $g$ contained in the quadric $\mathscr{Q}_{t}$. Similarly, $A_{0}^{q}$ is the only point of the (other) transversal $g^{q}$ contained in the quadric $\mathscr{Q}_{t}$. That is, when $\mathscr{O}$ is tangent to $\ell_{\infty}$, the quadrics ${\mathscr{Q}}_{t}^{\mbox{\tiny\char 73}}$ each meet $g$ in one point, namely ${[T]^{\mbox{\tiny\char 73}}}\cap g=A_{0}$. A similar argument using the conic with equation $f(x,y,z)=x^{2}-\delta y^{2}+z^{2}$, $\delta\in{{\mathbb{F}}_{q^{2}}}\backslash\{0\}$ for $q$ odd, and $\delta x^{2}+y^{2}+z^{2}+yx=0$, $\delta\in{{\mathbb{F}}_{q^{2}}}$ for $q$ even completes the other cases. $\square$

The proof of Theorem 4.1, and the 1-1 correspondence between points $\bar{P}$ of $\ell_{\infty}$ and points ${P}=[P]{{}^{\mbox{\tiny\char 73}}}\cap g$ of the transversal $g$, allow us to identify the points of the quadric ${\mathscr{Q}}_{t}^{\mbox{\tiny\char 73}}$ on $g$.

Corollary 4.2

Let $\mathscr{O}$ be a non-degenerate conic in ${\rm PG}(2,q^{2})$, then

$\bar{P}\in\mathscr{O}\cap\ell_{\infty}$* if and only if in ${\rm PG}(4,q^{2})$, ${P}\in g$;* 2. 2.

$\bar{P}$* is a point in the intersection of the extension of $\mathscr{O}$ and the extension of $\ell_{\infty}$ to ${\rm PG}(2,q^{4})$ if and only if in ${\rm PG}(4,q^{4})$, $P\in g^{\mbox{\tiny\char 72}}\backslash g$.*

Next we consider the set $[\mathscr{O}]$ extended to ${\rm PG}(4,q^{2})$ and ${\rm PG}(4,q^{4})$, and determine the exact intersection with the hyperplane at infinity.

Theorem 4.3

Let $\mathscr{O}$ be a non-degenerate conic in ${\rm PG}(2,q^{2})$.

Suppose $\mathscr{O}$ is secant to $\ell_{\infty}$, so $\mathscr{O}\cap\ell_{\infty}=\{\bar{P},\,\bar{Q}\}$, then

1. (a)

in ${\rm PG}(4,q)$, $[\mathscr{O}]\cap\Sigma_{\infty}=\{[P],[Q]\}$; 2. (b)

in ${\rm PG}(4,q^{2})$, ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\{{[P]^{\mbox{\tiny\char 73}}},{[Q]^{\mbox{\tiny\char 73}}},PQ^{q},P^{q}Q\}$; 3. (c)

in ${\rm PG}(4,q^{4})$, ${[{\mathscr{O}}]^{\mbox{\tiny\char 72}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{{[P]^{\mbox{\tiny\char 72}}},\ {[Q]^{\mbox{\tiny\char 72}}},\ (PQ^{q})^{{}^{\mbox{\tiny\char 72}}},\ (P^{q}Q)^{{}^{\mbox{\tiny\char 72}}}\}$. 2. 2.

Suppose $\mathscr{O}$ is tangent to $\ell_{\infty}$, so $\mathscr{O}\cap\ell_{\infty}=\{\bar{P}\}$, then

1. (a)

in ${\rm PG}(4,q)$, $[\mathscr{O}]\cap\Sigma_{\infty}=\{[P]\}$; 2. (b)

in ${\rm PG}(4,q^{2})$, ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\{{[P]^{\mbox{\tiny\char 73}}}\}$, 3. (c)

in ${\rm PG}(4,q^{4})$, ${[{\mathscr{O}}]^{\mbox{\tiny\char 72}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{{[P]^{\mbox{\tiny\char 72}}}\}$. 3. 3.

Suppose $\mathscr{O}$ is exterior to $\ell_{\infty}$, so in the extension to ${\rm PG}(2,q^{4})$, the extension of $\mathscr{O}$ meets the extension of $\ell_{\infty}$ in two points $\{\bar{P},\bar{P}^{q^{2}}\}$. Then

1. (a)

in ${\rm PG}(4,q)$, $[\mathscr{O}]\cap\Sigma_{\infty}=\emptyset$; 2. (b)

in ${\rm PG}(4,q^{2})$, ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\emptyset$; 3. (c)

in ${\rm PG}(4,q^{4})$, {[{\mathscr{O}}]^{\mbox{\tiny\char 72}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{\ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}},\ \ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}^{q},\ \ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}^{q^{2}},\ \ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}^{q^{3}}\}, where \ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}=PP^{q}.

Proof   As noted above, $[\mathscr{O}]=\mathscr{Q}_{\infty}\cap\mathscr{Q}_{0}$, for quadrics $\mathscr{Q}_{\infty},\mathscr{Q}_{0}$, and this correspondence is exact, so $[\mathscr{O}]$ meets $\Sigma_{\infty}$ in either the empty set, or in 1 or 2 spread lines (corresponding respectively to $\mathscr{O}$ meeting $\ell_{\infty}$ in 0, 1 or 2 points). The cases $\mathscr{O}$ tangent, secant and exterior to $\ell_{\infty}$, $q$ odd and even, are proved separately using the same conic equations as in the proof of Theorem 4.1. We omit the calculations, noting that we rely on [11, Table 2] to show that the intersection of the two quadrics in the 3-space ${\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ is a set of four lines, possibly repeated. $\square$

We have shown that in ${\rm PG}(4,q^{2})$, the set ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}$ contains an extended spread line ${[P]^{\mbox{\tiny\char 73}}}$ if and only if in ${\rm PG}(2,q^{2})$, the point $\bar{P}\in\mathscr{O}\cap\ell_{\infty}$. We will need the next corollary which considers whether the set ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}$ can contain a point of any other extended spread line.

