Editing Projective line

{{More citations needed|date=December 2009}}
In [[projective geometry]] and [[mathematics]] more generally, a '''projective line''' is, roughly speaking, the extension of a usual [[line (geometry)|line]] by a point called a ''[[point at infinity]]''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a [[projective plane]] meet in exactly one point (there is no "parallel" case).

There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a [[field (mathematics)|field]] ''K'', commonly denoted '''P'''<sup>1</sup>(''K''), as the set of one-dimensional [[Linear subspace|subspaces]] of a two-dimensional ''K''-[[vector space]]. This definition is a special instance of the general definition of a [[projective space]].

The projective line over the [[real number|reals]] is a [[manifold]]; see ''[[Real projective line]]'' for details.

== Homogeneous coordinates ==
An arbitrary point in the projective line '''P'''<sup>1</sup>(''K'') may be represented by an [[equivalence class]] of ''[[homogeneous coordinates]]'', which take the form of a pair
: <math>[x_1 : x_2]</math>
of elements of ''K'' that are not both zero. Two such pairs are [[equivalence relation|equivalent]] if they differ by an overall nonzero factor ''λ'':
: <math>[x_1 : x_2] \sim [\lambda x_1 : \lambda x_2].</math>

== Line extended by a point at infinity ==
The projective line may be identified with the line ''K'' extended by a [[point at infinity]]. More precisely,
the line ''K'' may be identified with the subset of '''P'''<sup>1</sup>(''K'') given by
: <math>\left\{[x : 1] \in \mathbf P^1(K) \mid x \in K\right\}.</math>
This subset covers all points in '''P'''<sup>1</sup>(''K'') except one, which is called the ''point at infinity'':
: <math>\infty = [1 : 0].</math>
This allows to extend the arithmetic on ''K'' to '''P'''<sup>1</sup>(''K'') by the formulas
: <math>\frac {1}{0}=\infty,\qquad \frac {1}{\infty}=0,</math>
: <math>x\cdot \infty = \infty \quad \text{if}\quad x\not= 0</math>
: <math>x+ \infty = \infty \quad \text{if}\quad x\not= \infty</math>

Translating this arithmetic in terms of homogeneous coordinates gives, when {{nowrap|{{bracket|0 : 0}}}} does not occur:
: <math>[x_1 : x_2] + [y_1 : y_2] = [(x_1 y_2 + y_1 x_2) : x_2 y_2],</math>
: <math>[x_1 : x_2] \cdot [y_1 : y_2] = [x_1 y_1 : x_2 y_2],</math>
: <math>[x_1 : x_2]^{-1} = [x_2 : x_1].</math>

== Examples ==
=== Real projective line ===
{{Main|Real projective line}}

The projective line over the [[real number]]s is called the '''real projective line'''.  It may also be thought of as the line ''K'' together with an idealised ''[[point at infinity]]'' ∞; the point connects to both ends of ''K'' creating a closed loop or topological circle.

An example is obtained by projecting points in '''R'''<sup>2</sup> onto the [[unit circle]] and then [[Quotient space (topology)|identifying]] [[diametrically opposite]] points. In terms of [[group theory]] we can take the quotient by the [[subgroup]] {{nowrap|{{mset|1, −1}}}} under multiplication.

Compare the [[extended real number line]], which distinguishes ∞ and −∞.

=== Complex projective line: the Riemann sphere ===
Adding a point at infinity to the [[complex plane]] results in a space that is topologically a [[sphere]]. Hence the complex projective line is also known as the '''[[Riemann sphere]]''' (or sometimes the ''Gauss sphere''). It is in constant use in [[complex analysis]], [[algebraic geometry]] and [[complex manifold]] theory, as the simplest example of a [[compact Riemann surface]].

=== For a finite field ===
The projective line over a [[finite field]] ''F''<sub>''q''</sub> of ''q'' elements has {{nowrap|''q'' + 1}} points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates {{nowrap|[''x'' : ''y'']}}, ''q'' of these points have the form:
: {{math|[''a'' : 1]}} for each {{mvar|''a''}} in {{mvar|''F''<sub>''q''</sub>}},

and the remaining [[point at infinity|point ''at infinity'']] may be represented as {{nowrap|{{bracket|1 : 0}}}}.

== Symmetry group ==
Quite generally, the group of [[homography|homographies]] with [[coefficient]]s in ''K'' acts on the projective line '''P'''<sup>1</sup>(''K''). This [[Group action (mathematics)|group action]] is [[Group action (mathematics)#Types of actions|transitive]], so that '''P'''<sup>1</sup>(''K'') is a [[homogeneous space]] for the group, often written PGL<sub>2</sub>(''K'') to emphasise the projective nature of these transformations. ''Transitivity'' says that there exists a homography that will transform any point ''Q'' to any other point ''R''. The ''point at infinity'' on '''P'''<sup>1</sup>(''K'') is therefore an ''artifact'' of choice of coordinates: [[homogeneous coordinates]]
: <math>[X : Y] \sim [\lambda X : \lambda Y]</math>
express a one-dimensional subspace by a single non-zero point {{nowrap|(''X'', ''Y'')}} lying in it, but the symmetries of the projective line can move the point {{nowrap|1=∞ = {{bracket|1 : 0}}}} to any other, and it is in no way distinguished.

