A characterisation of F_q-conics of PG(2,q^3)

TL;DR

This paper characterizes F_q-conics in PG(2,q^3) by relating them to 3-special normal rational curves in PG(6,q) through the Bruck-Bose representation, using geometric interactions between different projective space representations.

Contribution

It provides a new geometric characterization of F_q-conics in PG(2,q^3) via their correspondence to 3-special normal rational curves in PG(6,q).

Findings

F_q-conics correspond to 3-special normal rational curves in PG(6,q)

Characterization of which normal rational curves relate to F_q-conics

Use of geometric interactions between Bruck-Bose and Bose representations

Abstract

This article considers an F_q-conic contained in an F_q-subplane of PG(2,q^3), and shows that it corresponds to a normal rational curve in the Bruck-Bose representation in PG(6,q). This article then characterises which normal rational curves of PG(6,q) correspond via the Bruck-Bose representation to F_q-conics of PG(2,q^3). The normal rational curves of interest are called 3-special, which relates to how the extension of the normal rational curve meets the transversal lines of the regular 2-spread of the Bruck-Bose representation. This article uses geometric arguments that exploit the interaction between the Bruck-Bose representation of PG(2,q^3) in PG(6,q), and the Bose representation of PG(2,q^3) in PG(8,q).

Equations

$$\{(1,\theta ,\dots ,{\theta }^r)|\theta \in {\mathbb{F}}_q\}\cup \{(0,\dots ,0,1)\},$$

$${\bar{\mathsf{c}}}_{\pi }:\mathrm{PG}(2,q^3)$$

$$\bar{X}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

$${\bar{\mathsf{c}}}_b(\bar{X})$$

$$\bar{\ell }\cap {\bar{\ell }}^{{\bar{\mathsf{c}}}_{\pi }},\bar{\ell }\cap {\bar{\ell }}^{{\bar{\mathsf{c}}}_{\pi }^2},{\bar{\ell }}^{{\bar{\mathsf{c}}}_{\pi }}\cap {\bar{\ell }}^{{\bar{\mathsf{c}}}_{\pi }^2}.$$

$$X=(x_0,\dots ,x_{3s+2})⟼X^q=(x_0^q,\dots ,x_{3s+2}^q).$$

$$\bar{P}\in {\ell }_{\infty }⟷P\in g.$$

$$\begin{array}[]{ccccccccccc}{\rm PG}(2,q^{3})&\cong&\Gamma&\cong&{\cal I}_{\mbox{\scriptsize\sf Bose}}\\ \bar{X}&\longleftrightarrow&X&\longleftrightarrow&\mbox{$\llbracket X\rrbracket$}=\langle X,X^{q},X^{q^{2}}\rangle\cap{\rm PG}(8,q).\end{array}$$

$$\llparenthesis X\rrparenthesis_{\pi}=\big{\langle}\,X,(X^{{\mathsf{c}}^{5}_{\pi}})^{q},(X^{\mathsf{c}^{4}_{\pi}})^{q^{2}}\,\big{\rangle}.$$

$$\begin{array}[]{ccccccccccc}{\rm PG}(2,q^{3})&\cong&\Gamma&\cong&{\cal I}_{\mbox{\scriptsize\sf Bose}}&\cong&{\cal I}_{\mbox{\scriptsize\sf BB}}\\ \bar{P}&\longleftrightarrow&P&\longleftrightarrow&\mbox{$\llbracket P\rrbracket$}=\langle\,P,\ P^{q},\ P^{q^{2}}\,\rangle\cap{\rm PG}(8,q)&\longleftrightarrow&[P]=\mbox{$\llbracket P\rrbracket$}\cap\Sigma_{6,q}.\end{array}$$

$$\mathsf{e}:X=(x_0,\dots ,x_n)\mapsto X^{\mathsf{e}}=(x_0^r,\dots ,x_n^r).$$

$$\displaystyle{\cal V}(\mbox{$\llbracket{\cal C}\rrbracket$})^{\!\mbox{\tiny\char 73}}\cap\Gamma={\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}.$$

$$\displaystyle{\cal V}(\mbox{$\llbracket{\cal C}\rrbracket$})^{\!\mbox{\tiny\char 73}}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$$

$$w(P)\times o(P)=3s(P).$$

Full text

Keywords: Bruck-Bose representation, Bose representation, $\mathbb{F}_{q}$-subplanes, $\mathbb{F}_{q}$-conics, normal rational curves. AMS code: 51E20

1 Introduction

An $\mathbb{F}_{q}$-plane of ${\rm PG}(2,q^{n})$ is a subplane of ${\rm PG}(2,q^{n})$ which has order $q$. An $\mathbb{F}_{q}$-conic is a non-degenerate conic in an $\mathbb{F}_{q}$-subplane, and so is projectively equivalent to a non-degenerate conic of ${\rm PG}(2,q)$. The representation of $\mathbb{F}_{q}$-conics in the Bruck-Bose representation when $n=2$ was first looked at in [18]; and the following complete characterisation was given in [7].

Result 1.1

An $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{2})$ corresponds in the ${\rm PG}(4,q)$ Bruck-Bose representation to a 2-special normal rational curve. Further, a normal rational curve in ${\rm PG}(4,q)$ corresponds to an $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{2})$ if and only if it is 2-special.

The notion of 2-special describes how the extension of a normal rational curve meets the transversal lines of the Bruck-Bose spread. Let ${\rm PG}(4,q)$ have hyperplane at infinity $\Sigma_{\infty}$ and regular 1-spread $S$ in $\Sigma_{\infty}$. Let ${\cal N}_{r}$ be an $r$-dimensional normal rational curve, $r\leq 4$, that is not contained in $\Sigma_{\infty}$, then ${\cal N}_{r}$ meets $\Sigma_{\infty}$ in $r$ points $P_{1},\ldots,P_{r}$, possibly repeated, possibly in an extension. We say a point $P$ in an extension of $\Sigma_{\infty}$ has weight $w(P)=1$ if $P$ lies on an extended transversal line of the regular 1-spread $S$; otherwise, $w(P)=2$. The normal rational curve ${\cal N}_{r}$ is called * $2$-special* with respect to $S$ if $w(P_{1})+\ldots+w(P_{r})=4$.

This article looks at the representation of $\mathbb{F}_{q}$-conics in the Bruck-Bose representation when $n=3$. The main result is the following complete characterisation of $\mathbb{F}_{q}$-conics of ${\rm PG}(2,q^{3})$ in the ${\rm PG}(6,q)$ Bruck-Bose representation.

Theorem 1.2

Let ${\rm PG}(6,q)$ have hyperplane at infinity $\Sigma_{\infty}$ and regular 2-spread $\mathcal{S}$ in $\Sigma_{\infty}$. An $r$-dimensional normal rational curve ${\cal N}_{r}$, $r\leq 6$, corresponds via the Bruck-Bose representation to an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$ if and only if ${\cal N}_{r}$ is $3$-special.

The notion of 3-special normal rational curves is defined in Definitions 5.1 and 5.2. It relates to how the extension of the normal rational curve meets the three transversal lines of the Bruck-Bose regular 2-spread, and is a non-trivial generalisation of the idea of 2-special normal rational curves in ${\rm PG}(4,q)$.

This characterisation is proved using geometric arguments that exploit the interaction between the Bruck-Bose representation in ${\rm PG}(6,q)$ and the Bose representation in ${\rm PG}(8,q)$, and builds on previous work by the authors on these representations.

The article is set out as follows. We look at ${\rm PG}(2,q^{3})$ using several different models, namely the Bruck-Bose representation in ${\rm PG}(6,q)$ with the usual conventions; a more precise exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$; and the Bose representation in ${\rm PG}(8,q)$. In Section 2 we describe our notation which is carefully designed to differentiate between the different models. For easy reference, the notation is summarised in Section 2.10.

Section 3 begins with some preliminaries on $\mathbb{F}_{q}$-conics of ${\rm PG}(2,q^{3})$. Result 3.2 outlines eleven different cases for how an $\mathbb{F}_{q}$-conic sits in relation to the line at infinity $\ell_{\infty}$. The main result of Section 3 is Theorem 3.4, which uses the interplay between the Bose representation and the Bruck-Bose representation to show that an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$ corresponds in the ‘exact-at-infinity’ Bruck-Bose representation in ${\rm PG}(6,q)$ to an irreducible curve of degree $k$ and a linear component which is contained in the hyperplane at infinity.

In Section 4, we show that this irreducible curve is a normal rational curve, and we determine $k$ and describe the linear component in detail for the eleven different cases. Then, using the traditional convention for the Bruck-Bose representation, we ignore the linear component at infinity, and deduce that every $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$ corresponds to a normal rational curve of ${\rm PG}(6,q)$. The eleven cases can then be reduced to the following three cases: an $\mathbb{F}_{q}$-conic in an $\mathbb{F}_{q}$-subplane secant to $\ell_{\infty}$ corresponds to a non-degenerate conic of ${\rm PG}(6,q)$; an $\mathbb{F}_{q}$-conic in an $\mathbb{F}_{q}$-subplane tangent to $\ell_{\infty}$ corresponds to either a 4-dimensional or 6-dimensional normal rational curve of ${\rm PG}(6,q)$; and an $\mathbb{F}_{q}$-conic in an $\mathbb{F}_{q}$-subplane exterior to $\ell_{\infty}$ corresponds to either a 3-dimensional or 6-dimensional normal rational curve of ${\rm PG}(6,q)$. In each case, we determine the relationship between the normal rational curve and the transversal lines of the regular 2-spread of the Bruck-Bose representation.

We then consider the converse, and determine which normal rational curves of ${\rm PG}(6,q)$ correspond via the Bruck-Bose representation to $\mathbb{F}_{q}$-conics of ${\rm PG}(2,q^{3})$. In Section 5, we look at how a normal rational curve meets the transversal lines of the Bruck-Bose regular 2-spread, and define $3$-special normal rational curves. In Theorem 5.3 we classify the $3$-special normal rational curves of ${\rm PG}(6,q)$. In Section 6 we look at each possible $3$-special normal rational curve and show that it in each case it corresponds via the Bruck-Bose representation to an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$. This leads to a proof of the main result Theorem 1.2. We conclude in Section 7 with a discussion of the general case.

2 Preliminaries

An $r$-dimensional normal rational curve in ${\rm PG}(n,q)$, $q\geq r$, is a set of points lying in an $r$-space which is projectively equivalent to the set

[TABLE]

see [15]. We abbreviate this to an $r$-dim nrc. We will repeatedly use the geometrical property that an $r$-dim nrc ${\cal N}_{r}$ is a set of $q+1$ points in an $r$-space, such that no $t+2$ points of ${\cal N}_{r}$ lie in a $t$-space, $t=1,\ldots,r-1$. As we work with $6$-dim nrcs in this article, we assume $q\geq 6$ throughout.

2.1 Conjugacy with respect to an $\mathbb{F}_{q}$-subplane

Let $\mathbb{F}_{q}$ denote the finite field of prime power order $q$. An $\mathbb{F}_{q}$-subplane of ${\rm PG}(2,q^{3})$ is a subplane of ${\rm PG}(2,q^{3})$ which has order $q$, that is, a subplane which is isomorphic to ${\rm PG}(2,q)$. An $\mathbb{F}_{q}$-subline is a line of an $\mathbb{F}_{q}$-subplane, that is, isomorphic to ${\rm PG}(1,q)$. We will define conjugacy with respect to an $\mathbb{F}_{q}$-subplane or $\mathbb{F}_{q}$-subline.

Let $\bar{\pi}$ be an $\mathbb{F}_{q}$-subplane of ${\rm PG}(2,q^{3})$. Acting on the points of ${\rm PG}(2,q^{3})$ is a unique collineation group ${\bar{G}}_{\pi}\subseteq{\textup{P}\Gamma\textup{L}}(3,q^{3})$ which fixes $\bar{\pi}$ pointwise and has order 3. We need to distinguish between the two non-identity maps in ${\bar{G}}_{\pi}$, and as discussed in [8], we can without loss of generality write ${\bar{G}}_{\pi}=\langle\bar{\mathsf{c}}_{\pi}\rangle$ with

[TABLE]

with $B$ a $3\times 3$ non-singular matrix over $\mathbb{F}_{q^{3}}$. So ${\bar{\mathsf{c}}_{\pi}}^{3}=id$ and for $\bar{X}\in{\rm PG}(2,q^{3})\backslash\bar{\pi}$, the three points $\bar{X}$, $\bar{X}^{\bar{\mathsf{c}}_{\pi}}$, $\bar{X}^{\bar{\mathsf{c}}^{2}_{\pi}}$ are called conjugate with respect to $\bar{\pi}$. Note that $\bar{X}$, $\bar{X}^{\bar{\mathsf{c}}_{\pi}}$, $\bar{X}^{\bar{\mathsf{c}}^{2}_{\pi}}$ are collinear if and only if $\bar{X}$ lies on an extended line of $\bar{\pi}$. Moreover, we can uniquely extend the plane ${\rm PG}(2,q^{3})$ to ${\rm PG}(2,q^{6})$, and the collineation $\bar{\mathsf{c}}_{\pi}\in{\textup{P}\Gamma\textup{L}}(3,q^{3})$ has a natural extension to a collineation of ${\textup{P}\Gamma\textup{L}}(3,q^{6})$ acting on points of ${\rm PG}(2,q^{6})$; we use the same notation $\bar{\mathsf{c}}_{\pi}$ for this (extended) collineation. The collineation $\bar{\mathsf{c}}_{\pi}$ has order 3 when acting on ${\rm PG}(2,q^{3})$, and order 6 when acting on ${\rm PG}(2,q^{6})$. Under the collineation $\bar{\mathsf{c}}_{\pi}$, a point $\bar{X}\in{\rm PG}(2,q^{6})$ lies in an orbit of size: $1$ if $\bar{X}\in\bar{\pi}$; $3$ if $\bar{X}\in{\rm PG}(2,q^{3})\backslash\bar{\pi}$; $2$ or $6$ if $\bar{X}\in{\rm PG}(2,q^{6})\backslash{\rm PG}(2,q^{3})$.

Similarly, if $\bar{b}$ is an $\mathbb{F}_{q}$-subline of a line $\bar{\ell}_{b}$ of ${\rm PG}(2,q^{3})$, then acting on the points of $\bar{\ell}_{b}$ is a unique collineation group $\bar{G}_{b}\subseteq{\textup{P}\Gamma\textup{L}}(2,q^{3})$ of order 3 which fixes $\bar{b}$ pointwise. Moreover, ${\bar{G}}_{\pi}$ restricted to acting on $\bar{\ell}_{b}$ is isomorphic to $\bar{G}_{b}$ if and only if $\bar{b}$ is a line of $\bar{\pi}$. Without loss of generality we can write ${\bar{G}}_{b}=\langle\bar{\mathsf{c}}_{b}\rangle$ where for a point $\bar{X}\in\bar{\ell}_{b}$,

[TABLE]

with $D$ a non-singular matrix over $\mathbb{F}_{q^{3}}$, so $\bar{\mathsf{c}}_{b}^{3}=id$.

2.2 Carrier points

An $\mathbb{F}_{q}$-subplane exterior to a line $\ell$ determines two carrier points on $\ell$ as follows. Consider the collineation group $\bar{G}={\rm PGL}(3,q^{3})$ acting on ${\rm PG}(2,q^{3})$. Let $\bar{G}_{\pi,\ell}$ be the subgroup of $\bar{G}$ fixing an $\mathbb{F}_{q}$-subplane $\bar{\pi}$, and a line $\bar{\ell}$ exterior to $\bar{\pi}$. Then $\bar{G}_{\pi,\ell}$ is cyclic of order $q^{2}+q+1$, acts regularly on the points and on the lines of $\bar{\pi}$, and fixes exactly three points of ${\rm PG}(2,q^{3})$, called the $(\bar{\pi},\bar{\ell})$-carriers of $\bar{\pi}$, two of which lie on $\bar{\ell}$. That is, the $(\bar{\pi},\bar{\ell})$-carriers are the three fixed points

[TABLE]

2.3 Variety-extensions

Let $f_{1}(x_{0},\ldots,x_{n})=0,\ldots,f_{k}(x_{0},\ldots,x_{n})=0$ be $k$ homogeneous $\mathbb{F}_{q}$-equations (that is, all coefficients of $f_{i}$ lie in $\mathbb{F}_{q}$). These equations give rise to a variety in ${\rm PG}(n,q)$, the pointset of the variety is denoted $V(f_{1},\ldots,f_{k})$ and is the set of all points in ${\rm PG}(n,q)$ which satisfy all $k$ equations. In this article we primarily work with quadrics, that is the case where $f_{i}$ are homogeneous of degree $2$. The pointset of a variety in ${\rm PG}(n,q)$ has a natural extension to the pointset of a variety in the cubic extension ${\rm PG}(n,q^{3})$ and to ${\rm PG}(n,q^{6})$ (a sextic extension of ${\rm PG}(n,q)$, and a quadratic extension of ${\rm PG}(n,q^{3})$). Let $\mathcal{K}=V(f_{1},\ldots,f_{k})$, then ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ denotes the set of points in ${\rm PG}(n,q^{3})$ which satisfy the (same) $k$ equations $f_{1}(x_{0},\ldots,x_{n})=0,\ldots,f_{k}(x_{0},\ldots,x_{n})=0$. Similarly, ${\mathcal{K}^{\mbox{\tiny\char 72}}}$ denotes the set of points in ${\rm PG}(n,q^{6})$ which satisfy the $k$ equations $f_{1}(x_{0},\ldots,x_{n})=0,\ldots,f_{k}(x_{0},\ldots,x_{n})=0$.

In particular, if $\Pi_{r}$ is an $r$-dimensional subspace of ${\rm PG}(n,q)$, then ${\Pi}^{\mbox{\tiny\char 73}}_{r}$ is the natural extension to an $r$-dimensional subspace of ${\rm PG}(n,q^{3})$, and ${\Pi}^{\mbox{\tiny\char 72}}_{r}$ is the natural extension to ${\rm PG}(n,q^{6})$. Note that if $\Sigma_{r}$ is an $r$-dimensional subspace of ${\rm PG}(n,q^{3})$ (possibly disjoint from ${\rm PG}(n,q)$), then we denote the (quadratic) extension of $\Sigma_{r}$ to ${\rm PG}(n,q^{6})$ by ${\Sigma}^{\mbox{\tiny\char 72}}_{r}$.

In this article we use the ${{}^{\mbox{\tiny\char 73}}}$ and ${{}^{\mbox{\tiny\char 72}}}$ notations for varieties in the Bruck-Bose and Bose representations, that is, when $n=5,6,8$. We do not use the ${{}^{\mbox{\tiny\char 73}}},{{}^{\mbox{\tiny\char 72}}}$ notation in ${\rm PG}(2,q^{3})$.

2.4 Regular 2-spreads

We will work with the Desarguesian plane ${\rm PG}(2,q^{3})$, and both the Bruck-Bose and Bose representations, so we work with 2-spreads of ${\rm PG}(5,q)$ and ${\rm PG}(8,q)$ respectively. A 2-spread of ${\rm PG}(n,q)$, $n=3s+2$, is a set of planes that partition the points of ${\rm PG}(n,q)$. We use the following construction of a regular 2-spread of ${\rm PG}(3s+2,q)$, see [12]. Embed ${\rm PG}(3s+2,q)$ in ${\rm PG}(3s+2,q^{3})$ and consider the automorphic collineation in ${\textup{P}\Gamma\textup{L}}(3s+3,q^{3})$ of order 3 that fixes ${\rm PG}(3s+2,q)$ pointwise,

[TABLE]

Let $\Pi$ be an $s$-space in ${\rm PG}(3s+2,q^{3})$ which is disjoint from ${\rm PG}(3s+2,q)$, such that $\Pi,\Pi^{q},\Pi^{q^{2}}$ span ${\rm PG}(3s+2,q^{3})$. For a point $X\in\Pi$, the plane $\langle X,X^{q},X^{q^{2}}\rangle$ meets ${\rm PG}(3s+2,q)$ in a plane. The planes $\langle X,X^{q},X^{q^{2}}\rangle\cap{\rm PG}(3s+2,q)$ for $X\in\Pi$ form a 2-spread of ${\rm PG}(3s+2,q)$. The $s$-spaces $\Pi$, $\Pi^{q}$ and $\Pi^{q^{2}}$ are called the three transversal spaces of the spread. A regular 2-spread is any 2-spread of ${\rm PG}(3s+2,q)$ constructed in this way.

