A projective line is a projective space of dimension 1.
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” $1/0$).
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
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)).
(Buekenhout) A projective line is a set $\ell$ of cardinality $\ge 3$ together with the following.
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 $\sigma(r)=s$.
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.
There are, however, plenty of projective lines not arising from projective planes. For instance, we might set $\Lambda(p,q) = 1$ for all $p,q$.
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
Conversely, every Desarguesian projective line arises from a division ring in this way. Fix three points $0,1,\infty \in \ell$ and define
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)
Last revised on May 15, 2019 at 07:49:07. See the history of this page for a list of all contributions to it.