Much more is true, in that some transformation can take any given [[Distinct (mathematics)|distinct]] points ''Q''<sub>''i''</sub> for {{nowrap|1=''i'' = 1, 2, 3}} to any other 3-tuple ''R''<sub>''i''</sub> of distinct points (''triple transitivity''). This amount of specification 'uses up' the three dimensions of PGL<sub>2</sub>(''K''); in other words, the group action is [[Group action (mathematics)|sharply 3-transitive]]. The computational aspect of this is the [[cross-ratio]]. Indeed, a generalized converse is true: a sharply 3-transitive group action is always (isomorphic to) a generalized form of a PGL<sub>2</sub>(''K'') action on a projective line, replacing "field" by "KT-field" (generalizing the inverse to a weaker kind of involution), and "PGL" by a corresponding generalization of projective linear maps.<ref>[https://mathoverflow.net/q/66865 Action of PGL(2) on Projective Space] – see comment and cited paper.</ref>

== As algebraic curve ==
The projective line is a fundamental example of an [[algebraic curve]]. From the point of view of algebraic geometry, '''P'''<sup>1</sup>(''K'') is a [[Algebraic curve#Singularities|non-singular]] curve of [[genus (mathematics)|genus]] 0. If ''K'' is [[algebraically closed]], it is the unique such curve over ''K'', up to [[rational equivalence]]. In general a (non-singular) curve of genus 0 is rationally equivalent over ''K'' to a [[conic]] ''C'', which is itself birationally equivalent to projective line if and only if ''C'' has a point defined over ''K''; geometrically such a point ''P'' can be used as origin to make explicit the birational equivalence.

The [[function field of an algebraic variety|function field]] of the projective line is the field ''K''(''T'') of [[rational function]]s over ''K'', in a single indeterminate ''T''. The [[field automorphism]]s of ''K''(''T'') over ''K'' are precisely  the group PGL<sub>2</sub>(''K'') discussed above.

Any function field ''K''(''V'') of an [[algebraic variety]] ''V'' over ''K'', other than a single point, has a subfield isomorphic with ''K''(''T''). From the point of view of [[birational geometry]], this means that there will be a [[rational map]] from ''V'' to '''P'''<sup>1</sup>(''K''), that is not constant. The image will omit only finitely many points of '''P'''<sup>1</sup>(''K''), and the inverse image of a typical point ''P'' will be of dimension {{nowrap|dim ''V'' − 1}}. This is the beginning of methods in algebraic geometry that are inductive on dimension. The rational maps play a role analogous to the [[meromorphic function]]s of [[complex analysis]], and indeed in the case of [[compact Riemann surface]]s the two concepts coincide.

If ''V'' is now taken to be of dimension 1, we get a picture of a typical algebraic curve ''C'' presented 'over' '''P'''<sup>1</sup>(''K''). Assuming ''C'' is non-singular (which is no loss of generality starting with ''K''(''C'')), it can be shown that such a rational map from ''C'' to '''P'''<sup>1</sup>(''K'') will in fact be everywhere defined. (That is not the case if there are singularities, since for example a ''[[double point]]'' where a curve ''crosses itself'' may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is [[Ramification (mathematics)|ramification]].

Many curves, for example [[hyperelliptic curve]]s, may be presented abstractly, as [[ramified cover]]s of the projective line. According to the [[Riemann–Hurwitz formula]], the genus then depends only on the type of ramification.

A '''rational curve''' is a curve that is [[birational equivalence|birationally equivalent]] to a projective line (see [[rational variety]]); its [[genus (mathematics)|genus]] is 0. A [[rational normal curve]] in projective space '''P'''<sup>''n''</sup> is a rational curve that lies in no proper linear subspace; it is known that there is only one example (up to projective equivalence),<ref>{{citation|title=Algebraic Geometry: A First Course|volume=133|series=Graduate Texts in Mathematics|first=Joe|last=Harris|publisher=Springer|year=1992|isbn=9780387977164|url=https://books.google.com/books?id=_XxZdhbtf1sC&pg=PA10}}.</ref> given parametrically in homogeneous coordinates as
: [1 : ''t'' : ''t''<sup>2</sup> : ... : ''t''<sup>''n''</sup>].

See ''[[Twisted cubic]]'' for the first interesting case.

== See also ==
* [[Algebraic curve]]
* [[Cross-ratio]]
* [[Möbius transformation]]
* [[Projective line over a ring]]
* [[Projectively extended real line]]
* [[Projective range]]
* [[Wheel theory]]

== References ==
{{reflist}}

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