2.5 The Bruck-Bose representation of ${\rm PG}(2,q^{3})$ in ${\rm PG}(6,q)$

We use the linear representation of a finite translation plane $\mathcal{P}$ of dimension at most three over its kernel, an idea which was developed independently by André [1] and Bruck and Bose [10, 11]. We will use the vector space setting following Bruck and Bose. Let $\Sigma_{\infty}$ be a hyperplane of ${\rm PG}(6,q)$ and let $\mathcal{S}$ be a regular 2-spread of $\Sigma_{\infty}$. The phrase a subspace of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ is used to mean a subspace of ${\rm PG}(6,q)$ that is not contained in $\Sigma_{\infty}$. We define the following incidence structure ${\cal I}_{\mbox{\scriptsize\sf BB}}$. The points of ${\cal I}_{\mbox{\scriptsize\sf BB}}$ are the points of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ and the planes of the regular 2-spread $\mathcal{S}$. The lines of ${\cal I}_{\mbox{\scriptsize\sf BB}}$ are the 3-spaces of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ that contain an element of $\mathcal{S}$ and the line at infinity corresponds to the set of planes of $\mathcal{S}$. Incidence in ${\cal I}_{\mbox{\scriptsize\sf BB}}$ is induced by incidence in ${\rm PG}(6,q)$. Then the incidence structure ${\cal I}_{\mbox{\scriptsize\sf BB}}$ is isomorphic to $PG(2,q^{3})$.

Throughout this article, we let $\mathcal{S}$ denote a regular 2-spread of $\Sigma_{\infty}\cong{\rm PG}(5,q)$. The regular 2-spread $\mathcal{S}$ has three conjugate transversals lines in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\cong{\rm PG}(5,q^{3}) which we denote by $g,\ g^{q},\ g^{q^{2}}$. The transversal lines of the regular 2-spread $\mathcal{S}$ play an important role in characterising varieties of ${\rm PG}(2,q^{3})$.

We also need to consider the quadratic extension of {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, that is {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\cong{\rm PG}(5,q^{6}), and we denote the extension of the three transversal lines $g,\ g^{q},\ g^{q^{2}}$ to lines of ${\rm PG}(5,q^{6})$ by ${g^{\mbox{\tiny\char 72}}},\ {g^{\mbox{\tiny\char 72}}}^{q},\ {g^{\mbox{\tiny\char 72}}}^{q^{2}}$ respectively.

If $\bar{\mathcal{K}}$ is a set of points in ${\rm PG}(2,q^{3})$, then we denote the corresponding set of points in the Bruck-Bose representation by $[\mathcal{K}]$. So if $\bar{P}\in{\rm PG}(2,q^{3})\backslash\ell_{\infty}$, then $[P]$ is a point in ${\rm PG}(6,q)\backslash\Sigma_{\infty}$. If $\bar{P}\in\ell_{\infty}$, then $[P]$ is plane of the 2-spread $\mathcal{S}$ in $\Sigma_{\infty}\cong{\rm PG}(5,q)$. Further, in the extension to ${\rm PG}(5,q^{3})$, the plane ${[P]^{\mbox{\tiny\char 73}}}$ meets the transversal $g$ in a point $P={[P]^{\mbox{\tiny\char 73}}}\cap g$. That is, there is a 1-1 correspondence

[TABLE]

The representations of $\mathbb{F}_{q}$-sublines, and tangent and secant $\mathbb{F}_{q}$-subplanes of ${\rm PG}(2,q^{3})$ in ${\rm PG}(6,q)$ were determined in [3, 4]. We repeatedly use the following two cases.

Result 2.1

[3, 4]** Consider the Bruck-Bose representation of ${\rm PG}(2,q^{3})$ in ${\rm PG}(6,q)$.

An $\mathbb{F}_{q}$-subline of ${\rm PG}(2,q^{3})$ that meets $\ell_{\infty}$ in a point $\bar{T}$ corresponds to a line of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ meeting the spread plane $[T]$. 2. 2.

An $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$ secant to $\ell_{\infty}$ corresponds to a plane of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ that meets each plane of the 2-regulus of $\mathcal{S}$ corresponding to the $\mathbb{F}_{q}$-subline $\bar{\pi}\cap\ell_{\infty}$.

Moreover, the converse of each of these correspondences holds.

2.6 The exact-at-infinity Bruck-Bose representation

As per our usual convention for the Bruck-Bose representation, the correspondences in Result 2.1 are not necessarily exact at infinity. For example, we can restate Result 2.1(2) to be exact at infinity as follows.

Result 2.2

Let $\bar{\pi}$ be an $\mathbb{F}_{q}$-subplane of ${\rm PG}(2,q^{3})$ secant to $\ell_{\infty}$. In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, $[\pi]$ consists of the 2-regulus $\mathcal{R}$ of $\mathcal{S}$ that corresponds to the $\mathbb{F}_{q}$-subline $\bar{b}=\bar{\pi}\cap\ell_{\infty}$, together with a plane $\alpha$ of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ that meets each plane of $\mathcal{R}$ in a point. **

For clarity, we distinguish between this and the usual convention by including the phrase “the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$”.

2.7 The Bose representation of ${\rm PG}(2,q^{3})$ in ${\rm PG}(8,q)$

Bose [9] gave a construction to represent the Desarguesian plane ${\rm PG}(2,q^{2})$ in ${\rm PG}(5,q)$ using a regular $1$-spread. More generally, we can use the technique of field reduction to generalise this to represent the Desarguesian plane ${\rm PG}(2,q^{h})$ using a regular $(h-1)$-spread in ${\rm PG}(3h-1,q)$, see for example [16]. This idea goes back to Segre [20] who introduced Desarguesian spreads arising from field reduction.

We work with the following representation of ${\rm PG}(2,q^{3})$ using a regular 2-spread in ${\rm PG}(8,q)$. Let $\mathbb{S}$ be a regular 2-spread in ${\rm PG}(8,q)$. Let ${\cal I}_{\mbox{\scriptsize\sf Bose}}$ be the incidence structure with points the $q^{6}+q^{3}+1$ planes of ${\mathbb{S}}$; lines the 5-spaces of ${\rm PG}(8,q)$ that meet ${\mathbb{S}}$ in $q^{3}+1$ planes; and incidence is inclusion. The 5-spaces of ${\rm PG}(8,q)$ that meet ${\mathbb{S}}$ in $q^{3}+1$ planes form a dual spread $\mathbb{H}$ (that is, each 7-space of ${\rm PG}(8,q)$ contains a unique 5-space in $\mathbb{H}$). Then ${\cal I}_{\mbox{\scriptsize\sf Bose}}\cong{\rm PG}(2,q^{3})$, and this representation is called the Bose representation of ${\rm PG}(2,q^{3})$ in ${\rm PG}(8,q)$. The regular 2-spread $\mathbb{S}$ has three conjugate transversal planes which we denote throughout this article by $\Gamma$, $\Gamma^{q}$, $\Gamma^{q^{2}}$. Note that ${\cal I}_{\mbox{\scriptsize\sf Bose}}\cong\Gamma\cong{\rm PG}(2,q^{3})$.

We use the following notation. A point $\bar{X}$ in ${\rm PG}(2,q^{3})$ has Bose representation the plane of $\mathbb{S}$ denoted by $\llbracket X\rrbracket$. Further, $\bar{X}$ corresponds to a unique point of $\Gamma$ denoted $X$, where \mbox{\llbracket X\rrbracket}{{}^{\mbox{\tiny\char 73}}}\cap\Gamma=X and \mbox{\llbracket X\rrbracket}{{}^{\mbox{\tiny\char 73}}}=\langle X,X^{q},X^{q^{2}}\rangle. More generally, if $\bar{\mathcal{K}}$ is a set of points of ${\rm PG}(2,q^{3})$, then \llbracket\mathcal{K}\rrbracket=\{\mbox{\llbracket X\rrbracket}\,|\,\bar{X}\in\bar{\mathcal{K}}\} denotes the corresponding set of planes in the Bose representation in ${\rm PG}(8,q)$, and \mathcal{K}=\{\mbox{\llbracket X\rrbracket}{{}^{\mbox{\tiny\char 73}}}\cap\Gamma\,|\,\bar{X}\in\bar{\mathcal{K}}\} denotes the corresponding set of points of $\Gamma$.

So we have the following correspondences:

[TABLE]

2.8 $\mathbb{F}_{q}$-substructures in the Bose representation

We need to look at $\mathbb{F}_{q}$-sublines, $\mathbb{F}_{q}$-subplanes and $\mathbb{F}_{q}$-conics of ${\rm PG}(2,q^{3})$ in the Bose representation in ${\rm PG}(8,q)$. The Bose representation of $\mathbb{F}_{q}$-sublines and $\mathbb{F}_{q}$-subplanes is proved in [17] and also in [16, Theorem 2.6] using field reduction techniques. The Bose representation of conics and $\mathbb{F}_{q}$-conics of ${\rm PG}(2,q^{3})$ are determined in [8]. Further, [8] looks at these structures in the extension of ${\rm PG}(8,q)$ to ${\rm PG}(8,q^{3})$ and ${\rm PG}(8,q^{6})$. We briefly summarise the results we need in order to establish the notation we will use.

In ${\rm PG}(2,q^{3})$, let $\bar{\pi}$ be an $\mathbb{F}_{q}$-subplane and $\bar{b}$ an $\mathbb{F}_{q}$-subline of the line $\bar{\ell}_{b}$. Define ${\bar{\mathsf{c}}_{\pi}}$ and $\bar{\mathsf{c}}_{b}$ as in (2.1) and (2). Denote the corresponding maps which act on the points of the transversal plane $\Gamma$ and the line $\ell_{b}\subset\Gamma$ by ${\mathsf{c}_{\pi}}$ and $\mathsf{c}_{b}$ respectively. Note that ${\mathsf{c}_{\pi}}$ and $\mathsf{c}_{b}$ are not collineations of ${\rm PG}(8,q^{3})$, they only act on the points of $\Gamma$ and $\ell_{b}$ respectively. Further $\mathsf{c}_{\pi}$ and $\mathsf{c}_{b}$ induce collineations acting on the points of $\Gamma{{}^{\mbox{\tiny\char 72}}}$ and ${\ell^{\mbox{\tiny\char 72}}_{b}}$ respectively.

In ${\rm PG}(8,q^{6})$, we define the $\pi$-scroll-plane of a point $X$ in $\Gamma{{}^{\mbox{\tiny\char 72}}}$ to be the plane

[TABLE]

Note that if $X\in\pi$, then \llparenthesis X\rrparenthesis_{\pi}=\mbox{\llbracket X\rrbracket}{{}^{\mbox{\tiny\char 72}}}. Further, if $X\in\Gamma\backslash\pi$, then $\llparenthesis X\rrparenthesis_{\pi}$ simplifies to \llparenthesis X\rrparenthesis_{\pi}=\big{\langle}\,X,(X^{{\mathsf{c}}^{2}_{\pi}})^{q},(X^{\mathsf{c}_{\pi}})^{q^{2}}\,\big{\rangle}, and is disjoint from ${\rm PG}(8,q)$. The $b$-scroll-plane of a point $X\in{\ell^{\mbox{\tiny\char 72}}_{b}}$ in ${\rm PG}(8,q^{6})$ is the plane \llparenthesis X\rrparenthesis_{b}=\big{\langle}\,X,(X^{{\mathsf{c}}^{5}_{b}})^{q},(X^{\mathsf{c}^{4}_{b}})^{q^{2}}\,\big{\rangle} which lies in the 5-space $\langle\ell_{b},\ell_{b}^{q},\ell_{b}^{q^{2}}\rangle{{}^{\mbox{\tiny\char 72}}}$. We quote the $\mathbb{F}_{q}$-conic result which we will need here.

Result 2.3

[8]** Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in the $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. In ${\rm PG}(8,q)$, the planes of $\llbracket{\cal C}\rrbracket$ form a variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})={\cal V}^{6}_{3} which is the intersection of nine quadrics: {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})=\mathscr{Q}_{1}\cap\cdots\cap\mathscr{Q}_{9}. In ${\rm PG}(8,q^{3})$, the points of the variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 73}}={\mathscr{Q}}^{\mbox{\tiny\char 73}}_{1}\cap\cdots\cap{\mathscr{Q}}^{\mbox{\tiny\char 73}}_{9} coincide with the points on the planes $\{\llparenthesis X\rrparenthesis_{\pi}\,|\,X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\}$; and in ${\rm PG}(8,q^{6})$, the points of the variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 72}}={\mathscr{Q}}^{\mbox{\tiny\char 72}}_{1}\cap\cdots\cap{\mathscr{Q}}^{\mbox{\tiny\char 72}}_{9} coincide with the points on the planes $\{\llparenthesis X\rrparenthesis_{\pi}\,|\,X\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\}$.

2.9 The Bruck-Bose representation inside the Bose representation

We can construct the Bruck-Bose representation of ${\rm PG}(2,q^{3})$ by intersecting a 6-space with the Bose representation of ${\rm PG}(2,q^{3})$ in ${\rm PG}(8,q)$ as follows. Let $\Sigma_{6,q}$ be a 6-space of ${\rm PG}(8,q)$ that contains a unique 5-space of the dual spread $\mathbb{H}$, we denote this 5-space by $\Sigma_{\infty}$. The extension of $\Sigma_{6,q}$ to $\Sigma_{6,q}^{{}^{\mbox{\tiny\char 73}}}$ meets the transversal plane $\Gamma$ in a line $g$. Further, $\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)=\Sigma_{\infty}\cong{\rm PG}(5,q)$. The intersection of the Bose representation with $\Sigma_{6,q}$ gives the Bruck-Bose representation of ${\rm PG}(2,q^{3})$ in the 6-space $\Sigma_{6,q}$. That is, ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$. In particular the 2-spread $\mathcal{S}$ of the Bruck-Bose representation is contained in the 2-spread $\mathbb{S}$ of the Bose setting, that is $\mathcal{S}=\mathbb{S}\cap\Sigma_{\infty}$.

So we have the following correspondences:

[TABLE]

Note that when considering the Bruck-Bose representation as ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$, we obtain a representation which is exact on $\Sigma_{\infty}$, that is, we have the “exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$” described in Section 2.6.

2.10 Notation summary

This article works with $\mathbb{F}_{q}$-conics in ${\rm PG}(2,q^{3})$ in the planar setting, the ${\rm PG}(6,q)$ Bruck-Bose setting and the ${\rm PG}(8,q)$ Bose setting. We have designed the notation to help distinguish between these settings, and a summary of the notation used is given here.

• •

In ${\rm PG}(2,q^{3})$

• $\cdot$

Objects in ${\rm PG}(2,q^{3})$ are indicated with an overline $\bar{\phantom{x}}$.

• $\cdot$

$\bar{\cal C}$ denotes an $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{3})$, and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ denotes the unique $\mathbb{F}_{q^{3}}$-conic in ${\rm PG}(2,q^{3})$ containing $\bar{\cal C}$.

• $\cdot$

For an $\mathbb{F}_{q}$-subplane $\bar{\pi}$ contained in ${\rm PG}(2,q^{3})$, $\bar{\mathsf{c}}_{\pi}:\bar{X}\mapsto B\bar{X}^{q}$ (defined in (2.1)) generates the unique collineation subgroup of order 3 acting on ${\rm PG}(2,q^{3})$ which fixes $\bar{\pi}$ pointwise.

• $\cdot$

For an $\mathbb{F}_{q}$-subline $\bar{b}$ contained in a line $\bar{\ell}_{b}$ of ${\rm PG}(2,q^{3})$, ${\bar{\mathsf{c}}}_{b}:\bar{X}\mapsto D\bar{X}^{q}$ (defined in (2)) generates the unique collineation subgroup of order 3 acting on $\bar{\ell}_{b}$, which fixes $\bar{b}$ pointwise.

• $\cdot$

We can extend ${\rm PG}(2,q^{3})$ to ${\rm PG}(2,q^{6})$, and let $\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}$ denote the quadratic extension of $\ell_{\infty}$, and ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$ denote the unique $\mathbb{F}_{q^{6}}$-conic containing $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$.

• •

In ${\rm PG}(n,q)$, $n>2$

• $\cdot$

${\cal N}_{r}$ denotes an $r$-dim nrc.

• $\cdot$

If $\mathscr{Q}$ is a quadric in ${\rm PG}(n,q)$, denote the extension to ${\rm PG}(n,q^{3})$ by ${\mathscr{Q}^{\mbox{\tiny\char 73}}}$, and the extension to ${\rm PG}(n,q^{6})$ by $\mathscr{Q}{{}^{\mbox{\tiny\char 72}}}$.

• $\cdot$

If ${\cal N}$ is a normal rational curve in ${\rm PG}(n,q)$, denote the extension to ${\rm PG}(n,q^{3})$ by ${{\cal N}^{\mbox{\tiny\char 73}}}$, and the extension to ${\rm PG}(n,q^{6})$ by ${\cal N}{{}^{\mbox{\tiny\char 72}}}$.

• $\cdot$

If $X=(x_{0},\ldots,x_{n})$, then $X^{q}=(x_{0}^{q},\ldots,x_{n}^{q})$.

• $\cdot$

$\mathsf{e}$ denotes the conjugate map associated with the square extension from ${\rm PG}(n,r)$ to ${\rm PG}(n,r^{2})$, that is,

[TABLE]

• •

In ${\cal I}_{\mbox{\scriptsize\sf Bose}}$

• $\cdot$

$\mathbb{S}$ is a regular 2-spread in ${\rm PG}(8,q)$.

• $\cdot$

$\mathbb{S}$ has three transversal planes in ${\rm PG}(8,q^{3})$, denoted $\Gamma$, $\Gamma^{q}$, $\Gamma^{q^{2}}$.

• $\cdot$

A point $\bar{P}$ in ${\rm PG}(2,q^{3})$ corresponds to a point $P$ in the transversal plane $\Gamma$, and to a plane $\llbracket P\rrbracket$ of $\mathbb{S}$, where \mbox{\llbracket P\rrbracket}=\langle\,P,\,P^{q},\,P^{q^{2}}\,\rangle\cap{\rm PG}(8,q).

• •

In ${\cal I}_{\mbox{\scriptsize\sf BB}}$

• $\cdot$

$\mathcal{S}$ is a regular 2-spread in the 5-space at infinity $\Sigma_{\infty}\cong{\rm PG}(5,q)$.

• $\cdot$

$\mathcal{S}$ has transversal lines denoted $g,g^{q},g^{q^{2}}$ which lie in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash\Sigma_{\infty}.

• $\cdot$

For a point $\bar{X}\in{\rm PG}(2,q^{3})\backslash\ell_{\infty}$, we denote corresponding point of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ by $[X]$.

• $\cdot$

If $\bar{X}\in\ell_{\infty}$, then we denote the corresponding point on $g$ by $X$ and the corresponding spread plane by $[X]$, so $[X]=\langle\,X,\,X^{q},\,X^{q^{2}}\,\rangle\cap\Sigma_{\infty}$ and $X=[X]{{}^{\mbox{\tiny\char 73}}}\cap g$.

• •

in ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$

• $\cdot$

The 6-space ${\Sigma}^{\mbox{\tiny\char 73}}_{6,q}$ meets the transversal plane $\Gamma$ of $\mathbb{S}$ in the transversal line $g$ of $\mathcal{S}$.

• $\cdot$

For a point $\bar{X}\in{\rm PG}(2,q^{3})$, we have [X]=\mbox{\llbracket X\rrbracket}\cap\Sigma_{6,q}.

3 Preliminaries

3.1 $\mathbb{F}_{q}$-conics

We define an $\mathbb{F}_{q^{3}}$-conic in ${\rm PG}(2,q^{3})$ to be a non-degenerate conic of ${\rm PG}(2,q^{3})$. We define an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$ to be a non-degenerate conic of an $\mathbb{F}_{q}$-subplane of ${\rm PG}(2,q^{3})$. That is, an $\mathbb{F}_{q}$-conic is projectively equivalent to a set of points in ${\rm PG}(2,q)$ that satisfy a non-degenerate homogeneous quadratic equation over $\mathbb{F}_{q}$.

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in the $\mathbb{F}_{q}$-subplane $\bar{\pi}$, let $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ be the unique $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$ containing $\bar{\cal C}$, and let ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$ denote the unique $\mathbb{F}_{q^{6}}$-conic in the quadratic extension ${\rm PG}(2,q^{6})$ that contains $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$. Let $\bar{\mathsf{c}}_{\pi}$ be defined as in (2.1), the following useful property is straightforward to verify.

Result 3.1

In ${\rm PG}(2,q^{3})$, let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic. If $\bar{X}\in\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, then $\bar{X}^{\bar{\mathsf{c}}_{\pi}},\bar{X}^{\bar{\mathsf{c}}^{2}_{\pi}}\in\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$. Further, in the quadratic extension ${\rm PG}(2,q^{6})$, if $\bar{Y}\in{\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$, then $\bar{Y}^{\bar{\mathsf{c}}^{i}_{\pi}}\in{\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$, $i=1,\ldots,5$.

The Bruck-Bose representation of ${\rm PG}(2,q^{3})$ is constructed using a line at infinity $\ell_{\infty}$. In order to study $\mathbb{F}_{q}$-conics in the Bruck-Bose representation, we need to look at the position of an $\mathbb{F}_{q}$-conic in relation to $\ell_{\infty}$. There are a total of eleven different settings for an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$ in relation to $\ell_{\infty}$. These are described in the next result, the proof of which is straightforward and is omitted.

