\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{projective line}

\hypertarget{projective_lines}{}\section*{{Projective lines}}\label{projective_lines}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{analytic_projective_lines}{Analytic projective lines}\dotfill \pageref*{analytic_projective_lines} \linebreak
\noindent\hyperlink{synthetic_projective_lines}{Synthetic projective lines}\dotfill \pageref*{synthetic_projective_lines} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{lines_in_projective_planes}{Lines in projective planes}\dotfill \pageref*{lines_in_projective_planes} \linebreak
\noindent\hyperlink{trivial_examples}{Trivial examples}\dotfill \pageref*{trivial_examples} \linebreak
\noindent\hyperlink{analytic_examples}{Analytic examples}\dotfill \pageref*{analytic_examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A projective line is a [[projective space]] of [[dimension]] 1.

\hypertarget{analytic_projective_lines}{}\subsection*{{Analytic projective lines}}\label{analytic_projective_lines}

If $k$ is a field, the projective line over $k$ is typically denoted $\mathbb{P}^1(k)$. Set-theoretically it is a disjoint union $k \sqcup \{\infty\}$ where each $a \in k$ has homogeneous coordinates $[a, 1]$ and $\infty$ has homogeneous coordinates $[1, 0]$.

The classical case of a projective line is over the complex numbers $\mathbb{C}$, where $\mathbb{P}^1(\mathbb{C})$ is also known as the [[Riemann sphere]]. A [[meromorphic function]] on $\mathbb{C}$ may be naturally interpreted as a [[holomorphic function]] $\mathbb{C} \to \mathbb{P}^1(\mathbb{C})$.

In particular, a [[rational function]] $p/q \in \mathbb{C}(z)$ may be interpreted as a holomorphic function $[p/q]: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$; concretely, if $p(z), q(z)$ are relatively prime and of degrees $m, n$ respectively, then we may homogenize by setting $p(x, y) \coloneqq y^m p(x/y)$ and $q(x, y) \coloneqq y^n q(x/y)$, and define $[p/q]$ by the mapping on homogeneous coordinates $[x, y] \mapsto [p(x, y), q(x, y)]$. In fact, there is a bijective correspondence between such holomorphic endomaps on $\mathbb{P}^1(\mathbb{C})$ and rational functions on $\mathbb{C}$ (well, almost: the constant holomorphic map valued at $\infty$ corresponds to the illegitimate ``rational function'' $1/0$).

\hypertarget{synthetic_projective_lines}{}\subsection*{{Synthetic projective lines}}\label{synthetic_projective_lines}

It is possible to define a [[synthetic mathematics|synthetic]]/[[axiom|axiomatic]] notion of ``projective line'', somewhat analogously to the synthetic definition of [[projective plane]]. It is less obvious how to do this, since there is no relation of ``incidence'' inside a projective line. One approach, due to \hyperlink{Buekenhout74}{(Buekenhout)}, is to axiomatize the collection of ``central collineations'' of a projective line.

In general, a \emph{central collineation} of an $n$-dimensional [[projective space]] $\pi$ (with $n\ge 2$) is an automorphism $\sigma$ of $\pi$ such that

\begin{enumerate}%
\item there exists a point $O$, called the \emph{center}, such that $\sigma$ fixes every line through $O$ (setwise, i.e.$\backslash$ it sends every point on that line to a point on the same line), and
\item there exists a hyperplane $H$ (an $(n-1)$-dimensional subspace), called the \emph{axis}, which $\sigma$ fixes pointwise (i.e.$\backslash$ it sends every point on $H$ to itself).

\end{enumerate}
If $\pi$ has dimension $\ge 3$, so that it has nontrivial sub-projective-spaces of dimension $\ge 2$, then the restriction of a central collineation of $\pi$ to any such sub-projective-space containing the center $O$ and not contained in the axis $H$ is again a central collineation. Conversely, every central collineation of any subspace of such a $\pi$ is induced from a central collineation of $\pi$ itself (see for instance \hyperlink{BeutelspacherRosenbaum}{Beutelspacher-Rosenbaum, Theorem 3.1.10}). The latter fact uses Desargues' theorem, but this is true since $\pi$ must be of dimension $\ge 3$ to have any nontrivial sub-projective-spaces.

Indeed, by a theorem of Baer (\hyperlink{BeutelspacherRosenbaum}{Beutelspacher-Rosenbaum, Theorem 3.1.8}), whenever $\pi$ (of dimension $\ge 2$) is Desarguesian, a central collineation is uniquely determined by its axis, its center, and one more pair of corresponding points. Thus, given $O$ and $H$, the central collineations with center $O$ and axis $H$ act freely and transitively on $\pi \setminus (H\cup \{O\})$.

Of course, when $\pi$ is of dimension $\ge 2$, before we can talk about central collineations, we need to already know what the ``hyperplanes'' are. However, in the hypothetical 1-dimensional case, hyperplanes are just points, so that the center and axis are both points, and we can imagine giving structure to $\pi$ by \emph{axiomatizing} its central collineations instead of \emph{defining} them. This is done by the following definition, due to Buekenhout (\hyperlink{Buekenhout74}{paper}, \hyperlink{BuekenhoutCohen}{book (chapter 6)}).