Corollary 4.4

Let $\mathscr{O}$ be a non-degenerate conic in ${\rm PG}(2,q^{2})$. Let $\bar{L}$ be a point of $\ell_{\infty}$ not in $\mathscr{O}$. In ${\rm PG}(4,q^{2})$, the corresponding extended spread line ${[L]^{\mbox{\tiny\char 73}}}$ is disjoint from ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}$.

Proof   If $\mathscr{O}$ is secant to $\ell_{\infty}$, so $\mathscr{O}\cap\ell_{\infty}=\{\bar{P},\,\bar{Q}\}$, then by Theorem 4.3, ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ consists of the four lines ${[P]^{\mbox{\tiny\char 73}}},{[Q]^{\mbox{\tiny\char 73}}},PQ^{q},P^{q}Q$. Let ${[L]^{\mbox{\tiny\char 73}}}$ be an extended spread line, $\bar{L}\neq\bar{P},\,\bar{Q}$. Then ${[L]^{\mbox{\tiny\char 73}}},{[P]^{\mbox{\tiny\char 73}}},{[Q]^{\mbox{\tiny\char 73}}},PQ^{q},P^{q}Q$ are all lines of the hyperbolic congruence of $g,g^{q}$, and so do not meet off $g,g^{q}$, and hence are mutually skew. So ${[L]^{\mbox{\tiny\char 73}}}\cap{[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}=\emptyset$. If $\mathscr{O}$ is tangent to $\ell_{\infty}$, then by Theorem 4.3, ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}={[P]^{\mbox{\tiny\char 73}}}$. Hence ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}$ meets no other spread line. If $\mathscr{O}$ is exterior to $\ell_{\infty}$, then by Theorem 4.3, ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\emptyset$, so ${[{\mathscr{O}}]^{\mbox{\tiny\char 73}}}$ contains no point on any extended spread line, as required. $\square$

5 Conics of Baer subplanes

In this section we improve Result 2.5 by characterising the normal rational curves of ${\rm PG}(4,q)$ that correspond to conics of a Baer subplane of ${\rm PG}(2,q^{2})$. In particular, we show that if ${\cal C}$ is a conic contained in a tangent Baer subplane ${\mathscr{B}}$ of ${\rm PG}(2,q^{2})$, then in ${\rm PG}(4,q)$, the corresponding 3- or 4-dimensional normal rational curve $[{\cal C}]$ is $g$-special. Further, we show that any $g$-special 3- or 4-dimensional normal rational curve in ${\rm PG}(4,q)$ corresponds to a conic in a Baer subplane of ${\rm PG}(2,q^{2})$.

5.1 $\mathbb{F}_{q^{2}}$-conics and $\mathbb{F}_{q}$-conics

In this section we show that the notion of specialness is also intrinsic to the Bruck-Bose representation of conics in Baer subplanes. First we introduce some notation to easily distinguish between conics in ${\rm PG}(2,q^{2})$ and conics contained in a Baer subplane. An $\mathbb{F}_{q^{2}}$-conic in ${\rm PG}(2,q^{2})$ is a non-degenerate conic of ${\rm PG}(2,q^{2})$. Note that an $\mathbb{F}_{q^{2}}$-conic meets a Baer subplane ${\mathscr{B}}$ in either 0, 1, 2, 3 or 4 points, or in a non-degenerate conic of ${\mathscr{B}}$. We define an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{2})$ to be a non-degenerate conic in a Baer subplane of ${\rm PG}(2,q^{2})$. For the remainder of this article, ${\cal C}$ will denote an $\mathbb{F}_{q}$-conic. Further, we denote the unique $\mathbb{F}_{q^{2}}$-conic containing ${\cal C}$ by ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$. An $\mathbb{F}_{q^{2}}$-conic contains many $\mathbb{F}_{q}$-conics.

Lemma 5.1

Let $\mathscr{O}$ be an $\mathbb{F}_{q^{2}}$-conic in ${\rm PG}(2,q^{2})$. Any three points of $\mathscr{O}$ lie in a unique $\mathbb{F}_{q}$-conic that is contained in $\mathscr{O}$, so there are $q(q^{2}+1)$ $\mathbb{F}_{q}$-conics contained in $\mathscr{O}$.

Proof   The $\mathbb{F}_{q^{2}}$-conic $\mathscr{O}$ is equivalent to the line $\ell\cong{\rm PG}(1,q^{2})$, and subconics of $\mathscr{O}$ are equivalent to Baer sublines of $\ell$. Since three points of $\ell$ lie in a unique Baer subline of $\ell$, three points of $\mathscr{O}$ lie in a unique subconic ${\cal C}$. As ${\cal C}$ is a normal rational curve over ${{\mathbb{F}}_{q}}$, by [15, Theorem 21.1.1] there is a homography $\phi$ that maps ${\cal C}$ to ${\cal C}^{\prime}=\phi({\cal C})=\{(1,\theta,\theta^{2})\,|\,\theta\in{{\mathbb{F}}_{q}}\cup\{\infty\}\}$. As ${\cal C}^{\prime}$ lies in the Baer subplane ${\mathscr{B}}^{\prime}={\rm PG}(2,q)$, ${\cal C}$ lies in the Baer subplane $\phi^{-1}({\mathscr{B}}^{\prime})$, that is, ${\cal C}$ is an $\mathbb{F}_{q}$-conic. Straightforward counting shows that the number of $\mathbb{F}_{q}$-conics in $\mathscr{O}$ is $(q^{2}+1)q^{2}(q^{2}-1)/(q+1)q(q-1)=q(q^{2}+1)$. $\square$

Remark 5.2

Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{2})$, $q>4$, so there is a unique $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ with ${\cal C}\subset{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$. Then in ${\rm PG}(4,q)$, $[{\cal C}]\subset[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$. This is clearly true for the affine points. For the points at infinity, we recall Remark 2.6, if $\bar{T}\in{\cal C}\cap\ell_{\infty}\subseteq{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}$, then $[{\cal C}]$ meets the spread line $[T]$ in a point, while $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$ contains the spread line $[T]$.

