% EJC papers *must* begin with the following two lines. \documentclass[12pt]{article} \usepackage{e-jc} % Please remove all other commands that change parameters such as % margins or pagesizes. % only use standard LaTeX packages % only include packages that you actually need % we recommend these ams packages \usepackage{amsthm,amsmath,amssymb} % My list of Packages \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb} \usepackage{amscd} \usepackage{pdfsync} \usepackage{epsfig} \usepackage{graphicx,color} % we recommend the graphicx package for importing figures \usepackage{graphicx} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{cor}[theorem]{Corollary} \newtheorem{prop}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} %%Page layout % \textwidth 16cm % \textheight 23 cm % \topmargin -1cm % \oddsidemargin 0cm % \evensidemargin 0cm % \parskip 2mm % \setlength{\parindent}{0pt} % %%Theorems etc ... % \newtheorem{theorem}{\sc Theorem}[section] % \newtheorem{lemma}[theorem]{\sc Lemma} % \newtheorem{cor}[theorem]{\sc Corollary} % \newtheorem{prop}[theorem]{\sc Proposition} % \newtheorem{conj}[theorem]{\sc Conjecture} %\newtheorem{example}[theorem]{\sc Example} %\newtheorem{remark}[theorem]{\sc Remark} %\newtheorem{definition}[theorem]{\sc Definition} %\def\endmark{\hskip 2em$\square$\par} %\def\proof{\trivlist \item[\hskip \labelsep{\bf Proof\ }]} %\def\endproof{\null\hfill\endmark\endtrivlist} \usepackage{latexsym} \usepackage{url} \def\End{\mathrm{End}} \def\PG{\mathrm{PG}} \def\Aut{\mathrm{Aut}} \def\F{\mathbb{F}} \def\Fq{\mathbb{F}_q} \def\S{\mathbb{S}} \def\N{\mathbb{N}} \def\K{\mathbb{K}} \def\cD{\mathcal{D}} \def\R{\mathcal{R}} \def\V{\bigotimes_{i\in I} V_i} \def\Hom{\mathrm{Hom}} \def\GL{\mathrm{GL}} \def\Trk{\mathrm{Trk}} \def\P{\mathcal{P}} \def\L{\mathcal{L}} \def\I{\mathcal{I}} \def\dim{\mathrm{dim}} \def\id{\mathrm{id}} \def\cF{\mathcal{F}} \def\cB{\mathcal{B}} \def\cA{\mathcal{A}} \def\cH{\mathcal{H}} \DeclareMathOperator{\subl}{subl} % Le seguenti linee sono state aggiunte da Corrado \DeclareMathOperator{\wt}{wt} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\PGGL}{P\Gamma L} \DeclareMathOperator{\AGL}{AGL} \DeclareMathOperator{\AG}{AG} \DeclareMathOperator{\AGGL}{A\Gamma L} %============================================================================== % NUOVE AGGIUNTE 11-26/2/2014 %============================================================================== %\usepackage{showlabels} \def\cC{\mathcal{C}} \def\cI{\mathcal{I}} \def\cK{\mathcal{K}} \def\cL{\mathcal{L}} \def\cP{\mathcal{P}} \def\cR{\mathcal{R}} \def\cS{\mathcal{S}} \def\cV{\mathcal{V}} \DeclareMathOperator{\Prod}{Prod} \DeclareMathOperator{\QPG}{QPG} \usepackage{tikz} \usetikzlibrary{matrix,decorations.pathreplacing} \newcommand{\bigslant}[2]{{\raisebox{.2em}{$#1$}\left/\raisebox{-.2em}{$#2$}\right.}} \newcommand{\segre}[3]{\cS_{#1,#2,#3}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{\bf Subspaces intersecting each element of a regulus in one point, Andr\'e-Bruck-Bose representation and clubs} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Michel Lavrauw\thanks{The research of this author is supported by the Research Foundation Flanders-Belgium (FWO-Vlaanderen) and by a Progetto di Ateneo from Universit\`a di Padova (CPDA113797/11).} \\ \small Dipartimento di Tecnica e Gestione dei Sistemi Industriali,\\[-0.8ex] \small Universit\`a di Padova,\\[-0.8ex] \small Stradella S. Nicola 3, \\ \small 36100 Vicenza, Italy,\\ \small \tt e-mail: michel.lavrauw@unipd.it.\\ \and Corrado Zanella\thanks{The research of this author is supported by the Italian Ministry of Education, University and Research (PRIN 2012 project ``Strutture geometriche, combinatoria e loro applicazioni'').}\\ \small Dipartimento di Tecnica e Gestione dei Sistemi Industriali,\\[-0.8ex] \small Universit\`a di Padova,\\[-0.8ex] \small Stradella S. Nicola 3, \\ \small 36100 Vicenza, Italy,\\ \small \tt e-mail: corrado.zanella@unipd.it.