Result 3.2

In ${\rm PG}(2,q^{3})$, let $\ell_{\infty}$ denote the line at infinity. Let $\bar{\pi}$ be an $\mathbb{F}_{q}$-subplane and let $\bar{\mathsf{c}}_{\pi}$ be the collineation which fixes $\bar{\pi}$ pointwise as defined in (2.1). Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in $\bar{\pi}$, let $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ be the unique $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$ containing $\bar{\cal C}$, and in the extension to ${\rm PG}(2,q^{6})$, let ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$ be the unique $\mathbb{F}_{q^{6}}$-conic of ${\rm PG}(2,q^{6})$ that contains $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$. Let $\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}$ denote the extension of $\ell_{\infty}$ to ${\rm PG}(2,q^{6})$.

• •

Suppose $\bar{\pi}$ is secant to $\ell_{\infty}$, then either

1. [S1]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is secant to $\ell_{\infty}$: * $\bar{\cal C}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, $\bar{P}\neq\bar{Q}$, $\bar{P},\bar{Q}\in\bar{\pi}$.* 2. [S2]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is tangent to $\ell_{\infty}$: * $\bar{\cal C}\cap\ell_{\infty}=\{\bar{P}\}$, $\bar{P}\in\bar{\pi}$.* 3. [S3]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is exterior to $\ell_{\infty}$: * $\bar{\cal C}\cap\ell_{\infty}=\emptyset$, in which case $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, and ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. Moreover, $\bar{Q}=\bar{P}^{\bar{\mathsf{c}}_{\pi}}=\bar{P}^{q^{3}}$, $\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}}=\bar{P}$, and if $\bar{b}=\bar{\pi}\cap\ell_{\infty}$, then $\bar{P}^{\bar{\mathsf{c}}_{b}}=\bar{P}^{\bar{\mathsf{c}}_{\pi}}=\bar{Q}$.*

• •

Suppose $\bar{\pi}$ is tangent to $\ell_{\infty}$, with $\bar{T}=\bar{\pi}\cap\ell_{\infty}$. Then either

1. [T1]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is secant to $\ell_{\infty}$: * $\bar{T}\in\bar{\cal C}$, in which case $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{T},\bar{Q}\}$ with $\bar{T}\neq\bar{Q}$. Note that $\bar{T}=\bar{T}^{\bar{\mathsf{c}}_{\pi}}=\bar{T}^{\bar{\mathsf{c}}_{\pi}^{2}}$ and $\bar{Q}^{\bar{\mathsf{c}}_{\pi}},\bar{Q}^{\bar{\mathsf{c}}_{\pi}^{2}}\notin\ell_{\infty}$.* 2. [T2]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is secant to $\ell_{\infty}$: * $\bar{T}\notin\bar{\cal C}$ and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, $\bar{P}\neq\bar{Q}$. So $\bar{P},\bar{Q}$ both have orbit size 3 under $\bar{\mathsf{c}}_{\pi}$, and $\bar{P}^{\bar{\mathsf{c}}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}},\bar{Q}^{\bar{\mathsf{c}}_{\pi}},\bar{Q}^{\bar{\mathsf{c}}^{2}_{\pi}}\notin\ell_{\infty}$.* 3. [T3]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is tangent to $\ell_{\infty}$: * $\bar{T}\notin\bar{\cal C}$ and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P}\}$. If $q$ is even, then $\bar{T}$ is the nucleus of $\bar{\cal C}$. The point $\bar{P}$ has orbit size 3 under $\bar{\mathsf{c}}_{\pi}$, and $\bar{P}^{\bar{\mathsf{c}}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}}\notin\ell_{\infty}$.* 4. [T4]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is exterior to $\ell_{\infty}$: * $\bar{T}\notin\bar{\cal C}$ and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, then ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. In this case $\bar{Q}=\bar{P}^{q^{3}}=\bar{P}^{\bar{\mathsf{c}}^{3}_{\pi}}$, $\bar{P}$ has orbit size 6 under $\bar{\mathsf{c}}_{\pi}$, and $\bar{P},\bar{P}^{\bar{\mathsf{c}}^{3}_{\pi}}\in\ell_{\infty}$, $\bar{P}^{\bar{\mathsf{c}}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{4}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{5}_{\pi}}\notin\ell_{\infty}$.*

• •

Suppose $\bar{\pi}$ is exterior to $\ell_{\infty}$, denote the $(\bar{\pi},\ell_{\infty})$-carriers which lie on $\ell_{\infty}$ by $\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}$. Then either

1. [E1]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is secant to $\ell_{\infty}$: * $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}$.* 2. [E2]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is secant to $\ell_{\infty}$: * $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, with $\bar{P}\neq\bar{Q}$ and $\{\bar{P},\bar{Q}\}\cap\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}=\emptyset$. In this case $\bar{P},\bar{Q}$ both have orbit size 3 under $\bar{\mathsf{c}}_{\pi}$, and $\bar{P}^{\bar{\mathsf{c}}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}},\bar{Q}^{\bar{\mathsf{c}}_{\pi}},\bar{Q}^{\bar{\mathsf{c}}^{2}_{\pi}}\notin\ell_{\infty}$.* 3. [E3]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is tangent to $\ell_{\infty}$: * $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P}\}$. This case only occurs if $q$ is odd, in which case $\bar{P}\cap\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}=\emptyset$ and $\bar{P}$ has orbit size 3 under $\bar{\mathsf{c}}_{\pi}$, and $\bar{P}^{\bar{\mathsf{c}}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}}\notin\ell_{\infty}$.* 4. [E4]

$\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ is exterior to $\ell_{\infty}$: * $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, in which case ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. Further $\bar{Q}=\bar{P}^{q^{3}}=\bar{P}^{\bar{\mathsf{c}}^{3}_{\pi}}$, $\bar{P}$ has orbit size 6 under $\bar{\mathsf{c}}_{\pi}$, and $\bar{P},\bar{P}^{\bar{\mathsf{c}}^{3}_{\pi}}\in\ell_{\infty}$, $\bar{P}^{\bar{\mathsf{c}}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{2}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{4}_{\pi}},\bar{P}^{\bar{\mathsf{c}}^{5}_{\pi}}\notin\ell_{\infty}$.*

Remark There are $q^{2}+q+1$  $\mathbb{F}_{q}$-conics of the type described in [E1], forming a circumscribed bundle of conics of $\overline{\pi}$. There are $q^{2}+q+1\,$ $\mathbb{F}_{q}$-conics of the type described in [E3], forming an inscribed bundle of conics of $\overline{\pi}$.

3.2 T-planes

We begin with a lemma describing how a $\pi$-scroll-plane (defined in Section 2.8) meets a 5-space. In ${\rm PG}(8,q^{3})$, let $\Gamma,\Gamma^{q},\Gamma^{q^{2}}$ be the transversal planes of the Bose spread $\mathbb{S}$. We use the following terminology in ${\rm PG}(8,q^{3})$.

• •

A T-point is a point which lies in one of the transversal planes $\Gamma,\Gamma^{q},\Gamma^{q^{2}}$.

• •

A T-line is a line which meets two of the transversal planes.

• •

A T-plane is a plane that meets all three of the transversal planes.

Similarly, we can define T-points, T-lines and T-planes in the quadratic extension ${\rm PG}(8,q^{6})$ in terms of the extended transversal planes.

Lemma 3.3

In the Bose setting, let $\Gamma$ be a transversal plane of the Bose spread $\mathbb{S}$. Let $g$ be a line of $\Gamma$, let $\pi$ be an $\mathbb{F}_{q}$-subplane of $\Gamma$, and consider the 5-space $\Pi_{g}=\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)$.

In ${\rm PG}(8,q^{3})$, for a point $X\in\Gamma$, $\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ is either $\emptyset$, a T-point, a T-line, or a T-plane. 2. 2.

In ${\rm PG}(8,q^{6})$, for a point $X\in\Gamma{{}^{\mbox{\tiny\char 72}}}$, $\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 72}}_{g}$ is either $\emptyset$, a T-point, a T-line, or a T-plane.

Proof   Note that the lines $g,g^{q},g^{q^{2}}$ are transversal lines of a regular 2-spread $\mathcal{S}$ of $\Pi_{g}$, where $\mathcal{S}=\mathbb{S}\cap\Pi_{g}$. In ${\Pi}^{\mbox{\tiny\char 73}}_{g}$: a T-point is a point of one of the transversal lines $g,g^{q},g^{q^{2}}$; a T-line is a line which meets two of $g,g^{q},g^{q^{2}}$; and a T-plane is a plane that meets all three of $g,g^{q},g^{q^{2}}$. Similar to the proof of [8, Cor 2.2], we can show that two distinct T-planes in ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ meet in either $\emptyset$, a T-point, or a T-line.

Suppose there exists a $\pi$-scroll-plane where $\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ is not one of: $\emptyset$, a T-point, a T-line or a T-plane. Then either there is a point $L\in\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ with $L$ on a T-line and not a T-point; or there is a point $K\in\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ with $K$ not on a T-line. First suppose that $\llparenthesis X\rrparenthesis_{\pi}$ is not contained in ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ and that there is a point $L\in\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ with $L$ on a T-line $m$, $L$ not a T-point. Let $\beta$ be one of the T-planes of ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ containing the line $m$. Then $\beta$ is a T-plane contained in ${\Pi}^{\mbox{\tiny\char 73}}_{g}$, so $\beta\neq\llparenthesis X\rrparenthesis_{\pi}$ and $L\in\beta\cap\llparenthesis X\rrparenthesis_{\pi}$. This contradicts our statement that that two distinct T-planes in ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ meet in either $\emptyset$, a T-point, or a T-line. Now suppose that there is a point $K\in\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ with $K$ not on a T-line. Then as in the previous case, $K$ lies on a unique T-plane $\alpha\subset{\Pi}^{\mbox{\tiny\char 73}}_{g}$. Note that $\alpha\neq\llparenthesis X\rrparenthesis_{\pi}$ and $\alpha$ is also a T-plane. That is, $K$ lies on two distinct T-planes, a contradiction. This completes the proof of part 1, the proof of part 2 is similar. $\square$

3.3 A first description of $[{\cal C}]$

We will determine the representation of an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$ in the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$. Fundamental to determining this structure is the interplay between the Bose representation ${\cal I}_{\mbox{\scriptsize\sf Bose}}$ in ${\rm PG}(8,q)$ and the Bruck-Bose representation ${\cal I}_{\mbox{\scriptsize\sf BB}}$ in ${\rm PG}(6,q)$ considered as ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$. Using Result 2.3 we first prove the following coarse description in the exact-at-infinity Bruck-Bose representation. Recall that an affine point of ${\rm PG}(6,q)$ means a point in ${\rm PG}(6,q)\backslash\Sigma_{\infty}$.

Theorem 3.4

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{3})$. In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, the pointset of $[{\cal C}]$ forms a variety ${\cal V}([{\cal C}])$ which is the intersection of nine quadrics. Further, ${\cal V}([{\cal C}])$ is generically a curve of degree $6$.

Proof   We look at the ${\rm PG}(6,q)$ Bruck-Bose representation of $\bar{\cal C}$ using the ${\rm PG}(8,q)$ Bose representation. Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{3})$. In the Bose representation in ${\rm PG}(8,q)$, $\llbracket{\cal C}\rrbracket$ is a set of $q+1$ planes. By Result 2.3, the pointset of $\llbracket{\cal C}\rrbracket$ forms a variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})={\cal V}^{6}_{3} of dimension 3 and degree 6 which is the intersection of nine quadrics {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})=\mathscr{Q}_{1}\cap\cdots\cap\mathscr{Q}_{9}. Let $\Sigma_{6,q}$ be a 6-space whose extension to ${\rm PG}(8,q^{3})$ meets the transversal plane $\Gamma$ in the line $g$. So in the Bruck-Bose setting ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$, we have hyperplane at infinity $\Sigma_{\infty}=\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)$. The $\mathbb{F}_{q}$-conic $\bar{\cal C}$ corresponds to a set of points [{\cal C}]=\mbox{\llbracket{\cal C}\rrbracket}\cap\Sigma_{6,q} in this Bruck-Bose setting. These points form a variety {\cal V}([{\cal C}])={\cal V}(\mbox{\llbracket{\cal C}\rrbracket})\cap\Sigma_{6,q}=\mathscr{Q}_{1}\cap\cdots\cap\mathscr{Q}_{9}\cap\Sigma_{6,q}. The 6-space $\Sigma_{6,q}$ is a variety ${\cal V}^{1}_{6}$. Generically, we have ${\cal V}^{6}_{3}\cap{\cal V}^{1}_{6}={\cal V}^{6}_{1}$, so the variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})={\cal V}^{6}_{3} generically meets $\Sigma_{6,q}$ in a curve of degree 6, that is, ${\cal V}([{\cal C}])={\cal V}^{6}_{1}$. $\square$

3.4 Transversals

The next result determines how the exact-at-infinity Bruck-Bose representation of an $\mathbb{F}_{q}$-conic $\bar{\cal C}$ meets the transversal lines $g,g^{q},g^{q^{2}}$ of the regular 2-spread $\mathcal{S}$. Note that the variety ${\cal V}([{\cal C}]){{}^{\mbox{\tiny\char 73}}}$ may well contain planes or lines in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, and these may account for the intersections with $g$ described in the next theorem.

Theorem 3.5

Let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$ of ${\rm PG}(6,q)$, with transversal lines denoted $g,g^{q},g^{q^{2}}$. Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$, then in the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, ${\cal V}([{\cal C}]){{}^{\mbox{\tiny\char 73}}}\cap g={\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$. Hence $\bar{P}\in\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}$ if and only if $P\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$.

Proof   Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$, in the Bose representation we have from Result 2.3 that

[TABLE]

We interpret this in the Bruck-Bose representation. Let $g$ be a line of $\Gamma$, and $\Sigma_{6,q}$ a 6-space whose extension contains the 5-space $\langle g,g^{q},g^{q^{2}}\rangle$. By the proof of Theorem 3.4, {\cal V}([{\cal C}]){{}^{\mbox{\tiny\char 73}}}={\cal V}(\mbox{\llbracket{\cal C}\rrbracket}){{}^{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{6,q}, and intersecting with $g$ yields {\cal V}([{\cal C}]){{}^{\mbox{\tiny\char 73}}}\cap g={\cal V}(\mbox{\llbracket{\cal C}\rrbracket}){{}^{\mbox{\tiny\char 73}}}\cap g. Intersecting both sides of (3) with $g$, and equating gives ${\cal V}([{\cal C}]){{}^{\mbox{\tiny\char 73}}}\cap g={\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$. $\square$

The next result looks how $\pi$-scroll-planes of {\cal V}(\mbox{\llbracket{\cal C}\rrbracket}) meet the 5-space $\langle g,g^{q},g^{q^{2}}\rangle$.

Lemma 3.6

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in an $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. As defined in (2.1), let $\bar{\mathsf{c}}_{\pi}:\bar{X}\mapsto B\bar{X}^{q}$. In the Bose representation ${\rm PG}(8,q)$, let {\cal V}(\mbox{\llbracket{\cal C}\rrbracket}) be the variety of ${\rm PG}(8,q)$ whose pointset corresponds to the pointset of $\llbracket{\cal C}\rrbracket$. Let $g$ be a line of $\Gamma$ and consider the 5-space $\Pi_{g}=\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)$.

In ${\rm PG}(8,q^{3})$, {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 73}}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}=\ \{\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}\,|\,X^{\mathsf{c}^{i}_{\pi}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g,\textup{ for some }i\in\{0,1,2\}\}. 2. 2.

In ${\rm PG}(8,q^{6})$, {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 72}}\cap{\Pi}^{\mbox{\tiny\char 72}}_{g}=\{\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 72}}_{g}\,|\,X^{\mathsf{c}^{i}_{\pi}}\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap g^{\mbox{\tiny\char 72}},\textup{ for some }i\in\{0,\ldots,5\}\}.

Proof   First consider the extension to ${\rm PG}(8,q^{3})$. By Result 2.3, the points of the variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 73}} coincide with the points on the planes $\{\llparenthesis X\rrparenthesis_{\pi}\,|\,X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\}$, so

[TABLE]

Recall that $\llparenthesis X\rrparenthesis_{\pi}=\langle X,(X^{\mathsf{c}_{\pi}^{2}})^{q},(X^{\mathsf{c}_{\pi}})^{q^{2}}\rangle,$ so we are looking for points $X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ for which $\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}\neq\emptyset$. By Lemma 3.3, $\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ is either

• •

$\emptyset$, or

• •

$X$ or $(X^{\mathsf{c}_{\pi}^{2}})^{q}$ or $(X^{\mathsf{c}_{\pi}})^{q^{2}}$, or

• •

a line joining two of $X,(X^{\mathsf{c}_{\pi}^{2}})^{q},(X^{\mathsf{c}_{\pi}})^{q^{2}}$, or

• •

$\llparenthesis X\rrparenthesis_{\pi}$.

Hence we want points $X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ for which at least one of the points $X,(X^{\mathsf{c}_{\pi}^{2}})^{q},(X^{\mathsf{c}_{\pi}})^{q^{2}}$ lies in ${\Pi}^{\mbox{\tiny\char 73}}_{g}$.

As ${\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ lies in the transversal plane $\Gamma$ and ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ meets $\Gamma$ in the line $g$, a point $X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ lies in ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ if and only if $X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$. For a point $X^{\mathsf{c}_{\pi}^{2}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, the point $(X^{\mathsf{c}_{\pi}^{2}})^{q}$ lies in the transversal plane $\Gamma^{q}$. Further, ${\Pi}^{\mbox{\tiny\char 73}}_{g}$ meets $\Gamma^{q}$ in the line $g^{q}$. Hence $(X^{\mathsf{c}_{\pi}^{2}})^{q}\in({\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}})^{q}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ if and only if $X^{\mathsf{c}_{\pi}^{2}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$. Similarly, for a point $X^{\mathsf{c}_{\pi}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, we have $(X^{\mathsf{c}_{\pi}})^{q^{2}}\in({\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}})^{q^{2}}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}$ if and only if $X^{\mathsf{c}_{\pi}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$. That is, for $X\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, $\llparenthesis X\rrparenthesis_{\pi}\cap{\Pi}^{\mbox{\tiny\char 73}}_{g}\neq\emptyset$ if and only if at least one of $X,X^{\mathsf{c}_{\pi}},X^{\mathsf{c}^{2}_{\pi}}$ lies in ${\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g$, proving part 1. The case in ${\rm PG}(8,q^{6})$ is similar. $\square$

4 $\mathbb{F}_{q}$-conics in the exact-at-infinity Bruck-Bose representation

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in an $\mathbb{F}_{q}$-plane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. In this section we determine in more detail the structure of the variety ${\cal V}([{\cal C}])$ in the exact-at-infinity Bruck-Bose representation. We look at the three cases where $\bar{\pi}$ is secant, tangent and exterior to $\ell_{\infty}$ separately.

4.1 $\mathbb{F}_{q}$-conics in an $\mathbb{F}_{q}$-subplane secant to $\ell_{\infty}$

We begin by looking at $\mathbb{F}_{q}$-conics contained in an $\mathbb{F}_{q}$-subplane that is secant to $\ell_{\infty}$.

Let $\bar{\pi}$ be an $\mathbb{F}_{q}$-subplane of ${\rm PG}(2,q^{3})$ that is secant to $\ell_{\infty}$. So $\bar{b}=\bar{\pi}\cap\ell_{\infty}$ is an $\mathbb{F}_{q}$-subline. Recall the $b$-scroll-plane of a point was defined in the Bose representation in Section 2.8. As $\bar{b}$ is an $\mathbb{F}_{q}$-subline of $\ell_{\infty}$, we can similarly define the $b$-scroll-plane in the Bruck-Bose representation. In the Bruck-Bose setting, the $b$-scroll-plane of a point $X\in g$ (or $g^{\mbox{\tiny\char 72}}$) is the plane $\llparenthesis X\rrparenthesis_{b}=\langle X,({X}^{\mathsf{c}^{5}_{b}})^{q},({X}^{\mathsf{c}^{4}_{b}})^{q^{2}}\,\rangle$, which lies in the 5-space {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} (or {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}). This simplifies to \llparenthesis X\rrparenthesis_{b}=\mbox{\llbracket X\rrbracket}{{}^{\mbox{\tiny\char 73}}} for $X\in b$; and $\llparenthesis X\rrparenthesis_{b}=\langle X,({X}^{\mathsf{c}^{2}_{b}})^{q},({X}^{\mathsf{c}_{b}})^{q^{2}}\,\rangle$ for $X\in g\backslash b$.

Theorem 4.1

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in a secant $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, the curve ${\cal V}([{\cal C}])$ decomposes into a non-degenerate conic ${\cal N}_{2}$ (which lies in a plane of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ that meets $q+1$ spread planes), together with two planes in $\Sigma_{\infty}$ (possibly repeated, possibly in the extension {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash\Sigma_{\infty}).