5.2 Conics in secant Baer subplanes

In this section we consider the Bruck-Bose representation of $\mathbb{F}_{q}$-conics in secant Baer subplanes of ${\rm PG}(2,q^{2})$, in particular looking at the relationship with the lines of the hyperbolic congruence of $g,g^{q}$.

Theorem 5.3

Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic in a Baer subplane ${\mathscr{B}}$ secant to $\ell_{\infty}$. The $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets $\ell_{\infty}$ in two points $\bar{P},\,\bar{Q}$ (possibly equal). In ${\rm PG}(4,q)$, $[{\cal C}]$ is a non-degenerate conic in the plane $[{\mathscr{B}}]$, and $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]\cap\Sigma_{\infty}$ is the two spread lines $[P]$, $[Q]$.

If $\bar{P}=\bar{Q}$, then $\bar{P}\in{\mathscr{B}}$, and $[{\cal C}]$ meets $\Sigma_{\infty}$ in one point $[P]\cap[{\mathscr{B}}]$. 2. 2.

If $\bar{P}\neq\bar{Q}$ and $\bar{P},\,\bar{Q}\in{\mathscr{B}}$, then $[{\cal C}]$ meets $\Sigma_{\infty}$ in two points $[P]\cap[{\mathscr{B}}]$ and $[Q]\cap[{\mathscr{B}}]$. 3. 3.

If $\bar{P}\neq\bar{Q}$ and $\bar{P},\,\bar{Q}\notin{\mathscr{B}}$, then $[{\cal C}]$ is a $(PQ^{q})$-special conic.

Proof   By Results 2.3 and 2.5, in ${\rm PG}(4,q)$, $[{\mathscr{B}}]$ is a plane, and $[{\cal C}]$ is a conic in $[{\mathscr{B}}]$. Parts 1 and 2 follow immediately from the Bruck-Bose definition. For part 3, the set $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$ contains the spread lines $[P]$ and $[Q]$. The set $[{\mathscr{B}}]$ is a plane, and the line $m=[{\mathscr{B}}]\cap\Sigma_{\infty}$ meets $q+1$ spread lines, but does not meet the spread lines $[P]$, $[Q]$. The set $[{\cal C}]$ is a non-degenerate conic in $[{\mathscr{B}}]$ which does not meet $m$, and in the extension to ${\rm PG}(4,q^{2})$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets ${\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ in two points of the line ${m^{\mbox{\tiny\char 73}}}={[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$. In ${\rm PG}(2,q^{2})$, we have ${\cal C}={\mathscr{B}}\cap{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$, and in ${\rm PG}(4,q)$, $[{\cal C}]=[{\mathscr{B}}]\cap[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$. Moreover, in ${\rm PG}(4,q^{2})$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}={[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}$, hence ${[{\cal C}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\{{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}\}\cap\{[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}\}$. By Theorem 4.3, this is equal to $\{{m^{\mbox{\tiny\char 73}}}\}\cap\{g,g^{q},PQ^{q},P^{q}Q\}$. Now ${m^{\mbox{\tiny\char 73}}}$ does not meet $g$ (or $g^{q}$) as the only lines of $\Sigma_{\infty}$ whose extension meets $g$ are the lines of $\mathcal{S}$. Hence the two points of ${[{\cal C}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ lie in $PQ^{q}$ and $P^{q}Q$, that is, $[{\cal C}]$ is a $(PQ^{q})$-special conic of ${\rm PG}(4,q)$. $\square$

We now characterise $\mathbb{F}_{q}$-conics in secant Baer subplanes by showing that the converse is true.

Theorem 5.4

In ${\rm PG}(4,q)$, let $\alpha$ be a plane not containing a spread line, and let ${\cal N}$ be a non-degenerate conic in $\alpha$.

In ${\rm PG}(2,q^{2})$, there is a secant Baer subplane ${\mathscr{B}}$ with $[{\mathscr{B}}]=\alpha$, and an $\mathbb{F}_{q}$-conic ${\cal C}$ in ${\mathscr{B}}$ with $[{\cal C}]={\cal N}$. 2. 2.

If ${\cal N}$ meets $\Sigma_{\infty}$ in a point of the spread line $[T]$, then $\bar{T}\in{\cal C}$. 3. 3.

If ${\cal N}$ is a $(PQ^{q})$-special conic, then the $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ containing ${\cal C}$ meets $\ell_{\infty}$ in the points $\bar{P},\,\bar{Q}$.

Proof   Parts 1 and 2 follow from Result 2.3. For part 3, in ${\rm PG}(4,q)$, let ${\cal N}$ be a $(PQ^{q})$-special conic of ${\rm PG}(4,q)$ lying in a plane $\alpha$ that does not contain a spread line. By part 1, $[{\mathscr{B}}]=\alpha$ and $[{\cal C}]={\cal N}$ where in ${\rm PG}(2,q^{2})$, ${\mathscr{B}}$ is a secant Baer subplane containing the $\mathbb{F}_{q}$-conic ${\cal C}$. As ${\cal N}$ is a $(PQ^{q})$-special conic, ${\cal N}\cap\Sigma_{\infty}=\emptyset$, and in ${\rm PG}(4,q^{2})$, ${{\cal N}^{\mbox{\tiny\char 73}}}$ is a conic which meets the line $\alpha\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ is two points, one lying on each of $PQ^{q}$ and $P^{q}Q$. As ${\cal N}\cap\Sigma_{\infty}=\emptyset$, in ${\rm PG}(2,q^{2})$, the $\mathbb{F}_{q}$-conic ${\cal C}$ does not meet $\ell_{\infty}$, so the $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets $\ell_{\infty}$ in two points $\bar{A},\bar{B}\notin{\mathscr{B}}$. By Theorem 5.3, $[{\cal C}]={\cal N}$ is a $(AB^{q})$-special conic. Hence $\{\bar{A},\bar{B}\}=\{\bar{P},\,\bar{Q}\}$, so ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}=\{\bar{P},\,\bar{Q}\}$ as required. $\square$

5.3 Conics in tangent Baer subplanes

We now consider a Baer subplane ${\mathscr{B}}$ that is tangent to $\ell_{\infty}$ and look at $\mathbb{F}_{q}$-conics in ${\mathscr{B}}$. There are two cases to consider, namely whether the $\mathbb{F}_{q}$-conic contains the point ${\mathscr{B}}\cap\ell_{\infty}$ or not. In each case we generalise Result 2.5 by showing that the corresponding normal rational curve of ${\rm PG}(4,q)$ is $g$-special. Further, we characterise all $g$-special normal rational curves in ${\rm PG}(4,q)$ as corresponding to $\mathbb{F}_{q}$-conics in a tangent Baer subplane.

