- Projective line
In

mathematics , a**projective line**is a one-dimensionalprojective space . The projective line over a field "K", denoted**P**^{1}("K"), may be defined as the set of one-dimensionalsubspace s of the two-dimensionalvector space "K"^{2}(it does carry other geometric structures).For the generalisation to the

**projective line over an associative ring**, seeinversive ring geometry .**Homogeneous coordinates**An arbitrary point in the projective line

**P**^{1}("K") may be given in "homogeneous coordinates " by a pair:$[x\_1\; :\; x\_2]$of points in "K" which are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factor λ::$[x\_1\; :\; x\_2]\; =\; [lambda\; x\_1\; :\; lambda\; x\_2]\; .$The line "K" may be identified with the subset of**P**^{1}("K") given by:$left\{\; [x\; :\; 1]\; in\; mathbf\; P^1(K)\; mid\; x\; in\; K\; ight\}.$This subset covers all points in**P**^{1}("K") except one: the "point at infinity", which may be given as:$infty\; =\; [1\; :\; 0]\; .$**Examples****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**^{2}onto theunit circle and then identifyingdiametrically opposite points. In terms ofgroup theory we can take the quotient by thesubgroup {1,−1}.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 asphere . Hence the complex projective line is also known as the(or sometimes the "Gauss sphere"). It is in constant use inRiemann sphere complex analysis ,algebraic geometry andcomplex manifold theory, as the simplest example of acompact Riemann surface .**For a finite field**The case of "K" a

finite field "F" is also simple to understand. In this case if "F" has "q" elements, the projective line has:"q" + 1

elements. We can write all but one of the subspaces as

:"y" = "ax"

with "a" in "F"; this leaves out only the case of the line "x" = 0. For a finite field there is a definite loss if the projective line is taken to be this set, rather than an algebraic curve — one should at least see the underlying "infinite" set of points in an

algebraic closure as potentially "on" the line.**ymmetry group**Quite generally, the group of

Möbius transformation s withcoefficient s in "K" acts on the projective line**P**^{1}("K"). Thisgroup action is transitive, so that**P**^{1}("K") is ahomogeneous space for the group, often written "PGL_{2}(K)" to emphasise its definition as aprojective linear group . "Transitivity" says that any point "Q" may be transformed to any other point "R" by a Möbius transformation. The "point at infinity" on**P**^{1}("K") is therefore an "artefact" of choice of coordinates:homogeneous coordinates : ["X":"Y"] = ["tX":"tY"]

express a one-dimensional subspace by a single non-zero point ("X","Y") lying in it, but the symmetries of the projective line can move the point ∞ = [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 points "Q_{i}" for "i" = 1,2,3 to any other 3-tuple "R_{i}" of distinct points ("triple transitivity"). This amount of specification 'uses up' the three dimensions of "PGL_{2}(K)"; in other words, the group action is sharply 3-transitive. The computational aspect of this is thecross-ratio .**As algebraic curve**The projective line is a fundamental example of an

algebraic curve . From the point of view of algebraic geometry,**P**^{1}("K") is anon-singular curve of genus 0. If "K" isalgebraically closed , it is the unique such curve over "K", up to isomorphism. In general (non-singular) curves of genus 0 are isomorphic over "K" to aconic "C", which is the 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 clear the correspondence using lines through "P".The

function field of the projective line is the field "K"("T") ofrational function s over "K", in a single indeterminate "T". Thefield automorphism s of "K"("T") over "K" are precisely the group "PGL_{2}(K)" discussed above.One reason for the great importance of the projective line is that any function field "K"("V") of an

algebraic variety "V" over "K", other than a single point, will have a subfield isomorphic with "K"("T"). From the point of view ofbirational geometry , this means that there will be arational map from "V" to**P**^{1}("K"), that is not constant. The image will omit only finitely many points of**P**^{1}("K"), and the inverse image of a typical point "P" will be of dimension "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 themeromorphic function s ofcomplex analysis , and indeed in the case ofcompact 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**^{1}("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**^{1}("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 isramification .Many curves, for example

hyperelliptic curve s, are best presented abstractly, asramified cover s of the projective line. According to theRiemann-Hurwitz formula , the genus then depends only on the type of ramification.A

**rational curve**is a curve of genus 0, so any curve in the birational class of the projective line (seerational variety ). Arational normal curve in projective space "P"^{"n"}is a rational curve that lies in no proper linear subspace; it is known that there is essentially one example, given parametrically in homogeneous coordinates as: [1:"t":"t"

^{2}:...:"t"^{"n"}] .See

twisted cubic for the first interesting case.

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