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Buekenhout)} A \textbf{projective line} is a set $\ell$ of cardinality $\ge 3$ together with the following.

\begin{enumerate}%
\item For each $p,q\in \ell$, a group $\Lambda(p,q)$ whose elements are called \emph{central collineations with center $p$ and axis $q$}. Note that $q$ need not be distinct from $p$.
\item $\Lambda(p,q)$ acts on $\ell$ fixing $p$ and $q$, and if an element of it fixes a point $r \notin \{p,q\}$, then it is the identity. In particular, the action is [[faithful action|faithful]], and therefore embeds each $\Lambda(p,q)$ in the [[permutation group]] $Aut(\ell)$.
\item If $p\neq q$, then $\Lambda(p,q)$ and $\Lambda(q,p)$ commute with each other.
\item For any $\sigma\in\Lambda(r,s)$ and any $p,q$, we have $\sigma \Lambda(p,q) \sigma^{-1} = \Lambda(\sigma(p),\sigma(q))$.
\item The composite of two collineations with center $p$ (possibly with different axes) is again a collineation with center $p$, and dually the composite of two collineations with axis $q$ (possibly with different centers) is again a collineation with axis $q$. Thus we have two groups $\Lambda(p) = \bigcup_q \Lambda(p,q)$ and $\Lambda^\vee(q) = \bigcup_p\Lambda(p,q)$.

\end{enumerate}
A projective line is \textbf{Desarguesian} if in addition $\Lambda(p,q)$ acts [[transitive action|transitively]] on $\ell \setminus \{p,q\}$. In other words, if $r,s\in \ell\setminus \{p,q\}$, there is a (necessarily unique) $\sigma\in\Lambda(p,q)$ with $\sigma(r)=s$.

\end{defn}
\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\hypertarget{lines_in_projective_planes}{}\paragraph*{{Lines in projective planes}}\label{lines_in_projective_planes}

We saw above that when $\pi$ is of dimension $\ge 3$, then every central collineation of a subspace (of dimension $\ge 2$) is induced by some central collineation of $\pi$. Even though this required Desargues' theorem to prove, which might not be true in a projective plane (dimension $2$), we can still take the point of view that every ``central collineation'' of a line in a projective plane ought to be induced by a central collineation of the plane itself.

This yields the following construction: Given a projective plane $\pi$ (not necessarily Desarguesian), and a line $\ell$ in $\pi$, for any $p,q\in \ell$ define $\Lambda(p,q)$ to be the set of permutations of $\ell$ that are the restriction to $\ell$ of some central collineation of $\pi$ with center $p$ and axis containing $q$. It is straightforward to verify that this makes $\ell$ into a ``projective line'' in the above sense.

\hypertarget{trivial_examples}{}\paragraph*{{Trivial examples}}\label{trivial_examples}

There are, however, plenty of projective lines not arising from projective planes. For instance, we might set $\Lambda(p,q) = 1$ for all $p,q$.

\hypertarget{analytic_examples}{}\paragraph*{{Analytic examples}}\label{analytic_examples}

Let $k$ be a [[division ring]] and $V$ a 2-dimensional right [[vector space]] over $k$. Then $\mathbb{P}(V)$ has the structure of a Desarguesian projective line, where

\begin{itemize}%
\item if $p\neq q$, then $\Lambda(p,q)$ is the image in $PGL(V)$ of those automorphisms in $GL(V)$ fixing $q$ pointwise and $p$ setwise.
\item if $p= q$, then $\Lambda(p,q)$ is the image in $PGL(V)$ of those automorphisms $\alpha\in GL(V)$ such that $\alpha(p)=p$ and $\alpha(x)-x\in p$ for all $x\in V$.

\end{itemize}
Conversely, every Desarguesian projective line arises from a division ring in this way. Fix three points $0,1,\infty \in \ell$ and define

\begin{itemize}%
\item $k=\ell\setminus\{\infty\}$.
\item $a+b = t_b(a)$, where $t_b$ is the unique element of $\Lambda(\infty,\infty)$ with $t_b(0)=b$.
\item $a b = \lambda_a(b)$, where $\lambda_b$ is the unique element of $\Lambda(0,\infty)$ with $t_a(1)=a$.

\end{itemize}
It follows that every Desarguesian projective line can be embedded into a Desarguesian projective plane, and indeed a projective space of any dimension. See \hyperlink{BuekenhoutCohen}{Buekenhout-Cohen, Chapter 6} for details.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Albrecht Beutelspacher and Ute Rosenbaum, \emph{Projective Geometry: from foundations to applications}. Cambridge University Press, 1998. (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/beutel.pdf}{pdf})


\item Francis Buekenhout, \emph{Foundations of one Dimensional Projective Geometry based on Perspectivities}. Abhandlungen aus dem Mathematischen Seminar der Universit\"a{}t Hamburg, 43 (1975) 21-29. doi:\href{https://doi.org/10.1007/BF02995931}{10.1007/BF02995931}


\item Francis Buekenhout and Arjeh M. Cohen, \emph{Diagram Geometry: Related to Classical Groups and Buildings}. Springer, 2013, doi:\href{https://doi.org/10.1007/978-3-642-34453-4}{10.1007/978-3-642-34453-4} (\href{http://www.win.tue.nl/~amc/buek/}{author pdf})



\end{itemize}
[[!redirects projective lines]]



\end{document}