5.3.1 Conics in ${\mathscr{B}}$ containing the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$

We first look at an $\mathbb{F}_{q}$-conic ${\cal C}$ in a tangent Baer subplane ${\mathscr{B}}$, with ${\mathscr{B}}\cap\ell_{\infty}$ in ${\cal C}$.

Theorem 5.5

In ${\rm PG}(2,q^{2})$, $q>5$, let ${\mathscr{B}}$ be a tangent Baer subplane and ${\cal C}$ an $\mathbb{F}_{q}$-conic in ${\mathscr{B}}$ containing the point ${\mathscr{B}}\cap\ell_{\infty}$. Then in ${\rm PG}(4,q)$, $[{\cal C}]$ is a $g$-special twisted cubic.

Proof   Let ${\mathscr{B}}$ be a Baer subplane of ${\rm PG}(2,q^{2})$ that is tangent to $\ell_{\infty}$ in the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$. Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic of ${\mathscr{B}}$ that contains $\bar{T}$. By Result 2.5, in ${\rm PG}(4,q)$, ${\cal C}$ corresponds to a twisted cubic $[{\cal C}]$ that lies in a $3$-space denoted \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal C}}}}}}}. By Result 2.1, \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal C}}}}}}} meets the ruled cubic surface $[{\mathscr{B}}]$ in exactly the twisted cubic $[{\cal C}]$. We show that $[{\cal C}]$ is $g$-special. By Lemma 2.11, the $3$-space \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal C}}}}}}} corresponds to an $\ell_{\infty}$-Baer pencil $\mathcal{K}$ of ${\rm PG}(2,q^{2})$ that meets ${\mathscr{B}}$ in ${\cal C}$. By Theorem 2.13, $\mathcal{K}$ has vertex $\bar{P}\in{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$. Hence \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal C}}}}}}} contains the spread line $[P]$ (and this is the only spread line in \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal C}}}}}}}). Consider the extension of ${\rm PG}(4,q)$ to ${\rm PG}(4,q^{2})$. Note that Lemma 2.9 can be generalised to a 3-dimensional normal rational curve when $q>5$. Hence as $[{\mathscr{B}}]$ is the intersection of three quadrics [3], we have ${[{\cal C}]^{\mbox{\tiny\char 73}}}\subset{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$ in ${\rm PG}(4,q^{2})$. Thus by Corollary 2.2, the twisted cubic ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ contains a unique point of each generator line of the ruled cubic surface ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$. By Result 2.3, $[{\mathscr{B}}]$ is $g$-special, so the transversal lines $g,g^{q}$ of the regular spread $\mathcal{S}$ are generator lines of the extended ruled cubic surface ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$. Hence ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ contains a point of $g$ and $g^{q}$. Thus ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ contains the points corresponding to the vertex of $\mathcal{K}$, that is, the point g\cap{\Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal C}}}}}}}}^{\mbox{\tiny\char 73}}=g\cap{[P]^{\mbox{\tiny\char 73}}}=P and $P^{q}$. That is, the twisted cubic $[{\cal C}]$ is $g$-special. $\square$

The converse of Theorem 5.5 is also true.

Theorem 5.6

A $g$-special twisted cubic in ${\rm PG}(4,q)$ corresponds to an $\mathbb{F}_{q}$-conic in some tangent Baer subplane of ${\rm PG}(2,q^{2})$.

Proof   Let ${\cal N}$ be a $g$-special twisted cubic in ${\rm PG}(4,q)$, so in ${\rm PG}(4,q^{2})$, ${{\cal N}^{\mbox{\tiny\char 73}}}$ meets the transversal $g$ of $\mathcal{S}$ in a point $R$, and meets $g^{q}$ in the point $R^{q}$. The line $RR^{q}$ meets $\Sigma_{\infty}$ in a spread line denoted $[R]$, corresponding to the point $\bar{R}\in\ell_{\infty}$. Let \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal N}}}}}}} be the 3-space containing ${\cal N}$, and recall that a twisted cubic meets a plane in three points, possibly repeated, or in an extension. As ${\cal N}$ meets the plane \pi=\Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal N}}}}}}}\cap\Sigma_{\infty} in two points, $R,R^{q}$ over ${{\mathbb{F}}_{q^{2}}}$, ${\cal N}$ meets $\pi$ in one point $X$ over ${{\mathbb{F}}_{q}}$. Let $[T]$ be the spread line containing the point $X$, so [T]\notin\Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal N}}}}}}}. Let $[A],[B],[C]$ be three affine points of ${\cal N}$, and let $\alpha=\langle[A],[B],[C]\rangle$.

As $\alpha$ lies in the 3-space \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal N}}}}}}}, if $\alpha$ contained a spread line, it would contain $[R]$. However, if $\alpha$ contains $[R]$, then the plane $\langle[A],[B],[R]\rangle{{}^{\mbox{\tiny\char 73}}}$ would contain four points of ${{\cal N}^{\mbox{\tiny\char 73}}}$, namely $[A],[B],R,R^{q}$, a contradiction. If $\alpha$ contained the point $X$, then $\alpha$ would contain four points of ${\cal N}$, namely $X,[A],[B],[C]$, a contradiction. Hence $\alpha$ corresponds to a Baer subplane ${\mathscr{B}}_{\alpha}$ of ${\rm PG}(2,q^{2})$ that is secant to $\ell_{\infty}$, with $\bar{T}\notin{\mathscr{B}}_{\alpha}$. Hence the points $\{\bar{T},A,B,C\}$ form a quadrangle and so lie in a unique Baer subplane denoted ${\mathscr{B}}$. As ${\mathscr{B}}_{\alpha}$ is the unique Baer subplane containing $A,B,C$ and secant to $\ell_{\infty}$, we have ${\mathscr{B}}\neq{\mathscr{B}}_{\alpha}$, and ${\mathscr{B}}$ is tangent to $\ell_{\infty}$ at the point $\bar{T}$.