Proof   Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in a secant $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. By Result 2.2, in the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$: the affine points of $[\pi]$ are the affine points of a plane $\alpha_{\pi}$; and the points of $[\pi]$ in $\Sigma_{\infty}$ are the points of the $2$-regulus of $\mathcal{S}$ which $\alpha_{\pi}$ meets. Moreover $\bar{\pi}$ and $\alpha_{\pi}$ are in 1-1 correspondence. So corresponding to $\bar{\cal C}$ is a non-degenerate conic ${\cal N}_{2}$ in $\alpha_{\pi}$. So the variety ${\cal V}([{\cal C}])$ contains the conic ${\cal N}_{2}$. We show that the variety ${\cal V}([{\cal C}])$ reduces into ${\cal N}_{2}$ and two planes contained in $\Sigma_{\infty}$ (or an extension). There are three cases to consider, depending on how $\bar{\cal C}$ meets $\ell_{\infty}$. We label the cases to be consistent with Result 3.2.

Cases [S1]and [S2]. Suppose $\bar{\cal C}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, possibly $\bar{P}=\bar{Q}$. In the exact-at-infinity Bruck-Bose setting, $[{\cal C}]$ contains the spread planes $[P]$, $[Q]$. Moreover, in this case ${\cal N}_{2}$ meets $\Sigma_{\infty}$ in the real points $[P]\cap\alpha_{\pi}$ and $[Q]\cap\alpha_{\pi}$ (possibly repeated).

Case [S3]. Suppose $\bar{\cal C}\cap\ell_{\infty}=\emptyset$, so by Result 3.2, ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. To determine the component of ${\cal V}([{\cal C}])$ at infinity, we work in the Bose representation. In ${\rm PG}(8,q^{3})$: $\ell_{\infty}$ corresponds to a line of $\Gamma$ denoted by $g$; $\pi$ is an $\mathbb{F}_{q}$-subplane of $\Gamma$ that meets $g$ in an $\mathbb{F}_{q}$-subline; ${\cal C}$ is an $\mathbb{F}_{q}$-conic in $\pi$; ${\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g=\emptyset$; and ${\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap g^{\mbox{\tiny\char 72}}=\{P,Q\}\subset g^{\mbox{\tiny\char 72}}\backslash g$. Let $b=\pi\cap g$, and let $\mathsf{c}_{\pi}$, $\mathsf{c}_{b}$ be as defined in (2.1) and (2), then by Result 3.2, $Q=P^{\mathsf{c}_{\pi}}=P^{\mathsf{c}_{b}}$ and $P^{\mathsf{c}_{b}^{2}}=P$.

We now look at the Bruck-Bose setting, let $\Sigma_{6,q}$ be a 6-space of ${\rm PG}(8,q)$ that contains the 5-space $\Sigma_{\infty}=\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)$, and use the setting ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$. We wish to find {\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. By Lemma 3.6, this intersection is contained in the $\pi$-scroll-planes $\llparenthesis X\rrparenthesis_{\pi}$ of ${\rm PG}(8,q^{6})$ for which $X^{\mathsf{c}_{\pi}^{i}}\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap g^{\mbox{\tiny\char 72}}$. That is, the intersection is contained in the two $\pi$-scroll-planes $\llparenthesis P\rrparenthesis_{\pi}$ and $\llparenthesis Q\rrparenthesis_{\pi}$. We look at how these two $\pi$-scroll-planes meet {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. As $P,P^{\mathsf{c}_{b}}\in g^{\mbox{\tiny\char 72}}$ and $P^{\mathsf{c}^{2}_{b}}=P$, we have $\llparenthesis P\rrparenthesis_{\pi}=\llparenthesis P\rrparenthesis_{b}$ and $\llparenthesis Q\rrparenthesis_{\pi}=\llparenthesis Q\rrparenthesis_{b}$. So \llparenthesis P\rrparenthesis_{b}=\langle\,P,(P^{\mathsf{c}^{5}_{b}})^{q},(P^{\mathsf{c}_{b}^{4}})^{q^{2}}\,\big{\rangle}=\langle P,Q^{q},P^{q^{2}}\rangle, which lies in the 5-space {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\langle g^{\mbox{\tiny\char 72}},{{g}^{\mbox{\tiny\char 72}}}^{q},{{g}^{\mbox{\tiny\char 72}}}^{q^{2}}\rangle. Hence $\llparenthesis P\rrparenthesis_{b}$, and similarly $\llparenthesis Q\rrparenthesis_{b}$, lies in {\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. That is, the variety ${\cal V}([{\cal C}])$ decomposes into ${\cal N}_{2}$ and two planes at infinity, namely $\llparenthesis P\rrparenthesis_{b},\ \llparenthesis Q\rrparenthesis_{b}$. $\square$

We give a complete description of the Bruck-Bose representation of an $\mathbb{F}_{q}$-conic in a secant $\mathbb{F}_{q}$-subplane and how the variety meets $\Sigma_{\infty}$.

Corollary 4.2

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in a secant $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$, and suppose $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ meets $\ell_{\infty}$ in the two points $\{\bar{P},\bar{Q}\}$ (possibly repeating or in an extension). In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, ${\cal V}([{\cal C}])$ decomposes into a non-degenerate conic ${\cal N}_{2}$ (which lies in a plane of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ that meets $q+1$ spread planes), together with two planes at infinity. Further, we have the following.

If $\bar{\cal C}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, possibly $\bar{P}=\bar{Q}$, then ${\cal V}([{\cal C}])$ contains the two spread planes $[P],[Q]$. Moreover, ${\cal N}_{2}\cap\Sigma_{\infty}$ is two real points, one in $[P]$ and one in $[Q]$. 2. 2.

If $\bar{\cal C}\cap\ell_{\infty}=\emptyset$, then $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$ and ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. The variety-extension ${\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}$ contains the two planes $\llparenthesis P\rrparenthesis_{b}$ and $\llparenthesis Q\rrparenthesis_{b}$ of {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, where $\bar{b}=\bar{\pi}\cap\ell_{\infty}$. Moreover, {{\cal N}}^{\mbox{\tiny\char 72}}_{2}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} is two points, K,K^{q}\in{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, with $K$ not on a T-line.

Proof   The proof of Theorem 4.1 verifies part 1 and the first statement of part 2. To prove the second statement of part 2, consider the non-degenerate conic ${\cal N}_{2}$ in ${\cal V}([{\cal C}])$ and look at ${\cal N}_{2}\cap\Sigma_{\infty}$. Recall $\alpha$ is the plane containing ${\cal N}_{2}$, and as $\bar{\cal C}\cap\ell_{\infty}=\emptyset$, the line $m=\alpha\cap\Sigma_{\infty}$ is exterior to ${\cal N}_{2}$. Hence ${\cal N}_{2}$ meets $\Sigma_{\infty}$ in two points $K,L$ lying in the extension {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, Further, $K=m{{}^{\mbox{\tiny\char 72}}}\cap\llparenthesis P\rrparenthesis_{b}$, $L=m{{}^{\mbox{\tiny\char 72}}}\cap\llparenthesis Q\rrparenthesis_{b}$, and the points $K,L$ are conjugate with respect to the quadratic extension of ${\rm PG}(5,q^{3})$ to ${\rm PG}(5,q^{6})$, that is, $L=K^{q^{3}}$. Moreover, as $m^{q}=m$, $K^{q}\in m$ and $L=K^{q^{3}}=K^{q}$.

Suppose $K$ lies on a T-line $\ell$, and without loss of generality, suppose $\ell=XY^{q}$ for some $X,Y\in g^{\mbox{\tiny\char 72}}$. So $K^{q^{2}}=K\in\ell^{q^{2}}=X^{q^{2}}Y^{q^{3}}$. As $Y^{q^{3}}\in g^{\mbox{\tiny\char 72}}$, the plane $\langle K,g\rangle$ contains $\ell$ and $\ell^{q^{2}}$, so meets ${{g}^{\mbox{\tiny\char 72}}}^{q}$ and ${{g}^{\mbox{\tiny\char 72}}}^{q^{2}}$, contradicting $g,g^{q},g^{q^{2}}$ spanning 5-space. $\square$

4.2 $\mathbb{F}_{q}$-conics in an $\mathbb{F}_{q}$-subplane tangent to $\ell_{\infty}$

Next we look at $\mathbb{F}_{q}$-conics contained in an $\mathbb{F}_{q}$-subplane that is tangent to $\ell_{\infty}$.

Theorem 4.3

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in a tangent $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. Then in the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, the points of $[{\cal C}]$ form either a 6-dim nrc ${\cal N}_{6}$, or a 4-dim nrc ${\cal N}_{4}$ and a spread plane.

Proof   Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in a tangent $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$, and let $\bar{T}=\bar{\pi}\cap\ell_{\infty}$. We work in the Bose representation, so $\Gamma$ is a transversal plane of the Bose spread $\mathbb{S}$; the line at infinity $\ell_{\infty}$ corresponds to a line $g$ of $\Gamma$; and in $\Gamma$, ${\cal C}$ is an $\mathbb{F}_{q}$-conic in an $\mathbb{F}_{q}$-subplane $\pi$, with $T=\pi\cap g$. Let $\mathsf{c}_{\pi}$ be as defined in (2.1), so $\mathsf{c}_{\pi}$ fixes the points of $\pi$, has order 3 acting on the points of $\Gamma$, and has order 6 when acting on the points of the quadratic extension $\Gamma{{}^{\mbox{\tiny\char 72}}}$. Let $\Sigma_{6,q}$ be a 6-space of ${\rm PG}(8,q)$ that contains the 5-space $\Sigma_{\infty}=\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)$, then the Bruck-Bose setting is constructed as ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$, and $g$ is a transversal line of the Bruck-Bose spread $\mathcal{S}$.

We want to determine the structure of the variety ${\cal V}([{\cal C}])$ in the $6$-space $\Sigma_{6,q}$. First note that $\bar{\cal C}$ contains $t\in\{q,q+1\}$ affine points, so the variety ${\cal V}([{\cal C}])$ contains exactly $t$ affine points (that is, points in $\Sigma_{6,q}\backslash\Sigma_{\infty}$) and no three of these points are collinear. So ${\cal V}([{\cal C}])$ is not contained in a line. By Theorem 3.4, {\cal V}([{\cal C}])={\cal V}(\mbox{\llbracket{\cal C}\rrbracket})\cap\Sigma_{6,q} is generically a curve of degree 6. If this is a reducible curve, the components may lie in an extension. We will show that for our analysis, it suffices to work in either the extension to ${\rm PG}(8,q^{3})$ or to ${\rm PG}(8,q^{6})$, and describe ${\cal V}([{\cal C}])$ in the extension ${\rm PG}(8,q^{6})$. By Result 2.3, in ${\rm PG}(8,q^{6})$, the points of the variety {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 72}}={\mathscr{Q}}^{\mbox{\tiny\char 72}}_{1}\cap\cdots\cap{\mathscr{Q}}^{\mbox{\tiny\char 72}}_{9} coincide with the points on the $q^{6}+1$ planes $\{\llparenthesis X\rrparenthesis_{\pi}\,|\,X\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\}$. So the extension {\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}={\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 72}}\cap\Sigma_{6,q}^{{}^{\mbox{\tiny\char 72}}} consists of the subspaces $\llparenthesis X\rrparenthesis_{\pi}\cap\Sigma_{6,q}^{{}^{\mbox{\tiny\char 72}}}$, $X\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$.

Next we consider whether ${\cal V}([{\cal C}])$ can contain a line in some extension. We argue in the extension ${\rm PG}(8,q^{6})$ so that we can use our notation, however the argument works for any finite extension. Suppose ${\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}$ contains a line $\ell$ which is not contained in {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. As $\ell$ is contained in {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 72}}=\{\llparenthesis X\rrparenthesis_{\pi}\,|\,X\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\}, and each $\pi$-scroll-plane meets \Sigma_{6,q}^{{}^{\mbox{\tiny\char 72}}}\backslash{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in at most one point, the line $\ell$ is not contained in a $\pi$-scroll-plane. That is, the $q^{6}+1$ points of $\ell$ lie one in each of the $q^{6}+1$ $\pi$-scroll-planes $\{\llparenthesis X\rrparenthesis_{\pi}\,|\,X\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\}$. This contradicts the fact that any three planes in $\{\llparenthesis X\rrparenthesis_{\pi}\cap\Sigma_{6,q}^{{}^{\mbox{\tiny\char 72}}}\,|\,X\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\}$ span ${\rm PG}(8,q^{6})$ as ${\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$ is a non-degenerate conic. We conclude that if ${\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}$ contains a line $\ell$, then $\ell$ is contained in {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. It follows that if the variety ${\cal V}([{\cal C}])$ contains a plane $\alpha$ in some extension, then $\alpha$ is contained in the hyperplane at infinity.

We now proceed on a case by case basis. In each case, we first determine the intersection at infinity, namely {\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}={\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 72}}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} using Lemma 3.6 which describes the latter. We work in the extension to either ${\rm PG}(8,q^{3})$ or ${\rm PG}(8,q^{6})$ as informed by Lemma 3.6. There are four cases depending on how $\bar{\cal C}$ meets $\ell_{\infty}$. We look at each case separately, labelling to be consistent with Result 3.2.

Case [T1]. Suppose $\bar{T}$ is a point of $\bar{\cal C}$, so $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{T},\bar{Q}\}$ with $\bar{T}\neq\bar{Q}$. We first look at the intersection of ${\cal V}([{\cal C}])$ with $\Sigma_{\infty}$. In the ${\rm PG}(8,q^{3})$ Bose setting, consider the point $Q\in\Gamma$, as $Q\notin\pi$, $Q$ has orbit size 3 under $\mathsf{c}_{\pi}$. As $Q\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, ${Q^{\mathsf{c}_{\pi}}},{Q^{\mathsf{c}^{2}_{\pi}}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$. As $Q\in g$ and $g$ is not a line of $\pi$, we have ${Q^{\mathsf{c}_{\pi}}},{Q^{\mathsf{c}^{2}_{\pi}}}\notin g$ and so {Q^{\mathsf{c}_{\pi}}},{Q^{\mathsf{c}^{2}_{\pi}}}\notin{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. By Lemma 3.6, {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{\llparenthesis X\rrparenthesis_{\pi}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\,|\,X^{\mathsf{c}_{\pi}^{i}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g\textup{ for\ some }i=0,1,2\}. So to determine {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, we only need to consider four planes of {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 73}}, namely $\llparenthesis T\rrparenthesis_{\pi}$, $\llparenthesis Q\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$. As $T\in\pi$, the first plane is $\llparenthesis T\rrparenthesis_{\pi}=[T]{{}^{\mbox{\tiny\char 73}}}$, which lies in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, so lies in ${\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}$. Hence the spread plane $[T]$ lies in ${\cal V}([{\cal C}])$. As $Q\in\Gamma$, the second plane is $\llparenthesis Q\rrparenthesis_{\pi}=\langle Q,(Q^{\mathsf{c}^{2}_{\pi}})^{q},(Q^{\mathsf{c}_{\pi}})^{q^{2}}\rangle.$ As {Q^{\mathsf{c}_{\pi}}},{Q^{\mathsf{c}^{2}_{\pi}}}\notin{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, by Lemma 3.3, $\llparenthesis Q\rrparenthesis_{\pi}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in one point $Q$. Similarly $\llparenthesis{Q^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}=\langle\,{Q^{\mathsf{c}_{\pi}}},\ Q^{q},\,({Q^{\mathsf{c}^{2}_{\pi}}})^{q^{2}}\,\rangle=\llparenthesis Q\rrparenthesis_{\pi}^{q}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in one point $Q^{q}$; and $\llparenthesis{Q^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}=\langle\,{Q^{\mathsf{c}^{2}_{\pi}}},\ ({Q^{\mathsf{c}_{\pi}}})^{q},\,Q^{q^{2}}\,\rangle=\llparenthesis Q\rrparenthesis_{\pi}^{q^{2}}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in one point $Q^{q^{2}}$. So the variety ${\cal V}([{\cal C}])$ meets $\Sigma_{\infty}$ in the spread plane $[T]$ and (in the $\mathbb{F}_{q^{3}}$-extension) the three conjugate points Q,Q^{q},Q^{q^{2}}\in{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash\Sigma_{\infty}.

Next we look at the affine points of the variety ${\cal V}([{\cal C}])$ and show they are $q$ points of a $4$-dim nrc. In ${\rm PG}(2,q^{3})$, let $\bar{U}$ be a point on $\ell_{\infty}$ distinct from $\bar{T},\bar{Q}$. Let $\bar{\ell}$ be a line through $\bar{U}$ distinct from $\ell_{\infty}$. Denote the points of $\bar{\cal C}$ by $\bar{T}=\bar{A}_{0},\bar{A}_{1},\ldots,\bar{A}_{q}$ and let $\bar{b}=\{\bar{\ell}\cap\bar{Q}\bar{A}_{i}\,|\,i=0,\ldots,q\}$. For each $i=0,\ldots,q$, the line $\bar{Q}\bar{A}_{i}$ is tangent to $\bar{\cal C}$, so the set of points $\bar{b}$ is an $\mathbb{F}_{q}$-line that meets $\ell_{\infty}$ (it is the projection of an $\mathbb{F}_{q}$-conic onto a line). We look at this substructure in the Bruck-Bose representation in $\Sigma_{6,q}$. By [3, Thm 2.5], the affine points of $[b]$ are the affine points of a line $\ell_{b}$, and $\ell_{b}$ meets $\Sigma_{\infty}$ in a point of the spread plane $[U]$. Hence $\Sigma_{4}=\langle[Q],\ell_{b}\rangle$ is a $4$-space. Moreover, $\Sigma_{4}$ contains the affine points $[A_{1}],\ldots,[A_{q}]$. Let $\Sigma_{5}$ be a $5$-space containing $\Sigma_{4}$. By [3, Thm 2.7], the affine points of $[\pi]$ are the affine points of a scroll $\mathcal{X}_{\pi}$ that consists of $q+1$ generator lines that rule a non-degenerate conic (contained in $[T]$) and a twisted cubic (contained in a $3$-space disjoint from $[T]$); this scroll is studied in [2]. By [2, Thm 5.1], $\Sigma_{5}\cap\mathcal{X}_{\pi}$ is either a $5$-dim nrc; a $4$-dim nrc and a generator line; or a $3$-dim nrc and up to two generator lines. As $\Sigma_{5}\cap\mathcal{X}_{\pi}$ contains the $q$ points $[A_{1}],\ldots,[A_{q}]$ which lie in a $4$-space, $\Sigma_{5}\cap\mathcal{X}_{\pi}$ is not a $5$-dim nrc. Suppose $\Sigma_{5}\cap\mathcal{X}_{\pi}$ is a $3$-dim nrc ${\cal N}$ and up to two generator lines. In ${\rm PG}(2,q^{3})$, the points $\bar{A}_{1},\ldots,\bar{A}_{q}$ lie on $q$ distinct lines through $\bar{T}$, so in $\Sigma_{6,q}$, the points $[A_{1}],\ldots,[A_{q}]$ lie on distinct generator lines of $\mathcal{X}_{\pi}$. The $4$-space $\Sigma_{4}$ meets $[T]$ in a point, so contains at most one generator line of $\mathcal{X}_{\pi}$. So $\Sigma_{4}\cap\mathcal{X}_{\pi}$ is a $3$-dim nrc ${\cal N}$ and at most one generator line, so ${\cal N}$ contains at least $q-1$ of the points $[A_{1}],\ldots,[A_{q}]$. In the $\mathbb{F}_{q^{3}}$-extension, the nrc ${\cal N}$ and $\Sigma_{\infty}$ meet in three points. As these points lie in ${\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}$, ${{\cal N}^{\mbox{\tiny\char 73}}}$ contains the points $Q,Q^{q},Q^{q^{2}}$. Hence ${\cal N}$ lies in a $3$-space containing the spread plane $[Q]$. In ${\rm PG}(2,q^{3})$ this corresponds to a line through $\bar{Q}$ that contains $q-1$ of the points $\bar{A}_{1},\ldots,\bar{A}_{q}$, a contradiction. Hence $\Sigma_{5}\cap\mathcal{X}_{\pi}$ is a $4$-dim nrc ${\cal N}$ and a generator line $\ell$. Further, by [2, Cor 2.8], $\ell$ does not lie in the $4$-space containing ${\cal N}$. As $[A_{1}],\ldots,[A_{q}]$ is a set of $q$ points no three collinear, it follows that $\Sigma_{4}\cap\mathcal{X}_{\pi}$ is a $4$-dim nrc, with one point in the spread plane $[T]$, namely the point $\Sigma_{4}\cap[T]$. This nrc meets $\Sigma_{\infty}$ in four points over some extension, and these four points lie in the extension of ${\cal V}([{\cal C}])$. Thus the variety ${\cal V}([{\cal C}])$ decomposes into the spread plane $[T]$ and a $4$-dim nrc whose intersection with $\Sigma_{\infty}$ consists of one point of $[T]$ and the three (conjugate) points Q,Q^{q},Q^{q^{2}}\in{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash\Sigma_{\infty}.

