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projective line

Projective lines

Projective lines

Idea

A projective line is a projective space of dimension 1.

Analytic projective lines

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

The classical case of a projective line is over the complex numbers \mathbb{C}, where 1()\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 1()\mathbb{C} \to \mathbb{P}^1(\mathbb{C}).

In particular, a rational function p/q(z)p/q \in \mathbb{C}(z) may be interpreted as a holomorphic function [p/q]: 1() 1()[p/q]: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C}); concretely, if p(z),q(z)p(z), q(z) are relatively prime and of degrees m,nm, n respectively, then we may homogenize by setting p(x,y)y mp(x/y)p(x, y) \coloneqq y^m p(x/y) and q(x,y)y nq(x/y)q(x, y) \coloneqq y^n q(x/y), and define [p/q][p/q] by the mapping on homogeneous coordinates [x,y][p(x,y),q(x,y)][x, y] \mapsto [p(x, y), q(x, y)]. In fact, there is a bijective correspondence between such holomorphic endomaps on 1()\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” 1/01/0).

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 nn-dimensional projective space π\pi (with n2n\ge 2) is an automorphism σ\sigma of π\pi such that

  1. there exists a point OO, called the center, such that σ\sigma fixes every line through OO (setwise, i.e.\ it sends every point on that line to a point on the same line), and
  2. there exists a hyperplane HH (an (n1)(n-1)-dimensional subspace), called the axis, which σ\sigma fixes pointwise (i.e.\ it sends every point on HH to itself).

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

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

Of course, when π\pi is of dimension 2\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 π\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 3\ge 3 together with the following.

  1. For each p,qp,q\in \ell, a group Λ(p,q)\Lambda(p,q) whose elements are called central collineations with center pp and axis qq. Note that qq need not be distinct from pp.
  2. Λ(p,q)\Lambda(p,q) acts on \ell fixing pp and qq, and if an element of it fixes a point r{p,q}r \notin \{p,q\}, then it is the identity. In particular, the action is faithful, and therefore embeds each Λ(p,q)\Lambda(p,q) in the permutation group Aut()Aut(\ell).
  3. If pqp\neq q, then Λ(p,q)\Lambda(p,q) and Λ(q,p)\Lambda(q,p) commute with each other.
  4. For any σΛ(r,s)\sigma\in\Lambda(r,s) and any p,qp,q, we have σΛ(p,q)σ 1=Λ(σ(p),σ(q))\sigma \Lambda(p,q) \sigma^{-1} = \Lambda(\sigma(p),\sigma(q)).
  5. The composite of two collineations with center pp (possibly with different axes) is again a collineation with center pp, and dually the composite of two collineations with axis qq (possibly with different centers) is again a collineation with axis qq. Thus we have two groups Λ(p)= qΛ(p,q)\Lambda(p) = \bigcup_q \Lambda(p,q) and Λ (q)= pΛ(p,q)\Lambda^\vee(q) = \bigcup_p\Lambda(p,q).

A projective line is Desarguesian if in addition Λ(p,q)\Lambda(p,q) acts transitively on {p,q}\ell \setminus \{p,q\}. In other words, if r,s{p,q}r,s\in \ell\setminus \{p,q\}, there is a (necessarily unique) σΛ(p,q)\sigma\in\Lambda(p,q) with σ(r)=s\sigma(r)=s.

Examples

Lines in projective planes

We saw above that when π\pi is of dimension 3\ge 3, then every central collineation of a subspace (of dimension 2\ge 2) is induced by some central collineation of π\pi. Even though this required Desargues’ theorem to prove, which might not be true in a projective plane (dimension 22), 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 π\pi (not necessarily Desarguesian), and a line \ell in π\pi, for any p,qp,q\in \ell define Λ(p,q)\Lambda(p,q) to be the set of permutations of \ell that are the restriction to \ell of some central collineation of π\pi with center pp and axis containing qq. 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 Λ(p,q)=1\Lambda(p,q) = 1 for all p,qp,q.

Analytic examples

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

  • if pqp\neq q, then Λ(p,q)\Lambda(p,q) is the image in PGL(V)PGL(V) of those automorphisms in GL(V)GL(V) fixing qq pointwise and pp setwise.
  • if p=qp= q, then Λ(p,q)\Lambda(p,q) is the image in PGL(V)PGL(V) of those automorphisms αGL(V)\alpha\in GL(V) such that α(p)=p\alpha(p)=p and α(x)xp\alpha(x)-x\in p for all xVx\in V.

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

  • k={}k=\ell\setminus\{\infty\}.
  • a+b=t b(a)a+b = t_b(a), where t bt_b is the unique element of Λ(,)\Lambda(\infty,\infty) with t b(0)=bt_b(0)=b.
  • ab=λ a(b)a b = \lambda_a(b), where λ b\lambda_b is the unique element of Λ(0,)\Lambda(0,\infty) with t a(1)=at_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.

References

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

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

  • Francis Buekenhout and Arjeh M. Cohen, Diagram Geometry: Related to Classical Groups and Buildings. Springer, 2013 (author pdf)

Last revised on September 3, 2018 at 03:43:53. See the history of this page for a list of all contributions to it.