In ${\rm PG}(4,q)$, $[{\mathscr{B}}]$ is a ruled cubic surface with line directrix $[T]$. As $X,[A],[B],[C]$ are points of ${\cal N}$, no three are collinear, so $[A],[B],[C]$ lie on distinct generators of $[{\mathscr{B}}]$. Recall that \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal N}}}}}}} does not contain $[T]$, so by Result 2.1, \Pi_{\mbox{\raisebox{-0.1pt}{\scalebox{0.65}{{\boldmath{{\cal N}}}}}}} meets $[{\mathscr{B}}]$ in a twisted cubic, denoted ${\cal N}_{1}$. The argument in the proof of Theorem 5.5 shows that in the quadratic extension, ${{\cal N}}_{1}^{\mbox{\tiny\char 73}}$ contains the points $R$ and $R^{q}$. Hence ${{\cal N}^{\mbox{\tiny\char 73}}}$ and ${{\cal N}}_{1}^{\mbox{\tiny\char 73}}$ share six points, and so are equal. That is, ${\cal N}$ is a $g$-special twisted cubic contained in a $g$-special ruled cubic surface, and ${\cal N}$ meets $\Sigma_{\infty}$ in one point.

Straightforward counting shows that in ${\rm PG}(2,q^{2})$, the number of $\mathbb{F}_{q}$-conics in ${\mathscr{B}}$ that contain $\bar{T}$ is $q^{4}-q^{2}$. By Result 2.1, the number of $3$-spaces of ${\rm PG}(4,q)$ that meet the ruled cubic surface $[{\mathscr{B}}]$ in a twisted cubic is $q^{4}-q^{2}$. Hence they are in one to one correspondence. That is, ${\cal N}$ corresponds to an $\mathbb{F}_{q}$-conic in the tangent Baer subplane ${\mathscr{B}}$ as required. $\square$

Further, the proofs of Theorems 5.5 and 5.6 show that the points of $g$ on a $g$-special twisted cubic correspond to the points on $\ell_{\infty}$ contained in the corresponding $\mathbb{F}_{q^{2}}$-conic.

Corollary 5.7

Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic in a tangent Baer subplane ${\mathscr{B}}$ in ${\rm PG}(2,q^{2})$, $q>5$, with $\bar{T}={\mathscr{B}}\cap\ell_{\infty}\in{\cal C}$. The $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets $\ell_{\infty}$ in a point $\bar{P}\neq\bar{T}$ if and only if in ${\rm PG}(4,q^{2})$, the twisted cubic ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets the transversals of $\mathcal{S}$ in the points $P$, $P^{q}$.

5.3.2 Conics of ${\mathscr{B}}$ not containing the point $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$

We now look at an $\mathbb{F}_{q}$-conic ${\cal C}$ in a tangent Baer subplane ${\mathscr{B}}$, with ${\mathscr{B}}\cap\ell_{\infty}$ not in ${\cal C}$. The $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets $\ell_{\infty}$ in two distinct points (which may lie in ${\rm PG}(2,q^{4})$). We show that if these two points lie in ${\rm PG}(2,q^{2})$, then $[{\cal C}]$ is a $g$-special normal rational curve. Further, if the two points lie in the quadratic extension of ${\rm PG}(2,q^{2})$ to ${\rm PG}(2,q^{4})$, then $[{\cal C}]$ is an $g^{\mbox{\tiny\char 72}}$-special normal rational curve.

Theorem 5.8

In ${\rm PG}(2,q^{2})$, $q>7$, let ${\mathscr{B}}$ be a Baer subplane tangent to $\ell_{\infty}$ with $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$. Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic in ${\mathscr{B}}$, $\bar{T}\notin{\cal C}$. In ${\rm PG}(4,q)$, $[{\cal C}]$ is a $g$-special or $g^{\mbox{\tiny\char 72}}$-special 4-dimensional normal rational curve.

Proof   Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic in ${\mathscr{B}}$ not through $\bar{T}={\mathscr{B}}\cap\ell_{\infty}$, and consider the $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ containing ${\cal C}$. Then either (i) ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is secant to $\ell_{\infty}$ and ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}$ is two distinct points $\bar{P},\,\bar{Q}$; (ii) ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is tangent to $\ell_{\infty}$ and ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}$ is a repeated point $\bar{P}=\bar{Q}$; or (iii) ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is exterior to $\ell_{\infty}$ and in ${\rm PG}(2,q^{4})$, the extension of ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets the extension of $\ell_{\infty}$ in two points $\bar{P},\,\bar{Q}$ which are conjugate with respect to this extension from ${\rm PG}(2,q^{2})$ to ${\rm PG}(2,q^{4})$, that is, $\bar{Q}=\bar{P}^{q^{2}}$. By Result 2.5, as $\bar{T}\notin{\cal C}$, in ${\rm PG}(4,q)$, $[{\cal C}]$ is a 4-dimensional normal rational curve lying on the $g$-special ruled cubic surface $[{\mathscr{B}}]$, and $[{\cal C}]$ does not meet $\Sigma_{\infty}$. Thus it remains to show that in ${\rm PG}(4,q)$ $[{\cal C}]$ is a $g$-special or $g^{\mbox{\tiny\char 72}}$-special. We will show that in an appropriate extension of ${\rm PG}(4,q)$, the extension of the normal rational curve $[{\cal C}]$ contains the points ${P},{Q}$ of the (possibly extended) transversal $g$, giving the $g$-special property. Recall that a 4-dimensional normal rational curve of ${\rm PG}(4,q)$ meets the $3$-space $\Sigma_{\infty}$ in four points, possibly repeated or in an extension. As $[{\cal C}]$ is disjoint from $\Sigma_{\infty}$, either (a) in ${\rm PG}(4,q^{2})$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets ${\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ in four points of the form $X,X^{q},Y,Y^{q}$, possibly $X=Y$; or (b) in ${\rm PG}(4,q^{4})$, $[{\cal C}]^{\mbox{\tiny\char 72}}$ meets ${\Sigma}_{\infty}^{\mbox{\tiny\char 72}}$ in four points of form $X,X^{q},X^{q^{2}},X^{q^{3}}$.

