Projective lines

# Projective lines

## Idea

A projective line is a projective space of dimension 1.

## 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” $1/0$).

## Synthetic projective lines

It is possible to define a synthetic/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 (Buekenhout), is to axiomatize the collection of “central collineations” of a projective line.

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

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

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 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 (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 axiomatizing its central collineations instead of defining them. This is done by the following definition, due to Buekenhout (paper, book (chapter 6)).

###### Definition

(Buekenhout) A projective line is a set $\ell$ of cardinality $\ge 3$ together with the following.

1. For each $p,q\in \ell$, a group $\Lambda(p,q)$ whose elements are called central collineations with center $p$ and axis $q$. Note that $q$ need not be distinct from $p$.
2. $\Lambda(p,q)$ 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, and therefore embeds each $\Lambda(p,q)$ in the permutation group $Aut(\ell)$.
3. If $p\neq q$, then $\Lambda(p,q)$ and $\Lambda(q,p)$ commute with each other.
4. For any $\sigma\in\Lambda(r,s)$ and any $p,q$, we have $\sigma \Lambda(p,q) \sigma^{-1} = \Lambda(\sigma(p),\sigma(q))$.
5. 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 $\Lambda^\vee(q) = \bigcup_p\Lambda(p,q)$.

A projective line is Desarguesian if in addition $\Lambda(p,q)$ acts 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 $\sigma(r)=s$.

### Examples

#### 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.

#### 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 $p,q$.

#### 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

• 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.
• 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 $x\in V$.

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

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

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

• Albrecht Beutelspacher and Ute Rosenbaum, Projective Geometry: from foundations to applications. Cambridge University Press, 1998. (pdf)

• Francis Buekenhout, Foundations of one Dimensional Projective Geometry based on Perspectivities. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 43 (1975) 21-29. doi:10.1007/BF02995931

• Francis Buekenhout and Arjeh M. Cohen, Diagram Geometry: Related to Classical Groups and Buildings. Springer, 2013, doi:10.1007/978-3-642-34453-4 (author pdf)