\\ } % \date{\dateline{submission date}{acceptance date}\\ % \small Mathematics Subject Classifications: comma separated list of % MSC codes available from http://www.ams.org/mathscinet/freeTools.html} \date{\dateline{Sep ??, 2014}{Jan ??, 2016}\\ \small Mathematics Subject Classifications: 51E20} \begin{document} \maketitle % E-JC papers must include an abstract. The abstract should consist of a % succinct statement of background followed by a listing of the % principal new results that are to be found in the paper. The abstract % should be informative, clear, and as complete as possible. Phrases % like "we investigate..." or "we study..." should be kept to a minimum % in favor of "we prove that..." or "we show that...". Do not % include equation numbers, unexpanded citations (such as "[23]"), or % any other references to things in the paper that are not defined in % the abstract. The abstract will be distributed without the rest of the % paper so it must be entirely self-contained. \begin{abstract} In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\cR$ of $(n-1)$-subspaces in $\PG(2n-1,q)$, with respect to the subgroup of $\PGL(2n,q)$ fixing $\cR$. Such results have consequences on several aspects of finite geometry. First of all, a necessary condition for an $(n-1)$-subspace $U$ and a regulus $\cR$ of $(n-1)$-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results in \cite{BaJa12} on the Andr\'e-Bruck-Bose representation of a $q$-subline in $\PG(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs. % keywords are optional \bigskip\noindent \textbf{Keywords:} club; linear set; subplane; Andr\'e-Bruck-Bose representation; Segre variety \end{abstract} \section{Introduction}\label{sec:intro} The $(n-1)$-dimensional projective projective space over the field $F$ is denoted by $\PG(n-1,F)$ or $\PG(n-1,q)$ if $F$ is the finite field of order $q$ (denoted by $\F_q$). The set of nonzero elements of a field $F$ will be denoted by $F^*$, and similarly, the set of nonzero vectors of a vector space $V$ by $V^*$. If $L$ is an extension field $\F_q$, then the projective space defined by the $\F_q$-vector space induced by $L^d$ is also denoted by $\PG_q(L^d)$. For a (sets of) subspace(s) $R$ of a vector space or a projective space, the notation $\langle R\rangle$ is used to denote the subspace generated by (the elements of) $R$. In case there is any ambiguity about the coefficient field, then the notation $\langle R\rangle_q$ will be used, to denote that the considered subspace is the one generated over $\F_q$. In this case the terminology of {\it $\F_q$-span} will sometimes be used. For example, if $S$ is a set of two points on the projective line $\PG(1,q^2)$, then $\langle S\rangle_q$ denotes the $\F_q$-subline defined by $S$, while $\langle S\rangle_{q^2}$ coincides with the whole projective line $\PG(1,q^2)$. For further notation and general definitions employed in this paper the reader is referred to \cite{LaVaPrep,LaZa14b,Po10}. %Referee: Notation should be clearly explained, in particular: %1. for a regulus R; %2. F^*, define * For more information on Desarguesian spreads see \cite{BaLu11}. This paper is structured as follows. In Section \ref{sec:2} subspaces which intersect each element of a regulus in one point are studied and a result from \cite{Burau} is generalised. Section \ref{sec:3} contains one of the main results of this paper, determining the order of the normal rational curves obtained from $n$-dimensional subspaces on an external $(n-1)$-dimensional subspace with respect to a regulus in $\PG(2n-1,q)$, obtained from a point and a subline after applying the field reduction map to $\PG(1,q^n)$. This leads to a necessary condition on the existence of a Desarguesian spread containing a subspace and regulus (Corollary \ref{cor1}). The Andr\'e-Bruck-Bose representation of sublines and subplanes of a finite projective plane is studied\footnote{A different study of $\F_{q^k}$-sublines and $\F_{q^k}$-subplanes of $\PG(2, q^n)$ in this representation can be found in \cite{RoShVa2015}.