In ${\rm PG}(2,q^{2})$, we have ${\cal C}\subset{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$, so as discussed in Remark 5.2, in ${\rm PG}(4,q)$, $[{\cal C}]\subset[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$. By [4, Cor 3.3], $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$ is the exact intersection of two quadrics, so by Lemma 2.9: in ${\rm PG}(4,q^{2})$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}\subset[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}$; and in ${\rm PG}(4,q^{4})$, $[{\cal C}]^{\mbox{\tiny\char 72}}\subset[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}$. Similarly, as $[{\cal C}]\subset[{\mathscr{B}}]$ and $[{\mathscr{B}}]$ is the intersection of three quadrics [3], by Lemma 2.9, ${[{\cal C}]^{\mbox{\tiny\char 73}}}\subset{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$ and $[{\cal C}]^{\mbox{\tiny\char 72}}\subset[{\mathscr{B}}]^{\mbox{\tiny\char 72}}$. In ${\rm PG}(2,q^{2})$, we have ${\cal C}={\mathscr{B}}\cap{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$. As $[{\cal C}]$ is disjoint from $\Sigma_{\infty}$, in ${\rm PG}(4,q)$, we have $[{\cal C}]=[{\mathscr{B}}]\cap[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]$. We need to determine ${[{\cal C}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ in ${\rm PG}(4,q^{2})$, and $[{\cal C}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap[{\mathscr{B}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}$ in ${\rm PG}(4,q^{4})$.

First we determine ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ and $[{\mathscr{B}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}$. In ${\rm PG}(2,q^{2})$, $\bar{T}\in{\mathscr{B}}$, so in ${\rm PG}(4,q)$, $[T]\subset[{\mathscr{B}}]$, and $[{\mathscr{B}}]\cap\Sigma_{\infty}=[T]$. Hence in ${\rm PG}(4,q^{2})$, ${[T]^{\mbox{\tiny\char 73}}}\subset{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$, and in ${\rm PG}(4,q^{4})$, $[T]^{\mbox{\tiny\char 72}}\subset[{\mathscr{B}}]^{\mbox{\tiny\char 72}}$. By Result 2.3, $[{\mathscr{B}}]$ is a $g$-special ruled cubic surface, so the transversal lines $g,g^{q}$ lie in ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$. That is, $\{{[T]^{\mbox{\tiny\char 73}}},g,g^{q}\}$ lie in ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$, and using Result 2.1 in ${\rm PG}(4,q^{2})$, the 3-space ${\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ meets the ruled cubic surface ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}$ in exactly these three lines, so ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\{{[T]^{\mbox{\tiny\char 73}}},g,g^{q}\}$. Similarly, in ${\rm PG}(4,q^{4})$, the 3-space ${\Sigma}_{\infty}^{\mbox{\tiny\char 72}}$ meets the ruled cubic surface $[{\mathscr{B}}]^{\mbox{\tiny\char 72}}$ in the three lines $\{[T]^{\mbox{\tiny\char 72}},g^{\mbox{\tiny\char 72}},{g^{q}}^{\mbox{\tiny\char 72}}\}$.

Recall that Theorem 4.3 determines the intersection $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ and $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}$ for the three cases where ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is (i) secant, (ii) tangent or (iii) exterior to $\ell_{\infty}$ in ${\rm PG}(2,q^{2})$. For each case we determine $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ in ${\rm PG}(4,q^{2})$ and $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap[{\mathscr{B}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}$ in ${\rm PG}(4,q^{2})$.

In case (i), ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is secant to $\ell_{\infty}$, so by Theorem 4.3, $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\{{[P]^{\mbox{\tiny\char 73}}},{[Q]^{\mbox{\tiny\char 73}}},{P}Q^{q},\ P^{q}{Q}\}$. Now ${[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\{{[T]^{\mbox{\tiny\char 73}}},g,g^{q}\}$, and by Corollary 4.4, $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ does not meet ${[T]^{\mbox{\tiny\char 73}}}$. Hence $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ consists of the four points ${P},{Q},P^{q},Q^{q}$. Similarly, $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap[{\mathscr{B}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{{P},{Q},P^{q},Q^{q}\}$. As ${[{\cal C}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets $g$ in two distinct points, namely $P,Q$, and so $[{\cal C}]$ is a $g$-special normal rational curve.

In case (ii), ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is tangent to $\ell_{\infty}$, so by Theorem 4.3, \big{\{}[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}\big{\}}\cap\big{\{}{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}\big{\}}=\{{[P]^{\mbox{\tiny\char 73}}}\}\ \cap\ \{{[T]^{\mbox{\tiny\char 73}}},g,g^{q}\}=\{{P},P^{q}\}. Similarly, $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap[{\mathscr{B}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{{P},P^{q}\}$. Hence ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets $g$ in the repeated point ${P}$, and so $[{\cal C}]$ is a $g$-special normal rational curve.