Case [T2]. Suppose $\bar{T}\notin{\cal C}$, and that $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, $\bar{P}\neq\bar{Q}$. In the transversal plane $\Gamma$, we have $P,Q\in\Gamma\backslash\pi$, so $P,Q$ both have orbit size 3 under $\mathsf{c}_{\pi}$. As $P,Q\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, the four distinct points ${P^{\mathsf{c}_{\pi}}},{P^{\mathsf{c}^{2}_{\pi}}},{Q^{\mathsf{c}_{\pi}}},{Q^{\mathsf{c}^{2}_{\pi}}}$ lie in ${\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$. As $P,Q\in g$ and $g$ is not a line of $\pi$, the four points ${P^{\mathsf{c}_{\pi}}},{P^{\mathsf{c}^{2}_{\pi}}},{Q^{\mathsf{c}_{\pi}}},{Q^{\mathsf{c}^{2}_{\pi}}}$ do not lie on $g$, and so do not lie in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. By Lemma 3.6, {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{\llparenthesis X\rrparenthesis_{\pi}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\,|\,X^{\mathsf{c}^{i}_{\pi}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g\textup{ for some }i=0,1,2\}, so we only need consider the six $\pi$-scroll-planes, $\llparenthesis P\rrparenthesis_{\pi}$, $\llparenthesis{P^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{P^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis Q\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$. Exactly one of the T-points in $\llparenthesis P\rrparenthesis_{\pi}=\langle P,(P^{\mathsf{c}^{2}_{\pi}})^{q},(P^{\mathsf{c}_{\pi}})^{q^{2}}\rangle,$ lies in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, namely $P$. So by Lemma 3.3, $\llparenthesis P\rrparenthesis_{\pi}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the single point $P$. Similarly, for $i=1,2,3$, $\llparenthesis P^{\mathsf{c}_{\pi}^{i}}\rrparenthesis_{\pi}=\llparenthesis P\rrparenthesis_{\pi}^{q^{i}}=\langle P^{q^{i}},(P^{\mathsf{c}^{2}_{\pi}})^{q^{i+1}},(P^{\mathsf{c}_{\pi}})^{q^{i+2}}\rangle$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in one point $P^{q^{i}}$; and $\llparenthesis Q^{\mathsf{c}_{\pi}^{i}}\rrparenthesis_{\pi}=\llparenthesis Q\rrparenthesis_{\pi}^{q^{i}}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in one point $Q^{q^{i}}$. Thus {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} consists of the six points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$. Hence there is no linear component at infinity, and so, as argued above, ${\cal V}([{\cal C}])$ contains no plane. It follows from Theorem 3.4 that the variety ${\cal V}([{\cal C}])$ is a curve $\mathcal{K}={\cal V}^{6}_{1}$ of degree six. The $\mathbb{F}_{q}$-points of $\mathcal{K}$ are precisely the points of $[{\cal C}]$, that is, a set of $q+1$ affine points, no three collinear. Moreover, the points of $[{\cal C}]$ do not lie in a plane (since a plane of $\Sigma_{6,q}$ corresponds to either a subset of a line of ${\rm PG}(2,q^{3})$, or an $\mathbb{F}_{q}$-plane secant to $\ell_{\infty}$, and $\bar{\cal C}$ does not lie in either).

Suppose the curve $\mathcal{K}$ is reducible. As $\mathcal{K}$ contains $q+1$ affine ($\mathbb{F}_{q}$-rational) points that do not lie in a plane, $\mathcal{K}$ contains an irreducible component ${\cal A}$ which is a curve of degree $x$ with $3\leq x\leq 6$. As $\mathcal{K}$ is reducible, $\mathcal{K}$ also contains a component ${\cal B}$ of degree $y$ with $0<y\leq 3$. The curve ${\cal B}$ spans a subspace $\Sigma$ of dimension at most three. Moreover, $\Sigma$ lies in the extension $\Sigma_{6,q}^{{}^{\mbox{\tiny\char 72}}}\backslash\Sigma_{6,q}$. Hence the conjugate curves ${\cal B}^{q}$, ${\cal B}^{q^{2}}$ are distinct from ${\cal B}$, and are contained in $\mathcal{K}$. Adding degrees of ${\cal A},{\cal B},{\cal B}^{q},{\cal B}^{q^{2}}$, we conclude that $y=1$. As argued above, if ${\cal V}([{\cal C}])$ contains a line ${\cal B}$ over some extension, then ${\cal B}$ is contained in the hyperplane at infinity. However, \mathcal{K}^{{}^{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}, so $\mathcal{K}$ does not contain a line at infinity over any extension. We conclude that the curve $\mathcal{K}$ is irreducible.

Next we show that $\mathcal{K}$ does not lie in a $5$-space. In $\Sigma_{6,q}^{{}^{\mbox{\tiny\char 73}}}$, the curve ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ contains the six points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$, which span the $5$-space {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. As ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ also contains an affine point, ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ does not lie in a 5-space. Hence the curve ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ is irreducible and does not lie in a $5$-space so by [13, Prop 18.9], ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ is a 6-dim nrc. Hence the points of $[{\cal C}]$ form a 6-dim nrc.

Case [T3]. Suppose $\bar{T}\notin{\cal C}$, and that $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P}\}$. A similar argument to case [T2] shows that {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{P,P^{q},P^{q^{2}}\}. So there is no linear component at infinity, it follows from Theorem 3.4 that the variety ${\cal V}([{\cal C}])$ is a curve $\mathcal{K}={\cal V}^{6}_{1}$ of degree six. The curve ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in six points, so each point in $\{P,P^{q},P^{q^{2}}\}$ is a repeated intersection. A similar argument to case [T2] shows that $\mathcal{K}$ is irreducible. Suppose $\mathcal{K}$ lies in a 5-space $\Pi_{5}$ (we work to a contradiction to show this is not possible). In ${\rm PG}(2,q^{3})$, let $\bar{U}$ be a point on $\ell_{\infty}$ distinct from $\bar{T},\bar{P}$. Let $\bar{\ell}$ be a line through $\bar{U}$ distinct from $\ell_{\infty}$. Denote the points of $\bar{\cal C}$ by $\bar{A}_{1},\ldots,\bar{A}_{q+1}$ and let $\bar{b}=\{\bar{\ell}\cap\bar{P}\bar{A}_{i}\,|\,i=1,\ldots,q+1\}$. For each $i=1,\ldots,q+1$, the line $\bar{P}\bar{A}_{i}$ is tangent to $\bar{\cal C}$, so the set of points $\bar{b}$ is an $\mathbb{F}_{q}$-line (it is the projection of an $\mathbb{F}_{q}$-conic onto a line). Moreover, $\bar{T}\notin\bar{\cal C}$, so $\bar{b}$ is disjoint from $\ell_{\infty}$. We look at this substructure in the Bruck-Bose representation in $\Sigma_{6,q}$. The $5$-space $\Pi_{5}$ contains the spread plane $[P]$ and the points $[A_{i}]$, $i=1,\ldots,q+1$. So $\Pi_{5}$ contains the $3$-spaces corresponding to the lines $\bar{P}\bar{A}_{i}$, and hence contains the set $[b]$. The $3$-space $[\ell]$ meets $\Sigma_{\infty}$ in the spread plane $[U]$ and so is disjoint from $[P]$, hence $[\ell]\cap\Pi_{5}$ is a plane. That is, $[b]$ lies in a plane. This contradicts [3, Thm 2.5] which shows that $[b]$ is a 3-dim nrc in the 3-space $[\ell]$. Hence $\mathcal{K}$ does not lie in a $5$-space. That is, $\mathcal{K}$ is an irreducible curve of degree six that is not contained in a $5$-space, so by [13, Prop 18.9], $\mathcal{K}$ is a 6-dim nrc. Hence the points of $[{\cal C}]$ form a 6-dim nrc.

Case [T4]. Suppose $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, so ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. By Result 3.2, $\bar{Q}=\bar{P}^{q^{3}}=\bar{P}^{\bar{\mathsf{c}}_{\pi}^{3}}$. In the Bose setting in ${\rm PG}(8,q^{6})$, we have two points $P,Q\in\Gamma{{}^{\mbox{\tiny\char 72}}}\backslash\Gamma$, with $P,Q\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap g^{\mbox{\tiny\char 72}}$ and $Q=P^{q^{3}}=P^{\mathsf{c}_{\pi}^{3}}$. The six distinct points $P,P^{\mathsf{c}_{\pi}},P^{\mathsf{c}^{2}_{\pi}}$, $Q=P^{\mathsf{c}^{3}_{\pi}},P^{\mathsf{c}^{4}_{\pi}},P^{\mathsf{c}^{5}_{\pi}}$ all lie in ${\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$, while only two of them, namely $P,Q$ lie in {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. By Lemma 3.6, {\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{\llparenthesis X\rrparenthesis_{\pi}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\,|\,X^{\mathsf{c}^{i}_{\pi}}\in{\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap g^{\mbox{\tiny\char 72}}\textup{ for some }i=0,\ldots,5\}, so we only need consider the six $\pi$-scroll-planes, $\llparenthesis P^{\mathsf{c}^{i}_{\pi}}\rrparenthesis_{\pi}$, $i=0,\ldots,5$. As $\llparenthesis P^{\mathsf{c}_{\pi}^{i}}\rrparenthesis_{\pi}=\llparenthesis P\rrparenthesis_{\pi}^{q^{i}}=\langle P^{q^{i}},(P^{\mathsf{c}_{\pi}^{5}})^{q^{i+1}},(P^{\mathsf{c}_{\pi}^{4}})^{q^{i+2}}\rangle$, by Lemma 3.3, $\llparenthesis P^{\mathsf{c}^{i}_{\pi}}\rrparenthesis_{\pi}$ meets {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the point $P^{q^{i}}$, $i=0,\ldots,5$. Hence {\cal V}([{\cal C}])^{\!\mbox{\tiny\char 72}}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} consists of the six points $\{P,P^{q},P^{q^{2}},P^{q^{3}},P^{q^{4}},P^{q^{5}}\}$. So there is no linear component at infinity, it follows from Theorem 3.4 that the variety ${\cal V}([{\cal C}])$ is a curve $\mathcal{K}={\cal V}^{6}_{1}$ of degree six. A similar argument to case [T2] shows that the curve ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ is irreducible and does not lie in a $5$-space. So by [13, Prop 18.9], ${\mathcal{K}^{\mbox{\tiny\char 73}}}$ is a 6-dim nrc. Hence the points of $[{\cal C}]$ form a 6-dim nrc. $\square$

Full details of how these nrcs meet $\Sigma_{\infty}$ follow from the proof of Theorem 4.3.

Corollary 4.4

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in a tangent $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$, let $\bar{T}={\ell_{\infty}}\cap\bar{\pi}$. In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, we have the following.

If $\bar{T}\in\bar{\cal C}$, then $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{T},\bar{Q}\}$ for some $\bar{Q}\neq\bar{T}$. The pointset of $[{\cal C}]$ consists of the spread plane $[T]$ and a 4-dim nrc ${\cal N}_{4}$. Further, {{\cal N}}^{\mbox{\tiny\char 73}}_{4}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} consists of one real point of $[T]$, and the points $Q,Q^{q},Q^{q^{2}}$. 2. 2.

Otherwise $\bar{T}\notin\bar{\cal C}$, and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ meets $\ell_{\infty}$ in two points, $\bar{P},\bar{Q}$, possibly repeated, possibly in the quadratic extension. In this case ${\cal V}([{\cal C}])$ is a 6-dim nrc ${\cal N}_{6}$ of ${\rm PG}(6,q)$. Further ${{\cal N}}^{\mbox{\tiny\char 73}}_{6}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in six points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$, where $P,Q\in g$, possibly repeated, or possibly in the extension $g^{\mbox{\tiny\char 72}}\backslash g$, in which case $Q=P^{q^{3}}$.

4.3 $\mathbb{F}_{q}$-conics in an $\mathbb{F}_{q}$-subplane exterior to $\ell_{\infty}$

The final case to look at is $\mathbb{F}_{q}$-conics contained in an $\mathbb{F}_{q}$-subplane that is exterior to $\ell_{\infty}$. We show they correspond in the Bruck-Bose representation to either 3-dim or 6-dim nrcs, and determine how these nrcs meet $\Sigma_{\infty}$. A coordinate based proof was given in [5]. Here we give a geometric proof, moreover, the description in Corollary 4.6 is a stronger result than that in [5].

Theorem 4.5

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in an exterior $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$, either: ${\cal V}([{\cal C}])$ is a 6-dim nrc; or ${\cal V}([{\cal C}])$ is a 3-dim nrc of ${\rm PG}(6,q)$ and ${\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}$ consists of a 3-dim nrc and three T-lines of {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}.

Proof   Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in an exterior $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$ and let $\bar{\mathsf{c}}_{\pi}$ be as defined in (2.1). Denote the $(\bar{\pi},\ell_{\infty})$-carriers by $\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}},\bar{E}^{\bar{\mathsf{c}}^{2}_{\pi}}$, such that $\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\in\ell_{\infty}$. We work in the Bose representation, so the line at infinity $\ell_{\infty}$ corresponds to a line $g$ of the transversal plane $\Gamma$ of the regular 2-spread $\mathbb{S}$. Let $\Sigma_{6,q}$ be a 6-space of ${\rm PG}(8,q)$ that contains the 5-space $\Sigma_{\infty}=\langle g,g^{q},g^{q^{2}}\rangle\cap{\rm PG}(8,q)$, we use the Bruck-Bose setting ${\cal I}_{\mbox{\scriptsize\sf BB}}={\cal I}_{\mbox{\scriptsize\sf Bose}}\cap\Sigma_{6,q}$.

We want to determine the structure of the variety ${\cal V}([{\cal C}])$ in the $6$-space $\Sigma_{6,q}$. There are four cases depending on how $\bar{\cal C}$ meets $\ell_{\infty}$. We look at each case separately, labelling to be consistent with Result 3.2.

Case [E1]. Suppose that $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}$. By Result 3.1, we also have $\bar{E}^{\bar{\mathsf{c}}^{2}_{\pi}}\in\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$. By Lemma 3.6, {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{\llparenthesis X\rrparenthesis_{\pi}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\,|\,X^{\mathsf{c}^{i}_{\pi}}\in{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap g\textup{ for some }i=0,1,2\}. As the orbit of $\bar{E}$ under $\bar{\mathsf{c}}_{\pi}$ is $\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}},\bar{E}^{\bar{\mathsf{c}}^{2}_{\pi}}\}$, we only need consider the three $\pi$-scroll-planes, $\llparenthesis E\rrparenthesis_{\pi}$, $\llparenthesis E^{\mathsf{c}_{\pi}}\rrparenthesis_{\pi}$, $\llparenthesis E^{\mathsf{c}^{2}_{\pi}}\rrparenthesis_{\pi}$. First consider the plane $\llparenthesis E\rrparenthesis_{\pi}$. Exactly two of the T-points of $\llparenthesis E\rrparenthesis_{\pi}=\langle E,({E^{\mathsf{c}^{2}_{\pi}}})^{q},({E^{\mathsf{c}_{\pi}}})^{q^{2}}\rangle=\langle E,(E^{\mathsf{c}^{2}_{\pi}})^{q},(E^{\mathsf{c}_{\pi}})^{q^{2}}\rangle$ lie in {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, namely $E$ and $(E^{\mathsf{c}_{\pi}})^{q^{2}}$. Hence by Lemma 3.3, \llparenthesis E\rrparenthesis_{\pi}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} is the line $h=E(E^{\mathsf{c}_{\pi}})^{q^{2}}$. Similarly $\llparenthesis E^{\mathsf{c}_{\pi}}\rrparenthesis_{\pi}$ and $\llparenthesis E^{\mathsf{c}^{2}_{\pi}}\rrparenthesis_{\pi}$ meet {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the lines $h^{q}$ and $h^{q^{2}}$ respectively. Hence ${\cal V}([{\cal C}])$ meets $\Sigma_{\infty}$ in the three lines $h,h^{q},h^{q^{2}}$ of {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash\Sigma_{\infty}. Thus by Theorem 3.4, ${\cal V}([{\cal C}])$ reduces to an irreducible curve $\mathcal{K}={\cal V}^{3}_{1}$ of degree three and three lines which lie in the extension {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. Suppose $\mathcal{K}$ is contained in a plane $\alpha$ and let $\ell=\alpha\cap\Sigma_{\infty}$. If $\ell$ is contained in a spread plane $[X]$, then $\mathcal{K}$ is contained in the $3$-space $\Sigma_{3}=\langle[X],\alpha\rangle$. The $3$-space $\Sigma_{3}$ contains a spread plane, so corresponds to a line of ${\rm PG}(2,q^{3})$, implying that $\bar{\cal C}$ is contained in a line of ${\rm PG}(2,q^{3})$, a contradiction. Otherwise $\ell$ meets $q+1$ spread planes, and so by [3, Thm 2.2], $\alpha$ corresponds to an $\mathbb{F}_{q}$-plane of ${\rm PG}(2,q^{3})$ that is secant to $\ell_{\infty}$, implying that $\bar{\cal C}$ lies in an $\mathbb{F}_{q}$-plane secant to $\ell_{\infty}$, a contradiction. Hence $\mathcal{K}$ is not contained in a plane, so by [13, Prop 18.9], $\mathcal{K}$ is a $3$-dim nrc.

Case [E2]. Suppose that $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, with $\bar{P}\neq\bar{Q}$ and $\{\bar{P},\bar{Q}\}\cap\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}=\emptyset$. Similar to the proof of Case [T2] in Theorem 4.3, we are interested in six planes of {\cal V}(\mbox{\llbracket{\cal C}\rrbracket})^{\!\mbox{\tiny\char 73}}, namely $\llparenthesis P\rrparenthesis_{\pi}$, $\llparenthesis{P^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{P^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis Q\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$, moreover each meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in a point, and ${\cal V}([{\cal C}])$ is a 6-dim nrc whose extension meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the six points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$.

Case [E3]. Suppose that $\ell_{\infty}$ tangent to ${\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$, so by Result 3.2, $q$ is odd and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P}\}$, $\bar{P}$ not a $(\bar{\pi},\ell_{\infty})$-carrier. The proof in this case is identical to the proof of case [T3] in Theorem 4.3. The points of $[{\cal C}]$ form a 6-dim nrc that meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the points $P,P^{q},P^{q^{2}}$, each repeated.

Case [E4]. Suppose that $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, then by Result 3.2, ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}\subset\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}\backslash\ell_{\infty}$, $\bar{P}\neq\bar{Q}$. Moreover, by Result 3.2, $\bar{Q}=\bar{P}^{q^{3}}=\bar{P}^{\bar{\mathsf{c}}_{\pi}^{3}}$. So in the transversal plane $\Gamma$ in ${\rm PG}(8,q^{3})$, $P,Q\in g^{\mbox{\tiny\char 72}}\backslash g$ are points of ${\cal C}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}$, and $Q=P^{q^{3}}=P^{\mathsf{c}_{\pi}^{3}}$. Similar to the proof of Case [T4] in Theorem 4.3, we are interested in six $\pi$-scroll-planes, namely $\llparenthesis P\rrparenthesis_{\pi}$, $\llparenthesis{P^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{P^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis Q\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}_{\pi}}}\rrparenthesis_{\pi}$, $\llparenthesis{Q^{\mathsf{c}^{2}_{\pi}}}\rrparenthesis_{\pi}$; each meets {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in a point, and ${\cal V}([{\cal C}])$ is a 6-dim nrc that meets {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the six points $\{P,P^{q},P^{q^{2}},P^{q^{3}},P^{q^{4}},P^{q^{5}}\}$, where $P\in g^{\mbox{\tiny\char 72}}\backslash g$. $\square$

We can describe in detail how ${\cal V}([{\cal C}])$ meets $\Sigma_{\infty}$ in each case.

Corollary 4.6

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in an exterior $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$ and let $\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\in\ell_{\infty}$ be $(\bar{\pi},\ell_{\infty})$-carriers of $\bar{\pi}$. In the exact-at-infinity Bruck-Bose representation in ${\rm PG}(6,q)$ we have the following.

If $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}$, then ${\cal V}([{\cal C}])$ consists of a 3-dim nrc ${\cal N}_{3}$ of ${\rm PG}(6,q)$. Further, {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} is the three T-lines $h=E(E^{\mathsf{c}_{\pi}})^{q^{2}}$, $h^{q}$, $h^{q^{2}}$. Further {{\cal N}}^{\mbox{\tiny\char 73}}_{3}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{R,R^{q},R^{q^{2}}\} where $R\in h$, and $R$ is not contained in an extended spread plane. 2. 2.

