Projective line

In mathematics, a projective line is a one-dimensional projective space. The projective line over a field "K", denoted P¹("K"), may be defined as the set of one-dimensional subspaces of the two-dimensional vector space "K"² (it does carry other geometric structures).

For the generalisation to the projective line over an associative ring, see inversive ring geometry.

Homogeneous coordinates

An arbitrary point in the projective line P¹("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 P¹("K") given by:$left\{\; [x\; :\; 1]\; in\; mathbf\; P^1(K)\; mid\; x\; in\; K\; ight\}.$This subset covers all points in P¹("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 numbers 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 R² onto the unit circle and then identifying diametrically opposite points. In terms of group theory we can take the quotient by the subgroup {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 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 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 transformations with coefficients in "K" acts on the projective line P¹("K"). This group action is transitive, so that P¹("K") is a homogeneous space for the group, often written "PGL₂(K)" to emphasise its definition as a projective 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 P¹("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 "PGL₂(K)"; in other words, the group action is sharply 3-transitive. The computational aspect of this is the cross-ratio.

As algebraic curve

The projective line is a fundamental example of an algebraic curve. From the point of view of algebraic geometry, P¹("K") is a non-singular curve of genus 0. If "K" is algebraically 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 a conic "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") of rational functions over "K", in a single indeterminate "T". The field automorphisms of "K"("T") over "K" are precisely the group "PGL₂(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 of birational geometry, this means that there will be a rational map from "V" to P¹("K"), that is not constant. The image will omit only finitely many points of P¹("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 the meromorphic functions of complex analysis, and indeed in the case of compact Riemann surfaces 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' P¹("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 P¹("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.

Many curves, for example hyperelliptic curves, are best presented abstractly, as ramified covers 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 of genus 0, so any curve in the birational class of the projective line (see rational variety). A rational 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.