In case (iii), ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is exterior to $\ell_{\infty}$, so in ${\rm PG}(2,q^{4})$, the extension of ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets the extension of $\ell_{\infty}$ in two points $\bar{P},\,\bar{Q}$, where $\bar{Q}=\bar{P}^{q^{2}}$. By Theorem 4.3, $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\emptyset$ and [{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{\ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}},\ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}^{q},\ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}^{q^{2}},\ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}^{q^{3}}\}, where \ell_{\mbox{\raisebox{-0.15pt}{\scalebox{0.55}{{{!P}}}}}}=PP^{q}. Hence $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 73}}\cap{[{\mathscr{B}}]^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}=\emptyset$, and $[{\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}]^{\mbox{\tiny\char 72}}\cap[{\mathscr{B}}]^{\mbox{\tiny\char 72}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 72}}=\{{P},P^{q},{P}^{q^{2}},{P}^{q^{3}}\}$. So in this case the normal rational curve $[{\cal C}]$ meets $\Sigma_{\infty}$ in four points over ${{\mathbb{F}}_{q^{4}}}$. As $[{\cal C}]^{\mbox{\tiny\char 72}}$ meets $g^{\mbox{\tiny\char 72}}$ in two points (namely ${P}$ and ${P}^{q^{2}}={Q}$) $[{\cal C}]$ is an $g^{\mbox{\tiny\char 72}}$-special normal rational curve. $\square$

We now show that conversely, every $g$-special or $g^{\mbox{\tiny\char 72}}$-special normal rational curve corresponds to an $\mathbb{F}_{q}$-conic.

Theorem 5.9

Let ${\cal N}$ be a $g$-special or $g^{\mbox{\tiny\char 72}}$-special 4-dimensional normal rational curve in ${\rm PG}(4,q)$. Then ${\cal N}=[{\cal C}]$ where ${\cal C}$ is an $\mathbb{F}_{q}$-conic in a tangent Baer subplane of ${\rm PG}(2,q^{2})$.

Proof   Let ${\cal N}$ be a $g$-special 4-dimensional normal rational curve in ${\rm PG}(4,q)$. So there are two spread lines $[P]$, $[Q]$ (possibly equal) such that ${{\cal N}^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ consists of the four points $P=g\cap{[P]^{\mbox{\tiny\char 73}}}$, $P^{q}=g^{q}\cap{[P]^{\mbox{\tiny\char 73}}}$, $Q=g\cap{[Q]^{\mbox{\tiny\char 73}}}$, $Q^{q}=g^{q}\cap{[Q]^{\mbox{\tiny\char 73}}}$. Note that as ${{\cal N}^{\mbox{\tiny\char 73}}}$ meets ${\Sigma}_{\infty}^{\mbox{\tiny\char 73}}\backslash\Sigma_{\infty}$ in four points, ${\cal N}$ is disjoint from $\Sigma_{\infty}$. There are three cases to consider.

Case (i), suppose first that $[P]\neq[Q]$. Let $[A],[B],[C]$ be three points of ${\cal N}$, so $[A],[B],[C]\notin\Sigma_{\infty}$. If the plane $\alpha=\langle[A],[B],[C]\rangle$ contained a point of the spread line $[P]$, then the $3$-space $\langle\alpha,[P]\rangle{{}^{\mbox{\tiny\char 73}}}$ contains five points of ${{\cal N}^{\mbox{\tiny\char 73}}}$, namely $[A],[B],[C],P,P^{q}$, a contradiction. So $\alpha$ is disjoint from the spread lines $[P]$ and $[Q]$. If $\alpha$ contained a spread line $[X]$, then in ${\rm PG}(4,q^{2})$, $\langle{\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}},g\rangle$ is a 3-space that contains five points of ${{\cal N}^{\mbox{\tiny\char 73}}}$, namely $[A],[B],[C],P,Q$, a contradiction. So $\alpha$ corresponds to a Baer subplane ${\mathscr{B}}_{\alpha}$ of ${\rm PG}(2,q^{2})$ that is secant to $\ell_{\infty}$, and does not contain $\bar{P}$ or $\bar{Q}$.

Consider the corresponding points $\bar{P},\,\bar{Q},A,B,C$ in ${\rm PG}(2,q^{2})$. So $\bar{P},\,\bar{Q}\in\ell_{\infty}$ and $A,B,C\in{\rm PG}(2,q^{2})\backslash\ell_{\infty}$. Now $A,B,C$ are not collinear as $\alpha$ does not contain a spread line. So ${\mathscr{B}}_{\alpha}$ is the unique Baer subplane that contains $A,B,C$ and is secant to $\ell_{\infty}$. As $\bar{P},\,\bar{Q}\in\ell_{\infty}\backslash{\mathscr{B}}$ and $A,B,C\in{\mathscr{B}}\backslash\ell_{\infty}$, no three of $\bar{P},\,\bar{Q},A,B,C$ are collinear, hence they lie on a unique $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ . By Lemma 5.1, $A,B,C$ lie in a unique $\mathbb{F}_{q}$-conic ${\cal C}$ contained in ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$, and ${\cal C}$ lies in a Baer subplane ${\mathscr{B}}$.

Suppose ${\mathscr{B}}={\mathscr{B}}_{\alpha}$, then by Corollary 3.4, in ${\rm PG}(4,q^{2})$, the plane ${\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}}$ meets $PQ^{q}$. Note that the line $PQ^{q}$ contains two points of ${{\cal N}^{\mbox{\tiny\char 73}}}$, namely $P,Q^{q}$. Hence $\langle{\alpha{\mbox{\raisebox{0.9pt}{\scalebox{1.0}{{\tiny\char 73}}}}}},PQ^{q}\rangle$ is a 3-space of ${\rm PG}(4,q^{2})$ that contains five points of ${{\cal N}^{\mbox{\tiny\char 73}}}$, namely $[A],[B],[C],P,Q^{q}$, a contradiction. Thus ${\mathscr{B}}\neq{\mathscr{B}}_{\alpha}$.