*Otherwise $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ meets $\ell_{\infty}$ in two points, $\bar{P},\bar{Q}$, which are not $(\bar{\pi},\ell_{\infty}$)-carriers, possibly repeated, possibly in the quadratic extension. In this case ${\cal V}([{\cal C}])$ is a 6-dim nrc ${\cal N}_{6}$ of ${\rm PG}(6,q)$. Further ${{\cal N}}^{\mbox{\tiny\char 73}}_{6}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in six points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$, where $P,Q\in g$, possibly repeated, or possibly in the extension $g^{\mbox{\tiny\char 72}}\backslash g$, in which case $Q=P^{q^{3}}$. *

Proof   Part 2 is proved in the proof of Theorem 4.5. Consider part 1, where $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}$. By the proof of Case [E1] in Theorem 4.5, the variety ${\cal V}([{\cal C}])$ consists of a 3-dim nrc ${\cal N}_{3}$ of ${\rm PG}(6,q)$, and {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} is the three lines $h=E(E^{\mathsf{c}_{\pi}})^{q^{2}}$, $h^{q}$, $h^{q^{2}}$. As $\bar{\pi}$ is exterior to $\ell_{\infty}$, $\bar{\cal C}$ is exterior to $\ell_{\infty}$. Hence ${\cal V}([{\cal C}])$ contains $q+1$ affine points, so these are the points of ${\cal N}_{3}$, and so ${\cal N}_{3}$ is disjoint from $\Sigma_{\infty}$. Hence {{\cal N}}^{\mbox{\tiny\char 73}}_{3}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} is three points which are contained in $h\cup h^{q}\cup h^{q^{2}}$. As ${{\cal N}}^{\mbox{\tiny\char 73}}_{3}$ meets one of these lines, it meets all three, that is, let ${{\cal N}}^{\mbox{\tiny\char 73}}_{3}\cap h=R$, then $R^{q},R^{q^{2}}\in{{\cal N}}^{\mbox{\tiny\char 73}}_{3}$. We first show that $R$ is not a T-point. Suppose $R=E$, then $\langle R,R^{q},R^{q^{2}}\rangle=\langle E,E^{q},E^{q^{2}}\rangle$, and so the 3-space containing ${\cal N}_{3}$ meets $\Sigma_{\infty}$ in the spread plane \mbox{\llbracket E\rrbracket}=\langle E,E^{q},E^{q^{2}}\rangle\cap\Sigma_{\infty}. However {\cal V}([{\cal C}])^{\!{\mbox{\tiny\char 73}}}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} contains exactly the lines $h,h^{q},h^{q^{2}}$, which are not contained in the plane $\langle E,E^{q},E^{q^{2}}\rangle$, a contradiction. So $R\neq E$, and similarly $R\neq(E^{\mathsf{c}_{\pi}})^{q^{2}}$. Hence $R$ is not a T-point. The point $R$ lies in the T-plane $\llparenthesis E\rrparenthesis_{\pi}$ which contains the line $h$. It is shown in [8, Corollary 2.2] that two distinct T-planes meet in either $\emptyset$, a T-point, or a T-line, hence $R$ does not lie on an extended spread plane. $\square$

For the interested reader, we observe that in [5], the authors use the notation $g_{\mathbb{C}}=E^{\mathsf{c}_{\pi}}E^{q}$. In this article, we have the line $h=E(E^{\mathsf{c}_{\pi}})^{q^{2}}$. That is, the relationship between these two notations is $h=g_{\mathbb{C}}^{q^{2}}$.

4.4 $\mathbb{F}_{q}$-conics using the usual Bruck-Bose convention

As noted in Section 2.6, the usual convention is that the Bruck-Bose representation is not necessarily exact-at-infinity. That is, we do not usually include lines or planes contained in $\Sigma_{\infty}$ in our description. So, ignoring the linear component of ${\cal V}([{\cal C}])$ (which lies in $\Sigma_{\infty}$ or an extension of $\Sigma_{\infty}$), we have the following result.

Corollary 4.7

Let ${\cal C}$ be an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$. Then in the Bruck-Bose ${\rm PG}(6,q)$ representation, $[{\cal C}]$ is a $k$-dim nrc, $k=2$, $3$, $4$ or $6$.

5 Defining $3$-special normal rational curves in ${\rm PG}(6,q)$

We aim to characterise which nrcs of ${\rm PG}(6,q)$ correspond to $\mathbb{F}_{q}$-conics. To this end, we define the notion of a $3$-special nrc in relation to a regular 2-spread $\mathcal{S}$ in a hyperplane of ${\rm PG}(6,q)$. We assume throughout that $q\geq 8$, so that there is a unique 6-dim nrc through any nine points of ${\rm PG}(6,q)$, no seven in a hyperplane.

We define a weight of a point P\in{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, which describes the position of $P$ in relation to the transversal lines of $\mathcal{S}$. Note that this is different to the notion of weights for linear sets. In [16], the weight of a point in a linear set relates to how the corresponding subspace meets the regular spread $\mathcal{S}$ of $\Sigma_{\infty}$. In this article, we use weight to describe how a point in {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} sits in relation to the transversal lines of the regular spread $\mathcal{S}$.

Definition 5.1

Let $\mathcal{S}$ be a 2-regular spread in ${\rm PG}(5,q)$ embedded in ${\rm PG}(5,q^{6})$. Let $\sigma\in{\textup{P}\Gamma\textup{L}}(6,q^{6})$ be the collineation $\,\sigma\colon X=(x_{0},\ldots,x_{5})\mapsto X^{q}=(x_{0}^{q},\ldots,x_{5}^{q})$. Let $P$ be a point in ${\rm PG}(5,q^{6})$.

The weight of $P$, denoted $w(P)$, is defined to be:

• •

$w(P)=1$* if $P$ lies in one of the (extended) transversal lines of $\mathcal{S}$;*

• •

$w(P)=2$* if $w(P)\neq 1$ and $P$ lies on a line that meets two of the (extended) transversal lines of $\mathcal{S}$;*

• •

$w(P)=3$* otherwise.* 2. 2.

The orbit-size of $P$, denoted $o(P)$, is the size of the orbit of $P$ under the collineation $\sigma$. 3. 3.

*Let $s(P)=1$ if $P$ lies in an extended plane of $\mathcal{S}$, otherwise $s(P)=2$. * 4. 4.

The point $P$ is called an $\mathcal{S}$-good point if

[TABLE]

Note that if $P\in{\rm PG}(5,q)$, then $o(P)=1$; if $P\in{\rm PG}(5,q^{3})\backslash{\rm PG}(5,q)$, then $o(P)=3$; if $P\in{\rm PG}(5,q^{6})\backslash{\rm PG}(5,q^{3})$, then either $o(P)=2$ or $o(P)=6$. Moreover, if $P\in{\rm PG}(5,q^{3})$ lies on one of the transversal lines $g$ of the regular 2-spread $\mathcal{S}$, then $o(P)=3$, and if $P\in g^{\mbox{\tiny\char 72}}\backslash g$, then $o(P)=6$.

Definition 5.2

Let ${\rm PG}(6,q)$ have hyperplane at infinity $\Sigma_{\infty}$ and let $\mathcal{S}$ be a regular 2-spread in $\Sigma_{\infty}$. Let ${\cal N}_{r}$, $r\leq 6$, be an $r$-dim nrc of ${\rm PG}(6,q)$ not contained in $\Sigma_{\infty}$. The curve ${\cal N}_{r}$ meets $\Sigma_{\infty}$ in $r$ points, denoted $\{P_{1},\ldots,P_{r}\}$ (possibly repeated or in an extension). We say ${\cal N}_{r}$ is $3$-special* with respect to $\mathcal{S}$ if $w(P_{1})+\cdots+w(P_{r})=6$ and each $P_{i}$ is $\mathcal{S}$-good.*

We next look at an $r$-dim $3$-special nrc ${\cal N}_{r}$ and determine the possibilities for $r$. In each case we describe the points $P_{1}\ldots,P_{r}$, noting any relationship with the transversal lines of the regular 2-spread $\mathcal{S}$.

Theorem 5.3

Let ${\rm PG}(6,q)$ have hyperplane at infinity $\Sigma_{\infty}$ and let $\mathcal{S}$ be a regular 2-spread in $\Sigma_{\infty}$ with transversal lines $g,g^{q},g^{q^{2}}$. Then an $r$-dim $3$-special nrc ${\cal N}_{r}$ in ${\rm PG}(6,q)$ is one of the following.

$r=6$* and {{\cal N}}^{\mbox{\tiny\char 73}}_{6}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\} for some $P,Q\in g$, possibly repeated.* 2. 2.

$r=6$* and {{\cal N}}^{\mbox{\tiny\char 72}}_{6}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{P,P^{q},P^{q^{2}},P^{q^{3}},P^{q^{4}},P^{q^{5}}\} for some $P\in g^{\mbox{\tiny\char 72}}\backslash g$.* 3. 3.

$r=4$* and ${\cal N}_{4}\cap\Sigma_{\infty}=\{T\}$ and {{\cal N}}^{\mbox{\tiny\char 73}}_{4}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{T,P,P^{q},P^{q^{2}}\}, $P\in g$, with $T\notin\langle P,P^{q},P^{q^{2}}\rangle$.* 4. 4.

$r=3$* and {{\cal N}}^{\mbox{\tiny\char 73}}_{3}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{R,R^{q},R^{q^{2}}\} where $R\in XY^{q}$, for some distinct $X,Y\in g$, $R\neq X,Y^{q}$.* 5. 5.

$r=2$* and ${\cal N}_{2}\cap\Sigma_{\infty}=\{P,Q\}$ with $P,Q\in\Sigma_{\infty}$ (possibly repeated).* 6. 6.

$r=2$* and {{\cal N}}^{\mbox{\tiny\char 72}}_{2}\cap{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}=\{P,P^{q^{3}}\} with P\in{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, $P$ not in an extended transversal line.*

Proof   Let ${\cal N}_{r}$ be an $r$-dim $3$-special nrc in ${\rm PG}(6,q)$, and let $\mathcal{K}=\{K_{1},\ldots,K_{r}\}$ denote the points of ${\cal N}_{r}$ at infinity, possibly repeated, possibly in an extension. As ${\cal N}_{r}$ is fixed by the collineation $\sigma\in{\textup{P}\Gamma\textup{L}}(6,q^{6})$ with $\sigma\colon X\mapsto X^{q}$, the set $\mathcal{K}$ is fixed by $\sigma$. That is, if $P\in\mathcal{K}$, then $P^{q},\ldots,P^{q^{5}}\in\mathcal{K}$, possibly repeated, so $o(P)$ divides 6. We consider the possibilities for $\mathcal{S}$-good points in $\mathcal{K}$.

Suppose $A\in\mathcal{K}$ has $w(A)=1$. So without loss of generality, either $A\in g$, in which case $s(A)=1$ and $o(A)=3$; or $A\in g^{\mbox{\tiny\char 72}}\backslash g$, in which case $s(A)=2$ and $o(A)=6$. Both these possibilities satisfy $w(A)o(A)=3s(A)$, so both give $\mathcal{S}$-good points. Suppose $B\in\mathcal{K}$ has $w(B)=2$. The point $B$ is $\mathcal{S}$-good if $2o(B)=3s(B)$, that is, if $s(B)=2$ and $o(B)=3$. As $s(B)=2$, without loss of generality we have $B$ is $\mathcal{S}$-good if $B\in XY^{q}$ for some $X\neq Y\in g$, $B\neq X,Y^{q}$ Suppose $C\in\mathcal{K}$ has $w(C)=3$. The point $C$ is $\mathcal{S}$-good if $3o(C)=3s(C)$, so there are two possibilities. Firstly, if $o(C)=s(C)=1$, then $C\in{\rm PG}(5,q)$. Secondly, if $o(C)=s(C)=2$, then $C\in{\rm PG}(5,q^{6})\backslash{\rm PG}(5,q^{3})$, $C$ is not in an extended transversal line, and the orbit of $C$ is $\{C,C^{q}\}$. In summary, we have five possible point orbits for an $\mathcal{S}$-good point in $\mathcal{K}$:

1. (a)

$\langle A\rangle=\{A,A^{q},A^{q^{2}}\}\subset\mathcal{K}$, $w(A)=w(A^{q})=w(A^{q^{2}})=1$, $A\in g$; 2. (b)

$\langle A\rangle=\{A,\ldots,A^{q^{5}}\}\subset\mathcal{K}$, $w(A)=\cdots=w(A^{q^{5}})=1$, $A\in g^{\mbox{\tiny\char 72}}\backslash g$; 3. (c)

$\langle B\rangle=\{B,B^{q},B^{q^{2}}\}\subset\mathcal{K}$, $w(B)=w(B^{q})=w(B^{q^{2}})=2$, $B\in XY^{q}$ for some $X,Y\in g$, $X\neq Y$, $B\neq X,Y^{q}$; 4. (d)

$\langle C\rangle=\{C\}\subset\mathcal{K}$, $w(C)=3$, $C\in{\rm PG}(5,q)$; 5. (e)

$\langle C\rangle=\{C,C^{q}\}\subset\mathcal{K}$, $w(C)=w(C^{q})=3$, $C\in{\rm PG}(5,q^{6})\backslash{\rm PG}(5,q^{3})$, $C$ is not in an extended transversal line.

We now consider how we can combine these possible orbits of $\mathcal{S}$-good points to satisfy $w(K_{1})+\cdots+w(K_{r})=6$. If $r=6$, then the six points $K_{1},\ldots,K_{6}$ are either two sets of type (a) orbits, giving case 1 of the result; or one type (b) orbit, giving case 2 of the result. If $r=5$, then there is no way to combine orbits of points of type (a),…,(e) to get five points whose weights sum to 6. If $r=4$, then the only way to combine orbits of type (a),…,(e) to get four points whose weights sum to 6 gives $\mathcal{K}=\{T,P,P^{q},P^{q^{2}}\}$, with $T$ type (d), and $P$ type (a). If $T\in\langle P,P^{q},P^{q^{2}}\rangle$, then the plane $\langle P,P^{q},P^{q^{2}}\rangle$ meets ${\cal N}_{4}$ in four points, contradicting ${\cal N}_{4}$ being a 4-dim nrc. This gives case 3 of the result. If $r=3$, then we need an orbit of type (c), giving case 4 of the result. If $r=2$, then there are two possibilities: either two points of type (d), giving case 5 of the result; or one orbit of type (e), giving case 6 of the result. The case $r=1$ cannot occur as no point has weight 6. Hence the six cases outlined in the statement of the result are the only possibilities. $\square$

6 Characterising $\mathbb{F}_{q}$-conics in the Bruck-Bose representation

The main result of this section is Theorem 1.2, which gives one unifying characterisation of the nrcs of ${\rm PG}(6,q)$ that correspond to $\mathbb{F}_{q}$-conics as the $3$-special nrcs. For the remainder of this section, $[{\cal C}]$ is a $k$-dim nrc of ${\rm PG}(6,q)$, and we denote the unique extension of this $k$-dim nrc to a $k$-dim nrc of ${\rm PG}(6,q^{3})$ and ${\rm PG}(6,q^{6})$ by $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ and $[{\cal C}]{{}^{\mbox{\tiny\char 72}}}$ respectively.

6.1 $\mathbb{F}_{q}$-conics

First we prove that an nrc of ${\rm PG}(6,q)$ corresponding to an $\mathbb{F}_{q}$-conic is $3$-special in the sense of Definition 5.2.

Lemma 6.1

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$. Then in the Bruck-Bose ${\rm PG}(6,q)$ representation, $[{\cal C}]$ is a $3$-special nrc.

Proof   Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic in the $\mathbb{F}_{q}$-subplane $\bar{\pi}$ of ${\rm PG}(2,q^{3})$. We consider the three cases where $\bar{\pi}$ is secant, tangent and exterior to $\ell_{\infty}$ separately. We label the cases to be consistent with Result 3.2.

First suppose that $\ell_{\infty}$ is a secant of $\bar{\pi}$. There are two cases to consider, depending on whether $\bar{\cal C}$ contains a point of $\ell_{\infty}$. Case [S1]. If $\bar{\cal C}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}\in\bar{\pi}$ (possibly $\bar{P}=\bar{Q}$), then by Corollary 4.2 and Corollary 4.7, $[{\cal C}]$ is a non-degenerate conic ${\cal N}_{2}$ that meets $\Sigma_{\infty}$ in two real points, $X\in[P]$ and $Y\in[Q]$, possibly $X=Y$. We have $w(X)=w(Y)=3$, $o(X)=o(Y)=1$ and $s(X)=s(Y)=1$. Hence $X$ and $Y$ are $\mathcal{S}$-good and their weights sum to six, so ${\cal N}_{2}$ is a $3$-special 2-dim nrc.

Case [S2]. If $\bar{\cal C}\cap\ell_{\infty}=\emptyset$, by Corollary 4.2 and Corollary 4.7, $[{\cal C}]$ is a non-degenerate conic ${\cal N}_{2}$ whose extension meets $\Sigma_{\infty}$ in two points K,K^{q}\in{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}. Further, $K,K^{q}$ are not on a transversal line of $\mathcal{S}$, or on a line that meets two transversals of $\mathcal{S}$. Hence $w(K)=w(K^{q})=3$. As K,K^{q}\notin{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, they do not lie in an extended spread plane and so $s(K)=s(K^{q})=2$. As $K,K^{q}$ lie on a line in the extension of ${\cal N}_{2}$, they lie in a quadratic extension of $\Sigma_{\infty}$, so over {\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, they have orbit size 2. Hence $K,K^{q}$ are $\mathcal{S}$-good, and their weights sum to six, so ${\cal N}_{2}$ is a $3$-special 2-dim nrc.

Now suppose $\ell_{\infty}$ is a tangent of $\bar{\pi}$, there are three cases to consider. Case [T1]. Suppose $\bar{T}=\ell_{\infty}\cap\bar{\pi}\in\bar{\cal C}$. As $\ell_{\infty}$ is not a line of $\bar{\pi}$, $\ell_{\infty}$ is not a tangent of $\bar{\cal C}$. Hence $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{T},\bar{L}\}$ for some $\bar{L}\neq\bar{T}$. Then by Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 4-dim nrc ${\cal N}_{4}$ which meets $\Sigma_{\infty}$ in one real point $Z\in[T]$ and three points $L,L^{q},L^{q^{2}}$ for some $L\in g$. We have $w(Z)=3$, $o(Z)=1$, $s(Z)=1$, $w(L)=1$, $o(L)=3$, $s(L)=1$, so each of the four points are $\mathcal{S}$-good, and their weights sum to six. Hence ${\cal N}_{4}$ is a $3$-special 4-dim nrc.

Case [T2]. Suppose $\ell_{\infty}\cap\bar{\pi}\notin\bar{\cal C}$ and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}$, possibly repeated. Then by Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is 6-dim nrc ${\cal N}_{6}$ whose extension meets $\Sigma_{\infty}$ in points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$, where $P,Q\in g$ (possibly $P=Q$). For $i=1,2,3$ we have $w(P^{q^{i}})=w(Q^{q^{i}})=1,$ $o(P^{q^{i}})=o(Q^{q^{i}})=3$ and $s(P^{q^{i}})=s(Q^{q^{i}})=1$. Hence each of the six points are $\mathcal{S}$-good, and the weights sum to six. Hence ${\cal N}_{6}$ is a $3$-special 6-dim nrc.

Case [T3]. Suppose $\ell_{\infty}\cap\bar{\pi}\notin\bar{\cal C}$ and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, then ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}$. By Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is 6-dim nrc ${\cal N}_{6}$ whose extension meets $\Sigma_{\infty}$ in the points $\{P,P^{q},P^{q^{2}},P^{q^{3}},P^{q^{4}},P^{q^{5}}\}$, where $P\in g^{\mbox{\tiny\char 72}}\backslash g$. We have $w(P^{q^{i}})=1,$ $o(P^{q^{i}})=6$ and $s(P^{q^{i}})=2$ for $i=1,\ldots,6$. Hence the weights sum to six, and each $P^{q^{i}}$ is $\mathcal{S}$-good. Hence ${\cal N}_{6}$ is a $3$-special 6-dim nrc.

Finally, suppose $\ell_{\infty}$ is exterior to $\bar{\pi}$. Denote the $(\bar{\pi},\ell_{\infty})$-carriers of $\bar{\pi}$ by $\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\in\ell_{\infty}$ and $\bar{E}^{\bar{\mathsf{c}}^{2}_{\pi}}\notin\ell_{\infty}$. There are three cases to consider, depending on how the $\mathbb{F}_{q^{3}}$-conic $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ meets $\ell_{\infty}$.

Case [E1]. Suppose $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}$, then by Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 3-dim nrc ${\cal N}_{3}$ that meets $\Sigma_{\infty}$ in three points, R,R^{q},R^{q^{2}}\in{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash\Sigma_{\infty} with $R\in E(E^{\mathsf{c}_{\pi}})^{q}$, $R\neq E,E^{\mathsf{c}_{\pi}}$, $R$ not a T-point and $R$ not in an extended spread plane. So $w(R)=2$, $o(R)=3$ and $s(R)=2$, hence $R$ and similarly $R^{q},R^{q^{2}}$ are $\mathcal{S}$-good. So ${\cal N}_{3}$ is $3$-special.