} in Section \ref{sec:4} and improvements are obtained with respect to the known results \cite{BFG,QuCa,Vi,BaJa12}. The results from the first sections of this paper are then applied to the classification problem for clubs of rank three in $\PG(1,q^n)$ in Section \ref{sec:clubs}. A study of the incidence structure of the clubs in $\PG(1,q^n)$ after field reduction yields to a partial classification, concluding that the orbits of clubs under $\PGL(2,q^n)$ are at least $k-1$, where $k$ stands for the number of divisors of $n$. The paper concludes with an appendix discussing a result motivated by Burau \cite{Burau} for the complex numbers: the result is extended to general algebraically closed fields; a new proof is provided; and counterexamples are given to some of the arguments used in the original proof. \section{Subspaces intersecting each element of a regulus in one point}\label{sec:2} Let $\cR$ be a regulus of subspaces in a projective space and let $S$ be any subspace of $\langle \cR\rangle$. Questions about the properties of the set of intersection points, which for reasons of simplicity of notation we will denote by $S\cap \cR$, often turn up while investigating objects in finite geometry. If $S$ intersects each element of the regulus $\cR$ in a point, then the intersection $S\cap \cR$ is a normal rational curve, see Lemma \ref{lemma:0}. This was already pointed out in \cite[p.173]{Burau} with a proof originally intended for complex projective spaces, but actually holding in a more general setting. The notation of \cite{Burau} will be partly adopted. The Segre variety representing the Cartesian product $\PG(n,F)\times\PG(m,F)$ in $\PG((n+1)(m+1)-1,F)$ is denoted by $\segre nmF$. It is well known that $\segre nmF$ contains two families $\segre nmF^I$ and $\segre nmF^{II}$ of maximal subspaces of dimensions $n$ and $m$, respectively. When convenient, the notation $S^I$ or $S^{II}$ will be used for a subspace belonging to the first or second family. The points of $\segre nmF$ may be represented as one-dimensional subspaces spanned by rank one $(m+1)\times(n+1)$ matrices. This is the standard example of a regular embedding of product spaces, see \cite{Za96}. Note that in the finite case it is possible to embed product spaces in projective spaces of smaller dimension (see e.g. \cite{LaShZa2013}). A regulus $\cR$ of $(n-1)$-dimensional subspaces can also be defined as $\segre {n-1}1F^I$. \begin{lemma}\label{lemma:0} Let $n>1$ be an integer, and $F$ a field. Let $S_t$ be a $t$-subspace of $\PG(2n-1,F)$ intersecting each $S^I\in\segre{n-1}1F^I$ in precisely one point. Define $\Phi=S_t\cap\segre{n-1}1F$, and assume $\langle\Phi\rangle=S_t$. Then $|F|\geq t$ and the following properties hold. \begin{itemize} \item[(i)] The set $\Phi$ is a normal rational curve of order $t$. \item[(ii)] Let $\Xi^I\in\segre{n-1}1F^I$. Then the set $S(\Phi,\Xi^I)$ of the intersections of $\Xi^I$ with all transversal lines $l^{II}$ such that $l^{II}\cap\Phi\neq\emptyset$ is a normal rational curve of order $t$ or $t-1$ if $|F|=t$, and of order $t-1$ if $|F|>t$. \item[(iii)] If $\Phi$ is contained in a subvariety $\cS_{t-1,1,F}$ of $\cS_{n-1,1,F}$, then homogeneous coordinates can be chosen such that $\Phi$ is represented parametrically by \begin{equation}\label{proj-1} \left\langle\begin{pmatrix} y_0^t& y_0^{t-1}y_1 &\ldots &y_0y_1^{t-1}\\ y_0^{t-1}y_1&y_0^{t-2}y_1^2&\ldots &y_1^t\end{pmatrix}\right\rangle,\quad (y_0,y_1)\in(F^2)^*, \end{equation} and $S(\Phi,\Xi^I)$, for $z_0$, $z_1$ depending only on $\Xi^I$, by \begin{equation}\label{proj-2} \left\langle\begin{pmatrix} y_0^{t-1}z_0& y_0^{t-2}y_1z_0 &\ldots &y_1^{t-1}z_0\\ y_0^{t-1}z_1&y_0^{t-2}y_1z_1&\ldots &y_1^{t-1}z_1\end{pmatrix}\right\rangle,\quad (y_0,y_1)\in(F^2)^*. \end{equation} \end{itemize} \end{lemma} \begin{proof} \textit{(i), (iii)} The proof in \cite[Sect.41 no.3]{Burau}, which is offered for $F=\mathbb C$, works exactly the same provided that $|F|>t$ or, more generally, that $\Phi$ is contained in some subvariety $\cS_{t-1,1,F}$ of $\cS_{n-1,1,F}$. In case $|F|\le t$, the size of $\Phi$ being $|F|+1$ implies $|F|=t$, so $\Phi$ is just a set of $t+1$ independent points in a subspace isomorphic to $\PG(t,t)$, hence $\Phi$ is a normal rational curve of order $t$. \textit{(ii)} The case $|F|>t$ is proved in \cite{Burau} immediately after the corollary at p.\ 175. If $|F|\leq t$, then $|F|=t$ and two cases are possible. If $\Phi$ is contained in some $\cS_{t-1,1,F}\subseteq\cS_{n-1,1,F}$, Burau's proof is still valid as was mentioned in case (ii); so, $S(\Phi,\Xi^I)$ is a normal rational curve of order $t-1=|F|-1$. Otherwise $S(\Phi,\Xi^I)$ is an independent $(t+1)$-set, hence a normal rational curve of order $|F|$. \end{proof} \begin{remark} \emph{ If $|F|=t$ both cases in Lemma \ref{lemma:0} \textit{(ii)} can occur. The following two examples use the Segre embedding $\sigma=\sigma_{t-1,1,F}$ of the product space $\PG(t-1,t) \times \PG(1,t)$ in $\PG(2t-1,t)$. Let $\{s_0,s_1,\ldots,s_t\}$ be the set of points on $\PG(1,t)$ and suppose $\{r_0,r_1,\ldots,r_t\}$ is a set of $t+1$ points in $\PG(t-1,t)$. Put $\Xi^I=\sigma(\PG(1,t)\times s_0)$ and $\Phi:=\{\sigma(r_i\times s_i)~:~ i=0,1,\ldots, t\}$. Then $\Phi$ consists of $t+1$ points on the Segre variety $\cS_{t-1,1,F}$. Depending on the set $\{r_0,r_1,\ldots,r_t\}$ one obtains the two cases described in Lemma \ref{lemma:0} \textit{(ii)}. \begin{itemize} \item[a.] If $\{r_0,r_1,\ldots,r_t\}$ is a frame of a hyperplane of $\PG(t-1,t)$ then $\Phi$ generates a $t$-dimensional subspace of $\PG(2t-1,t)$ intersecting $\cS_{t-1,1,F}$ in $\Phi$ and $S(\Phi,\Xi^I)$ is a normal rational curve of order $t-1$. \item[b.] If $\{r_0,r_1,\ldots,r_t\}$ generates $\PG(t-1,t)$ then $\Phi$ generates a $t$-dimensional subspace of $\PG(2t-1,t)$ intersecting $\cS_{t-1,1,F}$ in $\Phi$ and $S(\Phi,\Xi^I)$ is a normal rational curve of order $t$. \end{itemize} } \end{remark} \begin{remark}\label{proj-is-proj}\emph{ By (\ref{proj-1}) and (\ref{proj-2}), the map $\alpha:\,\Phi\rightarrow S(\Phi,\Xi^I)$ defined by the condition that $X$ and $X^\alpha$ are on a common line in $\cS_{n-1,1,F}^{II}$ is related to a projectivity between the parametrizing projective lines. Such an $\alpha$ is also called a {\it projectivity}. } \end{remark} \section{The order of normal rational curves contained in $\cS_{n-1,1,q}$}\label{sec:3} Here $n\ge2$ is an integer. The field reduction map $\cF_{m,n,q}$ from $\PG(m-1,q^n)$ to $\PG(mn-1,q)$ will also be denoted by $\cF$. If $S$ is a set of points, in $\PG(m-1,q^n)$, then $\cF(S)$ is a set of subspaces, whose union, as a set of points will be denoted by $\tilde\cF(S)$. %The \textit{order} of a normal rational curve is the dimension of its span. The $\F_{q^h}$-span of a subset $b$ of $\PG(d,q^n)$ is denoted by $\langle b\rangle_{q^h}$. \begin{prop}\label{prop:grado_pto} Let $b$ be a $q$-subline of $\PG(1,q^n)$, and let $\Theta$ be a point of $\PG(1,q^n)$. Let $(1,\zeta)$ and $(1,\zeta')$ be homogeneous coordinates of $\Theta$ with respect to two reference frames for $\langle b\rangle_{q^n}$, each of which consists of three points of $b$. Then $\Fq(\zeta)=\Fq(\zeta')$. \end{prop} \begin{proof} Homogeneous coordinates of a point in both reference frames, say $(x_0,x_1)$ and $(x_0',x_1')$, are related by an equation of the form $\rho(x_0'\ x_1')^T=A(x_0\ x_1)^T$, $\rho\in\F^*_{q^n}$, $A\in\GL(2,q)$. Hence $(\rho\ \rho\zeta')^T=A(1\ \zeta)^T$ and this implies $\zeta'\in\Fq(\zeta)$. The proof of $\zeta\in\Fq(\zeta')$ is similar. \end{proof} By Proposition\ \ref{prop:grado_pto}, given a $q$-subline $b$ in a finite projective space $\PG(d,q^n)$ and a point $\Theta\in\langle b\rangle_{q^n}$, with homogeneous coordinates $(1,\zeta)$ with respect to a reference frame of $\langle b\rangle_{q^n}$ consisting of three points of $b$, the {\it degree} of $\Theta$ over $b$, denoted by $[\Theta : b]$, is well-defined in terms of the field extension degree as follows: $[\Theta : b]= [\F_q (\zeta) : \F_q ]$. %For $\Theta\in b$ set $[\Theta:b]=1$. %We define the \textit{degree of a point $\Theta$ with coordinates $(1,\zeta)$ over a $q$-subline} $b$ %in a finite projective space $\PG(d,q^n)$ as %\begin{displaymath} %[\Theta:b]:= %\left \{ %\begin{array}{ll} %[\Fq(\zeta):\Fq] & ~\mbox{for a point $\Theta\in\langle b\rangle_{q^n}\setminus b$,}\\ %1 & ~\mbox{for $\Theta\in b$.} %\end{array} %\right. %\end{displaymath} %By Proposition\ \ref{prop:grado_pto}, this degree is well-defined. This $[\Theta:b]$ also equals the minimum integer $m$ such that a subgeometry $\Sigma\cong\PG(d,q^m)$ exists containing both $b$ and $\Theta$. \begin{prop}\label{zero-} Any $n$-subspace of $\PG(2n-1,q)$ containing an $(n-1)$-subspace $S^I\in\cS_{n-1,1,q}^I$ intersects $\cS_{n-1,1,q}$ in the union of $S^I$ and a line in $\cS_{n-1,1,q}^{II}$. \end{prop} \begin{theorem}\label{zero} Let $b$ be a $q$-subline of $\PG(1,q^n)$, and $\Theta\not\in b$ a point of $\PG(1,q^n)$. Then in $\PG(2n-1,q)$ any $n$-subspace $\cH$ containing $\cF(\Theta)$ intersects the Segre variety $\cS_{n-1,1,q}=\tilde{\cF}(b)$, in a normal rational curve whose order is $\min\{q,[\Theta:b]\}$. \end{theorem} \begin{proof} Set $L=\F_{q^n}$, $F=\F_q$. Without loss of generality, $\PG(2n-1,q)=\PG_q(L^2)$, $\cF(b)=\{L(x,y)\mid(x,y)\in(F^2)^*\}$\footnote{For $x,y\in L$, $F(x,y)=\langle(x,y)\rangle_q$, and $L(x,y)=\langle(x,y)\rangle_{q^n}$.}, and $\Theta=L(1,\xi)$ with $[F(\xi):F]=[\Theta:b]$. % and the map $g$ is associated with $(\alpha,a,b)\mapsto(a\alpha,b\alpha)$ for $\alpha\in L$ and % $(a,b)\in F^2$. The $n$-subspace $\cH$ intersects $L(1,0)$ in one point $Y$ of the form $Y=F(\theta,0)$, $\theta\in L^*$. For any $x\in F$, seeking for the intersection $\langle\cF(\Theta),Y\rangle_q\cap L(x,1)$, or \[ \langle L(1,\xi),F(\theta,0)\rangle_q\cap L(x,1) \] gives two equations in $\alpha,\beta\in L$: \[ \alpha+\theta=\beta x,\quad \alpha\xi=\beta, \] whence $\beta=\theta(x-\xi^{-1})^{-1}$. The intersection point is then $F\left(x\theta(x-\xi^{-1})^{-1},\theta(x-\xi^{-1})^{-1}\right)$. So, for $\Xi=L(0,1)$, the set of the intersections of $\Xi$ with all lines in $\cS_{n-1,1,q}^{II}$ which meet $\cH$ is \[ S(\cH\cap\cS_{n-1,1,q},\Xi)= \{F(0,\theta(x-\xi^{-1})^{-1})\mid x\in \cF_q\}\cup\{F(0,\theta)\}. \] This $S(\cH\cap\cS_{n-1,1,q},\Xi)$ is obtained by inversion from the line joining the points $F(0,\theta^{-1})$ and $F(0,\theta^{-1}\xi^{-1})$. By \cite[Theorem 5]{LaZa14}, $\cC_Y$ is a normal rational curve of order $\delta'=\min\{q,[F(\xi^{-1}):F]-1\}=\min\{q,[\Theta:b]-1\}$. Now apply lemma \ref{lemma:0} for $S_t=\langle\cH\cap\cS_{n-1,1,q}\rangle_q$:%, and $\cC=S(\cH\cap\cS_{n-1,1,q},\Xi)$: if $t\ge q$, then $t=q$ and $\delta'=q$ or $\delta'=q-1$, so $[\Theta:b]\ge q$ and $t=\min\{q,[\Theta:b]\}$. If on the contrary $t2$ by using the package FinInG \cite{FinInG} of GAP \cite{GAP} examples can be given of $(n-1)$-subspaces disjoint from $\cS_{n-1,1,q}$ contained in $n$-subspaces intersecting the Segre variety in normal rational curves of distinct orders. We include one explicit example. Let $q=4$, $\F_q=\F_2(\omega)$, with $\omega^2+\omega+1=0$. Let $\R$ be the regulus of $3$-dimensional subspaces of $\PG(7,4)$ obtained from the standard subline $\PG(1,q)$ in $\PG(1,q^4)$, and put $$S_3:=\langle ( 1, 0, 0, 0, \omega^2, 1, 0, 1 ), ( 0, 1, 0, 0, 1, \omega^2, 0, \omega^2 ), ( 0, 0, 1, 0, 0, \omega, 1, \omega ), ( 0, 0, 0, 1, \omega^2, \omega^2, \omega, 1 ) \rangle. $$ Then $S_3$ is a three-dimensional subspace disjoint from the regulus $\R$. Moreover, the 4-dimensional subspace $\langle S_3,(1,0,0,0,0,0,0,0)\rangle$ intersects the regulus $\R$ in a normal rational curve of order 4, while the 4-dimensional subspace $\langle S_3,(0,1,0,\omega^2,0,0,0,0)\rangle$ intersects $\R$ in a conic. \end{remark} \section{Andr\'e-Bruck-Bose representation}\label{sec:4} The Andr\'e-Bruck-Bose representation of a Desarguesian affine plane of order $q^n$ is related to the image of $\PG(2,q^n)$, under the field reduction map $\cF$, by means of the following straightforward result. \begin{prop}\label{bb} Let $\cD$ be the Desarguesian spread in $\PG(3n-1,q)$ obtained after applying the field reduction map $\cF$ to the set of points of $\PG(2,q^n)$, $ l_\infty$ a line in $\PG(2,q^n)$, and $\cK$ a $(2n)$-subspace of $\PG(3n-1,q)$, containing the spread $\cF( l_\infty)$. Take $\PG(2,q^n)\setminus l_\infty$ and $\cK\setminus\langle\cF( l_\infty)\rangle_q$ as representatives of $\AG(2,q^n)$ and $\AG(2n,q)$, respectively. Then the map $\varphi:\,\AG(2,q^n)\rightarrow\AG(2n,q)$ defined by $\varphi(X)=\cF(X)\cap\cK$ for any $X\in\AG(2,q^n)$ is a bijection, mapping lines of $\AG(2,q^n)$ into $n$-subspaces of $\AG(2n,q)$ whose $(n-1)$-subspaces at infinity belong to the spread $\cF( l_\infty)$. \end{prop} The notation in Proposition \ref{bb} is assumed to hold in the whole section. The following result improves \cite[Theorems 3.3 and 3.5]{BaJa12}, by determining the order of the involved normal rational curves. \begin{theorem}\label{thm:q-subline} Let $b$ be a $q$-subline of $\PG(2,q^n)$, not contained in $ l_\infty$. Set $\Theta=\langle b\rangle_{q^n}\cap l_\infty$. Then the Andr\'e-Bruck-Bose representation $\varphi(b\setminus l_\infty)$ is the affine part of a normal rational curve whose order is $\delta=\min\{q,[\Theta:b]\}$. More precisely, if $\delta=1$, then $\varphi(b\setminus l_\infty)$ is an affine line; if $\delta>1$, then $b\cap l_\infty=\emptyset$, and $\varphi(b)$ is a normal rational curve with no points at infinity. \end{theorem} \begin{proof} The intersection $\cH=\langle\cF(b)\rangle_q\cap\cK$ is an $n$-space containing $\cF(\Theta)$, and contained in the span of the Segre variety $\segre{n-1}1q=\tilde\cF(b)$, as defined at the start of Section \ref{sec:3}. The result follows from Proposition \ref{zero-} and Theorem \ref{zero}. \end{proof} The results in \cite[Theorems 3.3 and 3.5]{BaJa12} also characterize the normal rational curves arising from $q$-sublines in $\AG(2,q^n)$. In \cite{BFG,QuCa,Vi} for $n=2$ and \cite[Theorem 3.6 (a)(b)]{BaJa12} for any $n$ the Andr\'e-Bruck-Bose representation of a $q$-subplane tangent to a line at the infinity is described. Further properties are stated in the following theorem: \begin{theorem} Let $B$ be a $q$-subplane of $\PG(2,q^n)$ that is tangent to $l_\infty$ at the point $T$. Let $b$ be a line of $B$ not through $T$, $\Theta=\langle b\rangle_{q^n}\cap l_\infty$, and $\delta=\min\{q,[\Theta:b]\}$. Then there are a normal rational curve $\cC_0$ of order $\delta$ in the $n$-subspace $\varphi(\langle b\rangle_{q^n})$, a normal rational curve $\cC_1\subset\cF(T)$ of order $\delta'$, with \begin{equation}\label{eq:delta1} \delta'\ \left\{\begin{array}{ll}=[\Theta:b]-1&\mbox{ for }q>[\Theta:b]\\ \in\{q-1,q\}&\mbox{otherwise,}\end{array}\right. \end{equation} and a projectivity $\kappa:\,\cC_0\rightarrow\cC_1$ (in the sense of Remark \ref{proj-is-proj}), such that $\varphi(B\setminus l_\infty)$ is the ruled surface union of all lines $XX^\kappa$ for $X\in\cC_0$. \end{theorem} \begin{proof} By Theorem \ref{thm:q-subline}, $\cC_0:=\varphi(b)$ is a normal rational curve of order $\delta$ in the $n$-subspace $\varphi(\langle b\rangle_{q^n}\setminus l_\infty)$, and for any $P=\varphi(X)\in\cC_0$, the subline $TX$ of $B$ corresponds to an affine line $PP^\kappa$ with $P^\kappa\in\cF(T)$ at infinity. Define $\cC_1=\{P^\kappa\mid P\in\cC_0\}$. By the field reduction map $\cF=\cF_{3,n,q}$, the subplane $B$ is mapped to $\cF(B)$ which is the set of all maximal subspaces of the first family in $\cS_{n-1,2,q}\subset\PG(3n-1,q)$. Considering $\F_{q^n}$ as an $\F_q$-vector space, the homomorphism \[ \F_{q^n}\times \F_q^3\rightarrow \F_{q^n}\otimes \F_q^3 ~:~ (\lambda,v)\mapsto \lambda\otimes v \] corresponds to a projective embedding $g:\,\PG(n-1,q)\times B \rightarrow\cS_{n-1,2,q}$ whose image is $\cS_{n-1,2,q}$, and such that $\cF(X)=(\PG(n-1,q)\times X)^g$ for any point $X$ in $B$. It holds $\varphi(B\setminus l_\infty)=\cS_{n-1,2,q}\cap\cK\setminus\cF(T)$. For any point $U$ in $B$ define \[ \kappa_U:\,(X,Y)^g\in\cS_{n-1,2,q}\mapsto(X,U)^g\in\cF(U). \] Note that for any $Y\in B$, the restriction of $\kappa_U$ to $\cF(Y)$ is a projectivity. For any $U\in b $, using the notation from Lemma \ref{lemma:0} it holds $\cC_0^{\kappa_U}=S(\cC_0,\cF(U))$, and as a consequence, $\cC_0^{\kappa_U}$ is a normal rational curve of order $\delta'$ as in (\ref{eq:delta1}). Now, since for any $P\in\cC_0$, say $P=(X_P,Y_P)^g$, the points $P$, $P^\kappa$ and $P^{\kappa_T}$ are on the plane $(X_P\times B)^g\in\segre{n-1}2q^{II}$, and $P^\kappa, P^{\kappa_T}\in \cF(T)$, it follows that $P^\kappa=P^{\kappa_T}$. It also follows that $\cC_1=\cC_0^{{\kappa_{U}}{\kappa_{T}}}=S(\cC_0,\cF(U))^{\kappa_{T}}$, and hence $\cC_1$ is a normal rational curve of order $\delta'$ as in (\ref{eq:delta1}). Finally, $\kappa_{U}:\,\cC_0\rightarrow S(\cC_0,\cF(U))$ is a projectivity as defined in Remark \ref{proj-is-proj}, and hence so is $\kappa$. \end{proof} \section{On the classification of clubs}\label{sec:clubs} An \textit{$\Fq$-club} (or simply a club) in $\PG(1,q^n)$ is an $\Fq$-linear set of rank three, having a point of weight two, called the \textit{head} of the club. An $\Fq$-club has $q^2+1$ points, and the non-head points have weight one. From now on it will be assumed that $n>2$. The next proposition is a straightforward consequence of the representation of linear sets as projections of subgeometries \cite[Theorem 2]{LuPo04}. \begin{prop}\label{prop:LPclub} Let $L$ be an $\Fq$-club in $\PG(1,q^n)\subset\PG(2,q^n)$. Then there are a $q$-subplane $\Sigma$ of $\PG(2,q^n)$, a $q$-subline $b$ in $\Sigma$, and a point $\Theta\in\langle b\rangle_{q^n}\setminus b$, such that $L$ is the projection of $\Sigma$ from the center $\Theta$ onto the axis $\PG(1,q^n)$. \end{prop} As before the notation $\cF$ and $\tilde{\cF}$ is used, where $\cF=\cF_{2,n,q}$ denotes the field reduction map from $\PG(1,q^n)$ to $\PG(2n-1,q)$. \begin{prop}\label{prop:maximal_subspaces} Let $L$ be an $\Fq$-club of $\PG(1,q^n)$ with head $\Upsilon$. Then $\tilde\cF(L)$ contains two collections of subspaces, say $F_1$ and $F_2$, satisfying the following properties. \begin{itemize} \item[(i)] The subspaces in $F_1$ are $(n-1)$-dimensional, are pairwise disjoint, and any subspace in $F_1$ is disjoint from $\cF(\Upsilon)$. \item[(ii)] Any subspace in $F_2$ is a plane and intersects $\cF(\Upsilon)$ in precisely a line. \item[(iii)] Any point of $\cF(\Upsilon)$ belongs to exactly $q+1$ planes in $F_2$. \item[(iv)] If $L$ is not isomorphic to $\PG(1,q^2)$, and $l$ is any line of $\PG(2n-1,q)$ contained in $\tilde\cF(L)$, then $l$ is contained in $\cF(\Upsilon)$ or in a subspace in $F_1\cup F_2$. \end{itemize} \end{prop} \begin{proof} The assumptions imply the existence of $\Sigma$ and a $q$-subline $b$ in $\Sigma$ as in Proposition \ref{prop:LPclub}. The assertions are a consequence of the fact