Hence the Baer subplane ${\mathscr{B}}$ is tangent to $\ell_{\infty}$. As ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is secant to $\ell_{\infty}$, we are in case (i) of the proof of Theorem 5.8, hence in ${\rm PG}(4,q)$, $[{\cal C}]$ is a $g$-special 4-dimensional normal rational curve and ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ contains the seven points $A,B,C,P,P^{q},Q,Q^{q}$. As seven points lie on a unique 4-dimensional normal rational curve, we have ${{\cal N}^{\mbox{\tiny\char 73}}}={[{\cal C}]^{\mbox{\tiny\char 73}}}$ and so ${\cal N}=[{\cal C}]$. That is, the normal rational curve ${\cal N}$ corresponds in ${\rm PG}(2,q^{2})$ to an $\mathbb{F}_{q}$-conic ${\cal C}$ in the tangent Baer subplane ${\mathscr{B}}$ as required.

Case (ii), suppose $[P]=[Q]$, the proof is very similar to case (i). Let ${\cal N}$ be a 4-dimensional normal rational curve of ${\rm PG}(4,q)$ such that ${\cal N}\cap\Sigma_{\infty}=\emptyset$, and ${{\cal N}^{\mbox{\tiny\char 73}}}\cap{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ consists of two repeated points $P,P^{q}$. Let $[A],[B],[C]\in{\cal N}$ and $\alpha=\langle[A],[B],[C]\rangle$. Similar to case (i), $\alpha$ corresponds to a Baer subplane ${\mathscr{B}}_{\alpha}$ of ${\rm PG}(2,q^{2})$ that is secant to $\ell_{\infty}$, and does not contain $\bar{P}$. The points $\bar{P},A,B,C$ lie in a unique $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ that is tangent to $\ell_{\infty}$ at $\bar{P}$. By Lemma 5.1, $A,B,C$ lie in a unique $\mathbb{F}_{q}$-conic ${\cal C}$ contained in ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$, and ${\cal C}$ lies in a Baer subplane ${\mathscr{B}}$. If ${\mathscr{B}}={\mathscr{B}}_{\alpha}$, then $\bar{P}\notin{\cal C}$, and so ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets $\ell_{\infty}$ in two points, a contradiction. Hence ${\mathscr{B}}\neq{\mathscr{B}}_{\alpha}$ and ${\mathscr{B}}$ is tangent to $\ell_{\infty}$. As ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is tangent to $\ell_{\infty}$, we are in case (ii) of the proof of Theorem 5.8, hence in ${\rm PG}(4,q)$, $[{\cal C}]$ is a $g$-special 4-dimensional normal rational curve containing $A,B,C$, and ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets ${\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ twice at $P$ and twice at $P^{q}$. These conditions define a unique normal rational curve of ${\rm PG}(4,q^{2})$, and so ${\cal N}={\cal C}$ as required.

Case (iii), suppose ${\cal N}$ is an $g^{\mbox{\tiny\char 72}}$-special 4-dimensional normal rational curve. As ${\cal N}$ is a normal rational curve over ${{\mathbb{F}}_{q}}$, ${\cal N}$ meets ${\Sigma}_{\infty}^{\mbox{\tiny\char 72}}\backslash{\Sigma}_{\infty}^{\mbox{\tiny\char 73}}$ in four points which are conjugate with respect to the map $x\mapsto x^{q}$, $x\in{{\mathbb{F}}_{q}}$. That is, points of form $X,X^{q},X^{q^{2}},X^{q^{3}}$ with $X,X^{q^{2}}\in g^{\mbox{\tiny\char 72}}$ and $X^{q},X^{q^{3}}\in{g^{q}}^{\mbox{\tiny\char 72}}$. Recalling the 1-1 correspondence between points of $g^{\mbox{\tiny\char 72}}$ and points of the quadratic extension of $\ell_{\infty}$ to ${\rm PG}(2,q^{4})$, there are points $\bar{P},\,\bar{Q}$ on the quadratic extension of $\ell_{\infty}$ such that $P=X$, $Q=X^{q^{2}}$. The argument of case (i) now generalises by working in the quadratic extension of ${\rm PG}(2,q^{2})$ to ${\rm PG}(2,q^{4})$; and the quartic extension of ${\rm PG}(4,q)$ to ${\rm PG}(4,q^{4})$. $\square$

Moreover, the proofs of Theorem 5.8, 5.9 show that the normal rational curve corresponding to an $\mathbb{F}_{q}$-conic ${\cal C}$ meets the transversal $g$ of the regular spread $\mathcal{S}$ in points corresponding to the points ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}\cap\ell_{\infty}$. The three cases when ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is tangent, secant or exterior to $\ell_{\infty}$ are summarised in the next result.

Theorem 5.10

In ${\rm PG}(2,q^{2})$, $q>7$, let ${\mathscr{B}}$ be a Baer subplane tangent to $\ell_{\infty}$. Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic in ${\mathscr{B}}$ with ${\mathscr{B}}\cap\ell_{\infty}\notin{\cal C}$, so $[{\cal C}]$ is a 4-dimensional normal rational curve. The $\mathbb{F}_{q^{2}}$-conic ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ meets $\ell_{\infty}$ in two points denoted $\bar{P},\,\bar{Q}$, possibly equal or in an extension. The three possibilities when ${\cal C}^{{\rm\boldsymbol{\raisebox{0.2pt}{\scalebox{0.5}{+}}}}}$ is tangent, secant or exterior to $\ell_{\infty}$ are as follows.

$\bar{P}=\bar{Q}$* if and only if, in ${\rm PG}(4,q^{2})$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets the transversal $g$ of $\mathcal{S}$ in the point $P$.* 2. 2.

$\bar{P},\,\bar{Q}\in\ell_{\infty}$* if and only if, in ${\rm PG}(4,q^{2})$, ${[{\cal C}]^{\mbox{\tiny\char 73}}}$ meets the transversal $g$ of $\mathcal{S}$ in the two points $P,Q$.* 3. 3.

$\bar{P},\,\bar{Q}$* lie in the extension ${\rm PG}(2,q^{4})$ if and only if, in ${\rm PG}(4,q^{4})$, $[{\cal C}]^{\mbox{\tiny\char 72}}$ meets the extended transversal $g^{\mbox{\tiny\char 72}}$ in the two points $P$, $Q$.*