Case [E2]. Suppose $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{P},\bar{Q}\}\neq\{\bar{E},\bar{E}^{\bar{\mathsf{c}}_{\pi}}\}$, (possibly $\bar{P}=\bar{Q}$). Then by Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 6-dim nrc ${\cal N}_{6}$, whose extension meets $\Sigma_{\infty}$ in the six points $\{P,P^{q},P^{q^{2}},Q,Q^{q},Q^{q^{2}}\}$ where $P,Q\in g$ (possibly repeated). For $i=1,2,3$ we have $w(P^{q^{i}})=w(Q^{q^{i}})=1,$ $o(P^{q^{i}})=o(Q^{q^{i}})=3$ and $s(P^{q^{i}})=s(Q^{q^{i}})=1$. Hence each of the six points are $\mathcal{S}$-good, and the weights sum to six. Hence ${\cal N}_{6}$ is a $3$-special 6-dim nrc.

Case [E3]. Suppose $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\emptyset$, so ${\bar{\cal C}}^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}\cap\ell^{{\rm{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}\!{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}_{\infty}=\{\bar{P},\bar{Q}\}$. By Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 6-dim nrc ${\cal N}_{6}$, whose extension meets $\Sigma_{\infty}$ in the six points $\{P,P^{q},P^{q^{2}},P^{q^{3}},P^{q^{4}},P^{q^{5}}\}$, where $P\in g^{\mbox{\tiny\char 72}}\backslash g$. We have $w(P^{q^{i}})=1,$ $o(P^{q^{i}})=6$ and $s(P^{q^{i}})=2$ for $i=1,\ldots,6$. Hence the weights sum to six, and each $P^{q^{i}}$ is $\mathcal{S}$-good. Hence ${\cal N}_{6}$ is a $3$-special 6-dim nrc. $\square$

6.2 $3$-special nrcs

If ${\cal N}_{r}$ is a $3$-special nrc in ${\rm PG}(6,q)$, then by Theorem 5.3 there are four possibilities for $r$, namely $r=2,3,4,6$. The next four lemmas look at each of these possibilities, and show that in each case ${\cal N}_{r}$ corresponds to an $\mathbb{F}_{q}$-conic in the corresponding Bruck-Bose plane ${\rm PG}(2,q^{3})$.

Lemma 6.2

In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$. Let ${\cal N}_{6}$ be a $3$-special 6-dim nrc in ${\rm PG}(6,q)$, then ${\cal N}_{6}$ corresponds to an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$.

Proof   In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$, and let $g,g^{q},g^{q^{2}}$ be the three transversal lines of $\mathcal{S}$. Let ${\cal N}_{6}$ be a $3$-special 6-dim nrc in ${\rm PG}(6,q)$, then by Theorem 5.3, parts 1 and 2, ${\cal N}_{6}$ meets $\Sigma_{\infty}$ in six points $\{X,X^{q},X^{q^{2}},\,Y,Y^{q},Y^{q^{2}}\}$, and either (i) $X=Y\in g$, (ii) $X,Y\in g$ are distinct, or (iii) $X,Y\in g^{\mbox{\tiny\char 72}}\backslash g$ and $Y=X^{q^{3}}$. The points $X,Y$ correspond in ${\rm PG}(2,q^{3})$ to points on $\ell_{\infty}$ denoted $\bar{X},\bar{Y}$ respectively, and either (i) $\bar{X}=\bar{Y}\in\ell_{\infty}$, (ii) $\bar{X},\bar{Y}\in\ell_{\infty}$ are distinct, or (iii) $\bar{X},\bar{Y}$ lie in the quadratic extension ${\rm PG}(2,q^{6})$ of ${\rm PG}(2,q^{3})$, and they are conjugate with respect to this quadratic extension.

We first show that we can choose three affine points $[A],[B],[C]$ in ${\cal N}_{6}$ so that in ${\rm PG}(2,q^{3})$, the corresponding points $\{\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}\}$ are no three collinear. Let $[A],[B],[C]\in{\cal N}_{6}$, so $[A],[B],[C]$ are not collinear, and by Theorem 5.3, they are not in $\Sigma_{\infty}$. Hence $[\alpha]=\langle\,[A],[B],[C]\,\rangle$ is a plane that meets $\Sigma_{\infty}$ in a line. Suppose the line $[\alpha]\cap\Sigma_{\infty}$ is contained in a plane $[Z]$ of the regular 2-spread $\mathcal{S}$. As ${\cal N}_{6}$ is a 6-dim nrc, there is an affine point $C^{\prime}$ lying in ${\cal N}_{6}$ which is not in the 3-space $\langle\,[\alpha],[Z]\,\rangle$. Hence the plane $\langle\,[A],[B],[C^{\prime}]\,\rangle$ meets $\Sigma_{\infty}$ in a line which is not contained in a spread plane. That is, we can choose three affine points $[A],[B],[C]$ in ${\cal N}_{6}$ so that the plane $[\alpha]=\langle\,[A],[B],[C]\,\rangle$ meets $\Sigma_{\infty}$ in a line which is not contained in a spread plane. By Result 2.1, in ${\rm PG}(2,q^{3})$, $\bar{\alpha}$ is an $\mathbb{F}_{q}$-subplane secant to $\ell_{\infty}$. Suppose that the three points $\bar{A},\bar{B},\bar{C}\in\bar{\alpha}$ are collinear in ${\rm PG}(2,q^{3})$, then they lie on an $\mathbb{F}_{q}$-subline of $\bar{\alpha}$. Hence by Result 2.1, in ${\rm PG}(6,q)$ the points $[A],[B],[C]$ are collinear, a contradiction. Hence the three affine points $\bar{A},\bar{B},\bar{C}$ of ${\rm PG}(2,q^{3})$ are not collinear.

We now show that the two points $\bar{X},\bar{Y}$ are not in the secant $\mathbb{F}_{q}$-subplane $\bar{\alpha}$. Suppose $\bar{X}\in\bar{\alpha}$, so $\bar{X}\in{\rm PG}(2,q^{3})$. Then by Result 2.1, in ${\rm PG}(6,q)$, $[\alpha]$ meets the spread plane $[X]$ in a point. Hence $\langle\,[X],[\alpha]\,\rangle$ is a 4-space, which contains six points of ${\cal N}_{6}$, namely $\{[A],[B],[C],X,X^{q},X^{q^{2}}\}$, contradicting ${\cal N}_{6}$ being a nrc. Hence $\bar{X}\notin\bar{\alpha}$ and similarly $\bar{Y}\notin\bar{\alpha}$. That is, in ${\rm PG}(2,q^{3})$, $\bar{\alpha}$ is an $\mathbb{F}_{q}$-subplane secant to $\ell_{\infty}$, $\bar{A},\bar{B},\bar{C}$ are non-collinear affine points of $\bar{\alpha}$, and $\bar{X},\bar{Y}$ are points on the line at infinity which are not in $\bar{\alpha}$. Thus the five points $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ are no three collinear.

We now show that in each of the three possible cases (outlined above) for the points $\bar{X},\bar{Y}$, the five points $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ lie on a unique $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$. Case (i) suppose $\bar{X}=\bar{Y}$, then the four points $\bar{A},\bar{B},\bar{C},\bar{X}$ are no three collinear, hence they lie in a unique $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$ that is tangent to $\ell_{\infty}$ at the point $\bar{X}$. Case (ii) suppose $\bar{X},\bar{Y}\in\ell_{\infty}$ are distinct, then as $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ are five points of ${\rm PG}(2,q^{3})$, no three collinear, they lie in a unique $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$. Case (iii) suppose $\bar{X},\bar{Y}$ lie in ${\rm PG}(2,q^{6})\backslash{\rm PG}(2,q^{3})$, then the five points $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ lie in a unique $\mathbb{F}_{q^{6}}$-conic of ${\rm PG}(2,q^{6})$. As the points $\bar{X},\bar{Y}$ are conjugate with respect to the quadratic extension from ${\rm PG}(2,q^{3})$ to ${\rm PG}(2,q^{6})$, this $\mathbb{F}_{q^{6}}$-conic meets ${\rm PG}(2,q^{3})$ in an $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$. Thus in each case, the five points $\{\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}\}$ lie on a unique $\mathbb{F}_{q^{3}}$-conic of ${\rm PG}(2,q^{3})$ which we denote $\bar{\mathcal{O}}$.

Similar to [6, Lemma 5.1], we can show that the three points $\bar{A},\bar{B},\bar{C}$ lie on a unique $\mathbb{F}_{q}$-conic $\bar{\cal C}$ that is contained in $\bar{\mathcal{O}}$ (that is, $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}=\bar{\mathcal{O}}$). Denote the $\mathbb{F}_{q}$-subplane containing $\bar{\cal C}$ by $\bar{\pi}$. We now consider three cases depending on whether $\ell_{\infty}$ is secant, tangent or exterior to $\bar{\pi}$. In each case we either reach a contradiction, or deduce that ${\cal N}_{6}=[{\cal C}]$, and so ${\cal N}_{6}$ corresponds to an $\mathbb{F}_{q}$-conic in $\bar{\pi}$.

Case 1: suppose that $\bar{\pi}$ is secant to $\ell_{\infty}$. There is a unique $\mathbb{F}_{q}$-subplane secant to $\ell_{\infty}$ containing the three non-collinear points $\bar{A},\bar{B},\bar{C}$, so $\bar{\pi}=\bar{\alpha}$. That is, $\bar{\cal C}$ is an $\mathbb{F}_{q}$-conic in a secant $\mathbb{F}_{q}$-subplane $\bar{\pi}=\bar{\alpha}$, $\bar{A},\bar{B},\bar{C}\in\bar{\cal C}$ and $\bar{X},\bar{Y}\notin\bar{\pi}$. So by Result 3.2, $\bar{X},\bar{Y}$ lie in the quadratic extension ${\rm PG}(2,q^{6})\backslash{\rm PG}(2,q^{3})$. Let $\bar{b}=\bar{\pi}\cap\ell_{\infty}$ and let $\bar{\mathsf{c}}_{b}$ be as defined in (2). By Corollary 4.2 and Corollary 4.7, in ${\rm PG}(6,q)$, $[{\cal C}]$ is a conic in the plane $[\alpha]=[\pi]$ and $[{\cal C}]$ meets $\Sigma_{\infty}$ in two points K,K^{q^{3}}\in{\Sigma}^{\mbox{\tiny\char 72}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}\backslash{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}}, with $K\in\llparenthesis X\rrparenthesis_{b}$. By Result 3.2, $\bar{X}^{\bar{\mathsf{c}}_{b}}=\bar{Y}$, so $\llparenthesis X\rrparenthesis_{b}=\langle X,(X^{{{\mathsf{c}_{b}}^{2}}})^{q},(X^{\mathsf{c}_{b}})^{q^{2}}\rangle=\langle X,X^{q},Y^{q^{2}}\rangle$. As $[\pi]{{}^{\mbox{\tiny\char 72}}}$ meets the plane $\llparenthesis X\rrparenthesis_{b}$ in the point $K$, $\langle\,[\pi]{{}^{\mbox{\tiny\char 72}}},\llparenthesis X\rrparenthesis_{b}\rangle$ is a 4-space of ${\rm PG}(6,q^{6})$ that contains six points of ${{\cal N}}^{\mbox{\tiny\char 72}}_{6}$, namely $\{[A],[B],[C],X,X^{{{q}}},Y^{q^{2}}\}$, a contradiction. Hence this case does not occur.

Case 2: suppose that $\bar{\pi}$ is tangent to $\ell_{\infty}$. By Result 3.2, either (a) one of $\bar{X}$ or $\bar{Y}$ lies in $\bar{\cal C}$, in which case $\bar{X}\neq\bar{Y}$ or (b) neither $\bar{X}$ or $\bar{Y}$ lie in $\bar{\pi}$. Case 2(a): suppose that $\bar{\pi}\cap\ell_{\infty}=\bar{X}$, so $\bar{X}\in\bar{\cal C}$ and $\bar{Y}\in\ell_{\infty}\backslash\bar{X}$. Then by Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 4-dim nrc that contains one point of the spread plane $[X]$, and the extension $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ contains the six points $\{[A],[B],[C],\,Y,Y^{q},Y^{q^{2}}\}.$ Thus $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ and ${{\cal N}}^{\mbox{\tiny\char 73}}_{6}$ have these six points in common, and these six points lie in the 4-space containing $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$, a contradiction as no six points of ${{\cal N}}^{\mbox{\tiny\char 73}}_{6}$ lie in a 4-space. Hence this case does not occur.

Case 2(b): suppose $\bar{X},\bar{Y}\notin\bar{\pi}$. Recall that either $X,Y\in g$ possibly equal, or $X,Y\in g^{\mbox{\tiny\char 72}}\backslash g$. By Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 6-dim nrc whose extension contains the nine points $\{[A],[B],[C],X,X^{q},X^{q^{2}},\,Y,Y^{q},Y^{q^{2}}\}$. That is, $[{\cal C}]$ and ${\cal N}_{6}$ are 6-dim nrcs whose extensions to ${\rm PG}(6,q^{3})$ or ${\rm PG}(6,q^{6})$ share nine points. Hence by [14, Theorem 21.1.1] they are equal. Hence ${\cal N}_{6}$ corresponds in ${\rm PG}(2,q^{3})$ to the $\mathbb{F}_{q}$-conic $\bar{\cal C}$.

Case 3: suppose that $\bar{\pi}$ is exterior to $\ell_{\infty}$. There are two cases, either (a) $\bar{X},\bar{Y}$ are the $(\bar{\pi},\ell_{\infty}$)-carriers on $\ell_{\infty}$, or (b) they are not. Case 3(a): suppose that $\bar{X}=\bar{E}$, $\bar{Y}=\bar{E}^{{\bar{\mathsf{c}}_{\pi}}}$ are the $(\bar{\pi},\ell_{\infty}$)-carriers on $\ell_{\infty}$. Then by Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 3-dim nrc that contains the three points $\{[A],[B],[C]\}$, and whose extension meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in three points $\{R,R^{q},R^{q^{2}}\}$ where $R\in h=E(E^{{{\mathsf{c}}_{\pi}}})^{q^{2}}=XY^{q^{2}}.$ Let $\Pi_{3}=\langle\,[A],[B],[C],R,R^{q},R^{q^{2}}\rangle$ be the 3-space of ${\rm PG}(6,q^{3})$ containing the 3-dim nrc $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$. The two lines $h=XY^{q^{2}}$ and $h^{q}=X^{q}Y$ are skew and each meet $\Pi_{3}$ in a point. Hence $\Pi_{5}=\langle\,\Pi_{3},h,h^{q}\rangle$ is a 5-space which contains seven points of ${{\cal N}}^{\mbox{\tiny\char 73}}_{6}$, namely $\{[A],[B],[C],X,X^{q},Y^{q},Y^{q^{2}}\},$ a contradiction. Hence this case does not occur.

Case 3(b): suppose that the points $\bar{X},\bar{Y}$ are not $(\bar{\pi},\ell_{\infty})$-carriers. Recall either $X,Y\in g$ possibly equal, or $X,Y\in g^{\mbox{\tiny\char 72}}\backslash g$. By Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 6-dim nrc whose extension contains the nine points $\{[A],[B],[C],X,X^{q},X^{q^{2}},\,Y,Y^{q},Y^{q^{2}}\}$. That is, the extensions of $[{\cal C}]$ and ${\cal N}_{6}$ to ${\rm PG}(2,q^{3})$ or ${\rm PG}(2,q^{6})$ are 6-dim nrcs which share nine points. So by [14, Theorem 21.1.1] they are equal. Hence ${\cal N}_{6}$ corresponds in ${\rm PG}(2,q^{3})$ to the $\mathbb{F}_{q}$-conic $\bar{\cal C}$.

In summary, the only possibilities for $\bar{\pi}$ are given in Cases 2b) and 3b). In each case we show that ${\cal N}_{6}$ corresponds in ${\rm PG}(2,q^{3})$ to the $\mathbb{F}_{q}$-conic $\bar{\cal C}\subset\bar{\pi}$. That is, a $3$-special 6-dim nrc of ${\rm PG}(6,q)$ corresponds to an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$. $\square$

Lemma 6.3

In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$. Let ${\cal N}_{4}$ be a $3$-special 4-dim nrc in ${\rm PG}(6,q)$, then ${\cal N}_{4}$ corresponds in ${\rm PG}(2,q^{3})$ to an $\mathbb{F}_{q}$-conic contained in a tangent $\mathbb{F}_{q}$-subplane.

Proof   In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$, and let $g,g^{q},g^{q^{2}}$ be the transversals of $\mathcal{S}$. We use a notation in ${\rm PG}(6,q)$ that is consistent with our Bruck-Bose notation, as described in the first paragraph of the proof of Lemma 6.2.

Let ${\cal N}_{4}$ be a $3$-special 4-dim nrc in ${\rm PG}(6,q)$, so by Theorem 5.3, ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in four points $\{T,\,Y,Y^{q},Y^{q^{2}}\}$ for some $T\in\Sigma_{\infty}$, $Y\in g$, $T\notin\langle Y,Y^{q},Y^{q^{2}}\rangle$. The point $Y$ corresponds in ${\rm PG}(2,q^{3})$ to a point $\bar{Y}\in\ell_{\infty}$. The point $T$ lies in a unique plane of the 2-spread $\mathcal{S}$, denote this spread plane by [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}], and note that as $T\notin[Y]$, we have [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]\neq[Y]. The spread plane [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}] corresponds in ${\rm PG}(2,q^{3})$ to a point \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\in\ell_{\infty}, and \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\neq\bar{Y}. Let $\Pi_{4}$ denote the 4-space containing ${\cal N}_{4}$. So $\Pi_{4}$ meets $\Sigma_{\infty}$ in the 3-space $\langle T,[Y]\rangle$, that is, \Pi_{4}\cap[{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]=T.

We first show that we can choose three affine points $[A],[B],[C]$ in ${\cal N}_{4}$ so that in ${\rm PG}(2,q^{3})$, the five points \bar{A},\bar{B},\bar{C},\bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y} are no three collinear. The same argument as that in the proof of Lemma 6.2 shows that we can find three affine points $[A],[B],[C]$ in ${\cal N}_{4}$ so that the plane $[\alpha]=\langle[A],[B],[C]\rangle$ corresponds to a secant $\mathbb{F}_{q}$-subplane $\bar{\alpha}$ of ${\rm PG}(2,q^{3})$, and the three affine points $\bar{A},\bar{B},\bar{C}\in\bar{\alpha}$ are not collinear. In ${\rm PG}(6,q)$, suppose the planes $[\alpha]$ and [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}] meet. As $[\alpha]$ is contained in $\Pi_{4}$ (the 4-space containing ${\cal N}_{4}$), [\alpha]\cap[{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]=\Pi_{4}\cap[{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]=T. That is, $[\alpha]$ contains the four points $\{[A],[B],[C],T\}$ of ${\cal N}_{4}$, a contradiction. Hence $[\alpha]$ does not meet [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]. Hence a line joining any two of the points $[A],[B],[C]$ does not meet [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]. Hence by Result 2.1, \bar{A},\bar{B},\bar{C},\bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}} are four points of ${\rm PG}(2,q^{3})$, no three collinear. Now suppose the three points $\bar{A},\bar{B},\bar{Y}$ of ${\rm PG}(2,q^{3})$ are collinear. Then by Result 2.1, in ${\rm PG}(6,q)$, the line joining $[A]$ and $[B]$ meets $\Sigma_{\infty}$ in a point of the spread plane $[Y]$. Hence $\langle[A],[B],[Y]\rangle$ is a 3-space whose extension contains five points of ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$, namely $\{[A],[B],Y,Y^{q},Y^{q^{2}}\}$, a contradiction. So $\bar{A},\bar{B},\bar{Y}$ are not collinear, similarly $\bar{A},\bar{C},\bar{Y}$ are not collinear and $\bar{C},\bar{B},\bar{Y}$ are not collinear. Hence in ${\rm PG}(2,q^{3})$, the five points \bar{A},\bar{B},\bar{C},\bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y} are no three collinear.

As \bar{A},\bar{B},\bar{C},\bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y} are five points of ${\rm PG}(2,q^{3})$, no three collinear, they lie in a unique $\mathbb{F}_{q^{3}}$-conic $\bar{\mathcal{O}}$ of ${\rm PG}(2,q^{3})$. Similar to [6, Lemma 5.1], we can show that $\bar{A},\bar{B},\bar{C}$ lie on a unique $\mathbb{F}_{q}$-conic $\bar{\cal C}$ that is contained in $\bar{\mathcal{O}}$, that is $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}=\bar{\mathcal{O}}$. Denote the $\mathbb{F}_{q}$-subplane containing $\bar{\cal C}$ by $\bar{\pi}$. We consider three cases depending on whether $\ell_{\infty}$ is secant, tangent or exterior to $\bar{\pi}$. In each case we either reach a contradiction, or deduce that ${\cal N}_{4}=[{\cal C}]$ in which case ${\cal N}_{4}$ corresponds to the $\mathbb{F}_{q}$-conic in $\bar{\pi}$.