that $\tilde\cF(\Sigma)$ is a Segre variety $\cS_{n-1,2,q}$ in $\PG(3n-1,q)$. Let \[ p_1:\,\PG(2,q^n)\setminus\Theta\rightarrow\PG(1,q^n) \] be the projection with center $\Theta$, associated with \[ p_2:\PG(3n-1,q)\setminus\cF(\Theta)\rightarrow\PG(2n-1,q). \] The collections $F_1$ and $F_2$ are defined as follows: \[ F_1=\{\cF(p_1(X))\mid X\in\Sigma\setminus b\}=\cF(L)\setminus\cF(\Upsilon),\quad F_2=\{p_2(V^{II})\mid V^{II}\in\tilde\cF(\Sigma)^{II}\}. \] The assertion \textit{(i)} is straightforward, as well as $\dim(V)=2$ for any $V\in F_2$. For any $V^{II}\in\tilde\cF(\Sigma)^{II}$, the intersection $V^{II}\cap\langle\tilde\cF(b)\rangle_q$ is a line, and this with $p_2^{-1}(\cF(\Upsilon))=\langle\tilde\cF(b)\rangle_q\setminus\cF(\Theta)$ implies the second assertion in \textit{(ii)}. Next, let $P$ be a point in $\cF(\Upsilon)$. A plane $V= p_2(V^{II})$ contains $P$ if, and only if, $V^{II}$ intersects the $n$-subspace $\langle\cF(\Theta),P\rangle_q$, that is, $V^{II}$ intersects the normal rational curve $\cS_{n-1,2,q}\cap\langle\cF(\Theta),P\rangle_q$; this implies \textit{(iii)}. Assume that a line $l\subset\tilde\cF(L)$ exists which is neither contained in $\cF(\Upsilon)$, nor in a $T\in F_1\cup F_2$. Let $Q$ be a point in $l\setminus\cF(\Upsilon)$, and let $V\in F_2$ such that $Q\in V$. It holds $L=\cB(V)$. Then $\cB(l)$ is a $q$-subline of $L$. Suppose that a line $l'$ in $V$ exists such that $\cB(l')=\cB(l)$. Since $\cB(Q)\neq\cB(Q')$ for any $Q'\in V$, $Q'\neq Q$, the line $l'$ contains $Q$. Then $l$, $l'$ are two distinct transversal lines in $\cB(l)^{II}$, a contradiction. Hence $\cB(l')\neq \cB(l)$ for any line $l'$ in $V$, that is, $\cB(l)$ is a so-called \textit{irregular subline} \cite{LaVV10}. By \cite[Corollary 13]{LaVV10}, no irregular subline exists in $L$, and this contradiction implies \textit{(iv}). \end{proof} \begin{prop}\label{prop:struttura} Let $L$ be an $\Fq$-club with head $\Upsilon$. Let $\Theta$ be the point and $b$ be the subline as defined in Proposition \ref{prop:LPclub}. Then for any point $X$ in $\cF(\Upsilon)$, the intersection lines of $\cF(\Upsilon)$ with any $q$ distinct planes in $F_2$ containing $X$ span an $s$-dimensional subspace, where \begin{itemize} \item[(i)] $s=[\Theta:b]-1$ if $q>[\Theta:b]$; \item[(ii)] $s\in\{q-1,q\}$ if $q\le[\Theta:b]$. \end{itemize} \end{prop} \begin{proof} Let $p_2$ be the projection map as defined in the proof of Proposition \ref{prop:maximal_subspaces}, $X=p_2(P)$, and $\cH=\langle\cF(\Theta),P\rangle_q$. For any plane $V=p_2(V^{II})$, it holds $X\in V$ if, and only if $V^{II}\cap\cH\neq\emptyset$. The intersection $\cH\cap\tilde\cF(b)$ is a normal rational curve of order $\min\{q,[\Theta:b]\}$ (cf.\ Theorem \ref{zero}). Let $V_0=p_2(V_0^{II})$ be the unique plane of $F_2$ through $X$ distinct from the $q$ planes chosen in the assumptions (cf.\ Proposition \ref{prop:maximal_subspaces}). Let $Q=\tilde\cF(b)\cap V_0^{II}$; $\cB(Q)$ is an $(n-1)$-subspace of $\tilde\cF(b)^I$. Such $\cB(Q)$ is mapped onto $\cB(X)=\cF(\Upsilon)$ by $p_2$. Assume $V_i=p_2(V_i^{II})$, $i=1,2,\ldots,q$, are the $q$ planes chosen in the assumptions. Any $V_i^{II}$, $i=1,2,\ldots,q$, intersects $\cH$, hence $V_i^{II}\cap\cB(Q)$ is the intersection of $\cB(Q)$ with a transversal line of $\tilde\cF(b)$ intersecting the normal rational curve $\cH\cap\tilde\cF(b)$. By Lemma \ref{lemma:0} \textit{(ii)}, the set \[ S=\{V_i^{II}\cap\cB(Q)\mid i=1,2,\ldots,q\}\cup\{Q\} \] is a normal rational curve of order $s$ where $s$ takes the values as stated in $(i)$ and $(ii)$. Since $V_i\cap\cF(\Upsilon)$ is the line through $X$ and a point of $p_2(S)$, distinct from $X$, the span of the intersection lines is the same as the span of $p_2(S)$. \end{proof} \begin{theorem}\label{thm:num_orb} Let $\cI_{n,q}$ be the set of integers $h$ dividing $n$ and such that $1