Case 1: suppose $\bar{\pi}$ is secant to $\ell_{\infty}$. As \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y} lie in ${\rm PG}(2,q^{3})$, by Result 3.2, they lie in $\bar{\pi}$, so lie in $\bar{\cal C}$. By Result 2.1, in ${\rm PG}(6,q)$, $[\pi]$ is a plane of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ which meets both spread planes [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}], $[Y]$ in a point. Hence $\langle[\pi],[Y]\rangle$ is a 4-space whose extension contains the six points $\{[A],[B],[C],Y,Y^{q},Y^{q^{2}}\}$ of ${\cal N}_{4}$. Hence ${\cal N}_{4}$ lies in this 4-space, and so $[\pi]$ meets [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}] in the point \Pi_{4}\cap[{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]=T. Thus the plane $[\pi]$ contains the four points $\{[A],[B],[C],T\}$ of ${\cal N}_{4}$, a contradiction. So this case does not occur.

Case 2: suppose $\bar{\pi}$ is tangent to $\ell_{\infty}$. Recall \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\neq\bar{Y}, so there are three cases to consider, either (a) \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}=\bar{\pi}\cap\ell_{\infty}, (b) $\bar{Y}=\bar{\pi}\cap\ell_{\infty}$, or (c) \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y}\notin\bar{\pi}. Case 2(a): suppose that \bar{\pi}\cap\ell_{\infty}=\bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}, so \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\in\bar{\cal C}. Then by Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 4-dim nrc lying in a 4-space $\Sigma_{4}$, and $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the four points \{Y,Y^{q},Y^{q^{2}},[{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]\cap\Sigma_{4}\}. Now $\Pi_{4}$ (the 4-space containing ${\cal N}_{4}$) contains the three non-collinear affine points $[A],[B],[C]$ and the spread plane $[Y]$, and so $\Pi_{4}=\langle[A],[B],[C],[Y]\rangle$. As $\Sigma_{4}$ contains $[A],[B],[C]$ and $[Y]$, $\Sigma_{4}=\Pi_{4}$. Thus [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]\cap\Sigma_{4}=[{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]\cap\Pi_{4}=T. Hence $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ and ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$ are 4-dimensional nrcs with seven points in common, namely $\{[A],[B],[C],Y,Y^{q},Y^{q^{2}},T\}$, so by [14, Theorem 21.1.1] they are equal. That is ${\cal N}_{4}$ corresponds to the $\mathbb{F}_{q}$-conic ${\cal C}$.

Case 2(b): suppose $\bar{Y}\in\bar{\pi}$, so $\bar{Y}\in\bar{\cal C}$. Then by Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 4-dim nrc contained in a 4-space $\Sigma_{4}$ whose extension meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in the four points \{{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}^{q},{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}^{q^{2}},\Sigma_{4}\cap[Y]\}. As the two planes [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}] and $[\alpha]=\langle[A],[B],[C]\rangle$ lie in $\Sigma_{4}$, they meet in a point. This contradicts the argument above where we show that $[\alpha]$ does not meet the spread plane [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}]. So this case does not occur.

Case 2(c): suppose \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y}\notin\bar{\pi}. Recall {X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},Y\in g, {X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\neq Y, so by Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 6-dim nrc, $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ contains the nine points \{[A],[B],[C],{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}^{q},{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}^{q^{2}},\,Y,Y^{q},Y^{q^{2}}\}. That is, $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ and ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$ share six points, namely $\{[A],[B],[C],Y,Y^{q},Y^{q^{2}}\}.$ That is, $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ contains six points which lie in the 4-space containing ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$, contradicting $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ being a 6-dim nrc. Hence this case cannot occur.

Case 3: suppose $\bar{\pi}$ is exterior to $\ell_{\infty}$. There are two cases, either (a) \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y} are the $(\bar{\pi},\ell_{\infty}$)-carriers on $\ell_{\infty}$, or (b) they are not. Case 3(a): suppose that \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}=\bar{E}, $\bar{Y}=\bar{E}^{{\bar{\mathsf{c}}_{\pi}}}$ are the $(\bar{\pi},\ell_{\infty}$)-carriers on $\ell_{\infty}$. Then by Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 3-dim nrc whose extension meets {\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} in three points $\{R,R^{q},R^{q^{2}}\}$, where R\in h={X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}Y^{q^{2}}, and R\neq{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},Y^{q^{2}}. The line $h=XY^{q}$ lies in ${\rm PG}(6,q^{3})$ and does not meet ${\rm PG}(6,q)$. The lines $h,h^{q},h^{q^{2}}$ are transversals of a regular 2-spread $\mathcal{S}_{h}$ of $\Sigma_{\infty}$. Moreover, [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}],[Y] are planes of $\mathcal{S}_{h}$, thus the three planes [{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}],[Y], $\langle R,R^{q},R^{q^{2}}\rangle\cap\Sigma_{\infty}$ of $\mathcal{S}_{h}$ are pairwise disjoint. The two planes $[\alpha]=\langle[A],[B],[C]\rangle$ and $\langle R,R^{q},R^{q^{2}}\rangle\cap\Sigma_{\infty}$ lie in the 3-space containing $[{\cal C}]$. Hence $[\alpha]$ meets $\langle R,R^{q},R^{q^{2}}\rangle\cap\Sigma_{\infty}$ in a line $m$. As $[\alpha]$ lies in the 4-space $\Pi_{4}$ containing ${\cal N}_{4}$, the line $m$ lies in the 3-space $\Pi_{4}\cap\Sigma_{\infty}$. This 3-space contains the plane $[Y]$, so $m$ meets $[Y]$ in at least a point. This contradicts the two planes $[Y]$, $\langle R,R^{q},R^{q^{2}}\rangle\cap\Sigma_{\infty}$ being disjoint. Hence this case does not occur.

Case 3(b): suppose \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},\bar{Y} are not $(\bar{\pi},\ell_{\infty})$-carriers. Recall {X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\neq Y and {X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},Y\in g. By Corollaries 4.6 and 4.7, $[{\cal C}]$ is a 6-dim nrc and $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ contains the nine points \{[A],[B],[C],{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}},{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}^{q},{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}^{q^{2}},\,Y,Y^{q},Y^{q^{2}}\}. That is, $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ and ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$ share the six points $\{[A],[B],[C],Y,Y^{q},Y^{q^{2}}\}.$ These six points lie in the 4-space containing ${{\cal N}}^{\mbox{\tiny\char 73}}_{4}$, contradicting $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ being a 6-dim nrc. Hence this case cannot occur.

In summary, the only possibility for $\bar{\pi}$ is that described in case 2(a), where $\bar{\pi}$ is tangent to $\ell_{\infty}$ and \bar{X}_{\mbox{\raisebox{0.01pt}{\scalebox{0.8}{{\tiny\!T}}}}}\in\bar{\pi}. Hence if ${\cal N}_{4}$ is a $3$-special 4-dim nrc in ${\rm PG}(6,q)$, then ${\cal N}_{4}$ corresponds in ${\rm PG}(2,q^{3})$ to the $\mathbb{F}_{q}$-conic ${\cal C}$, and $\bar{\pi}$ is tangent to $\ell_{\infty}$. $\square$

Lemma 6.4

In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$. Let ${\cal N}_{3}$ be a $3$-special 3-dim nrc in ${\rm PG}(6,q)$, then ${\cal N}_{3}$ corresponds in ${\rm PG}(2,q^{3})$ to an $\mathbb{F}_{q}$-conic contained in an exterior $\mathbb{F}_{q}$-subplane.

Proof   In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$, and let $g,g^{q},g^{q^{2}}$ be the transversals of $\mathcal{S}$. We use a notation in ${\rm PG}(6,q)$ that is consistent with our Bruck-Bose notation, as described in the first paragraph of the proof of Lemma 6.2.

Let ${\cal N}_{3}$ be a $3$-special 3-dim nrc in ${\rm PG}(6,q)$. By Theorem 5.3, there are distinct points $X,Y\in g$ such that ${\cal N}_{3}\cap\Sigma_{\infty}=\{R,R^{q},R^{q^{2}}\}$, where $R\in XY^{q}$, $R\neq X,Y^{q}$.

We first show that we can choose three affine points $[A],[B],[C]$ in ${\cal N}_{3}$ so that in ${\rm PG}(2,q^{3})$, the five points $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ are no three collinear. The same argument as that in the proof of Lemma 6.2 shows that we can find three affine points $[A],[B],[C]$ in ${\cal N}_{3}$ so that the plane $[\alpha]=\langle[A],[B],[C]\rangle$ corresponds to a secant $\mathbb{F}_{q}$-subplane $\bar{\alpha}$ of ${\rm PG}(2,q^{3})$, and the three affine points $\bar{A},\bar{B},\bar{C}\in\bar{\alpha}$ are not collinear. We next show that the points $\bar{X},\bar{Y}$ are not in $\bar{\alpha}$. Let $\Pi_{3}$ be the 3-space containing ${\cal N}_{3}$. Then ${\Pi}^{\mbox{\tiny\char 73}}_{3}$ contains the six points $\{[A],[B],[C],R,R^{q},R^{q^{2}}\}$ and {\Pi}^{\mbox{\tiny\char 73}}_{3}\cap{\Sigma}^{\mbox{\tiny\char 73}}_{\mbox{\raisebox{0.4pt}{\scalebox{0.6}{\infty}}}} is the plane $\langle R,R^{q},R^{q^{2}}\rangle$. The line $h=XY^{q}$ lies in ${\rm PG}(6,q^{3})$ and does not meet ${\rm PG}(6,q)$. Hence the three lines $h$, $h^{q},h^{q^{2}}$ are the transversal lines of a regular 2-spread $\mathcal{S}_{h}$ of ${\rm PG}(6,q)$. The two regular 2-spreads $\mathcal{S},\mathcal{S}_{h}$ share exactly two planes, namely $[X],[Y]$. As $\mathcal{S}_{h}$ contains the three planes $[X],[Y]$ and $\langle R,R^{q},R^{q^{2}}\rangle\cap\Sigma_{\infty}$, these three planes are pairwise disjoint. Hence $\Pi_{3}$ does not meet $[X]$ or $[Y]$ and so the plane $[\alpha]$ does not meet $[X]$ or $[Y]$. Thus in ${\rm PG}(2,q^{3})$, $\bar{A},\bar{B},\bar{C}$ are non-collinear affine points in the secant $\mathbb{F}_{q}$-subplane $\bar{\alpha}$, and the points $\bar{X},\bar{Y}$ lie in $\ell_{\infty}\backslash\bar{\alpha}$. Hence the five points $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ are no three collinear.

As $\bar{A},\bar{B},\bar{C},\bar{X},\bar{Y}$ are five points of ${\rm PG}(2,q^{3})$, no three collinear, they lie in a unique $\mathbb{F}_{q^{3}}$-conic $\bar{\mathcal{O}}$ of ${\rm PG}(2,q^{3})$. Similar to [6, Lemma 5.1], we can show that the three points $\bar{A},\bar{B},\bar{C}$ lie on a unique $\mathbb{F}_{q}$-conic $\bar{\cal C}$ that is contained in $\bar{\mathcal{O}}$ (that is, $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}=\bar{\mathcal{O}}$). Denote the $\mathbb{F}_{q}$-subplane containing $\bar{\cal C}$ by $\bar{\pi}$. We consider three cases depending on whether $\bar{\pi}$ is secant, tangent or exterior to $\ell_{\infty}$. In each case we either reach a contradiction, or deduce that ${\cal N}_{3}=[{\cal C}]$, and so ${\cal N}_{3}$ corresponds to an $\mathbb{F}_{q}$-conic in $\bar{\pi}$.

Case 1: suppose $\bar{\pi}$ is secant to $\ell_{\infty}$. In this case we have $\bar{\cal C}\cap\ell_{\infty}=\emptyset$ and $\bar{\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}\cap\ell_{\infty}=\{\bar{X},\bar{Y}\}\subset{\rm PG}(2,q^{3})$. This contradicts Result 3.2 which shows that if $\bar{\cal C}$ lies in a secant $\mathbb{F}_{q}$-subplane and $\bar{\cal C}\cap\ell_{\infty}=\emptyset$, then the intersection of ${\cal C}^{{\rm\boldsymbol{{\raisebox{0.2pt}{\scalebox{0.5}{{+}}}}}}}$ with $\ell_{\infty}$ lies in ${\rm PG}(2,q^{6})\backslash{\rm PG}(2,q^{3})$. Hence this case cannot occur.

Case 2: suppose $\bar{\pi}$ is tangent to $\ell_{\infty}$. Recalling that $\bar{X}\neq\bar{Y}$, there are two cases to consider. Either (a) exactly one of $\bar{X},\bar{Y}$ lie in $\bar{\pi}$ or (b) $\bar{X},\bar{Y}\notin\bar{\pi}$. Case 2(a): suppose $\bar{X}\in\bar{\pi}$, then $\bar{X}\in\bar{\cal C}$. By Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 4-dim nrc whose extension contains the six points $\{[A],[B],[C],Y,Y^{q},Y^{q^{2}}\}$ and a point in the spread plane $[X]$. So the 4-space $\Pi_{4}$ containing $[{\cal C}]$ contains the two planes $[Y]$ and $[\alpha]=\langle[A],[B],[C]\rangle$, so these planes meet in at least a point, a contradiction as the two planes $[Y]$, $[\alpha]$ do not meet. Hence this case cannot occur. Similarly the case $\bar{Y}\in\bar{\pi}$ cannot occur.

Case 2(b): suppose $\bar{X},\bar{Y}\notin\bar{\pi}$. Recall $X\neq Y$ and $X,Y\in g$. By Corollary 4.4 and Corollary 4.7, $[{\cal C}]$ is a 6-dim nrc and $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$ contains the nine points $\{[A],[B],[C],X,X^{q},X^{q^{2}},Y,Y^{q},Y^{q^{2}}\}.$ Let $\Pi_{3}$ be the 3-space containing ${\cal N}_{3}$. So ${\Pi}^{\mbox{\tiny\char 73}}_{3}$ contains the six points $\{[A],[B],[C],R,R^{q},R^{q^{2}}\}$ where $R\in h=XY^{q^{2}}$. As $h,h^{q}$ are skew, and are not contained in ${\Pi}^{\mbox{\tiny\char 73}}_{3}$, $\langle{\Pi}^{\mbox{\tiny\char 73}}_{3},h,h^{q}\rangle$ is a 5-space of ${\rm PG}(6,q^{3})$ that contains seven points of the 6-dim nrc $[{\cal C}]{{}^{\mbox{\tiny\char 73}}}$, namely $\{[A],[B],[C],X,X^{q},Y,Y^{q^{2}}\}$, a contradiction. Hence this case cannot occur.

Case 3: suppose $\bar{\pi}$ is exterior to $\ell_{\infty}$. There are two cases, either (a) $\bar{X},\bar{Y}$ are the $(\bar{\pi},\ell_{\infty}$)-carriers on $\ell_{\infty}$, or (b) they are not. Case 3(a): suppose that $\bar{X}=\bar{E}$, $\bar{Y}=\bar{E}^{{\bar{\mathsf{c}}_{\pi}}}$ are the $(\bar{\pi},\ell_{\infty}$)-carriers on $\ell_{\infty}$. By Corollary 4.6 and Corollary 4.7, $[{\cal C}]$ is a 3-dim nrc in a 3-space $\Sigma_{3}$ whose extension contains the six points $\{[A],[B],[C],K,K^{q},K^{q^{2}}\}$, where $K\in h=XY^{q^{2}}$, $K\neq X,Y^{q^{2}}$. Recall that the 3-space ${\Pi}^{\mbox{\tiny\char 73}}_{3}$ containing ${{\cal N}}^{\mbox{\tiny\char 73}}_{3}$ contains the six points $\{[A],[B],[C],R,R^{q},R^{q^{2}}\}$ where $R\in h=XY^{q^{2}}$. We show that $R=K$. Let $\mathcal{S}_{h}$ be the regular 2-spread of $\Sigma_{\infty}$ with transversal lines $h,h^{q},h^{q^{2}}$. The planes $\langle K,K^{q},K^{q^{2}}\rangle\cap\Sigma_{\infty}=\Sigma_{3}\cap\Sigma_{\infty}$ and $\langle R,R^{q},R^{q^{2}}\rangle\cap\Sigma_{\infty}=\Pi_{3}\cap\Sigma_{\infty}$ are planes of $\mathcal{S}_{h}$, so are either equal or disjoint. The plane $[\alpha]=\langle[A],[B],[C]\rangle$ lies in both $\Sigma_{3}$ and $\Pi_{3}$, so $\Sigma_{3}$ and $\Pi_{3}$ have a non-empty intersection in $\Sigma_{\infty}$. That is, the two planes $\langle K,K^{q},K^{q^{2}}\rangle$, $\langle R,R^{q},R^{q^{2}}\rangle$ are not disjoint, so they are equal and hence $K=R$. Hence $[{\cal C}]$ and ${\cal N}_{3}$ are 3-dim nrcs whose extensions have six points in common, namely $\{[A],[B],[C],R,R^{q},R^{q^{2}}\}$. Hence by [14, Theorem 21.1.1] they are equal. Case 3(b): suppose $\bar{X},\bar{Y}$ are not $(\bar{\pi},\ell_{\infty})$-carriers, then a similar argument to that in Case 2(b) gives a contradiction, and so this case does not occur.

In summary, the only possibility is that described in case 3(a). Thus if ${\cal N}_{3}$ is a $3$-special 3-dim nrc in ${\rm PG}(6,q)$, then ${\cal N}_{3}$ corresponds in ${\rm PG}(2,q^{3})$ to an $\mathbb{F}_{q}$-conic contained in an exterior $\mathbb{F}_{q}$-subplane. $\square$

Lemma 6.5

In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$. Let ${\cal N}_{2}$ be a $3$-special 2-dim nrc in ${\rm PG}(6,q)$, then ${\cal N}_{2}$ corresponds in ${\rm PG}(2,q^{3})$ to an $\mathbb{F}_{q}$-conic contained in an $\mathbb{F}_{q}$-subplane secant to $\ell_{\infty}$.

Proof   In ${\rm PG}(6,q)$, let $\mathcal{S}$ be a regular 2-spread in the hyperplane at infinity $\Sigma_{\infty}$. By Theorem 4.1, an $\mathbb{F}_{q}$-conic in a secant $\mathbb{F}_{q}$-subplane corresponds to a 2-dim nrc. Let ${\cal N}_{2}$ be a $3$-special 2-dim nrc in ${\rm PG}(6,q)$. As there is a direct isomorphism between secant $\mathbb{F}_{q}$-subplanes of ${\rm PG}(2,q^{3})$ and planes of ${\rm PG}(6,q)\backslash\Sigma_{\infty}$ that meet $\Sigma_{\infty}$ in a line not contained in a spread plane, ${\cal N}_{2}$ corresponds to an $\mathbb{F}_{q}$-conic of a secant $\mathbb{F}_{q}$-subplane. $\square$

6.3 Proof of Theorem 1.2

Let $\bar{\cal C}$ be an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{3})$. Then by Lemma 6.1, $[{\cal C}]$ is a $3$-special nrc of ${\rm PG}(6,q)$. Conversely, let ${\cal N}_{r}$ be a $3$-special nrc in ${\rm PG}(6,q)$. By Theorem 5.3 there are four possibilities for $r$, namely $2,3,4,6$. Lemmas 6.2, 6.3, 6.4, and 6.5 show that in each of these cases, ${\cal N}_{r}$ corresponds to an $\mathbb{F}_{q}$-conic in ${\rm PG}(2,q^{3})$. This completes the proof of Theorem 1.2. $\square$

7 Conclusion

It seems likely that a similar characterisation holds for $\mathbb{F}_{q}$-conics of ${\rm PG}(2,q^{n})$ in the Bruck-Bose representation in ${\rm PG}(2n,q)$, using a regular $(n-1)$-spread denoted $\mathcal{S}$. However some of the details in the techniques used in the article do not generalise easily, and it seems likely that a different approach is needed.

In particular, let $\mathcal{S}$ denote the Bruck-Bose regular $(n-1)$-spread in $\Sigma_{\infty}$, the hyperplane at infinity of ${\rm PG}(2n,q)$. We conjecture that a $k$-dim normal rational curve ${\cal N}_{k}$ in ${\rm PG}(2n,q)$ corresponds via the Bruck-Bose representation to an $\mathbb{F}_{q}$-conic of ${\rm PG}(2,q^{n})$ if and only if ${\cal N}_{k}$ is $n$-special with respect to $\mathcal{S}$. The definition of $n$-special is conjectured to be a generalisation of Definitions 5.1 and 5.2, including the following: for a point $P$ in an extension of $\Sigma_{\infty}$, $w(P)=i$ if $i$ is the smallest integer such that $P$ lies in an $(i-1)$-space that meets $i$ transversal lines of $\mathcal{S}$; a point $P$ is $\mathcal{S}$-good if $w(P)\times o(P)=n\times s(P)$; and ${\cal N}_{k}$ is $n$-special if the $k$ points of ${\cal N}_{k}$ at infinity are $\mathcal{S}$-good, and their weights sum to $2n$.