Projective line
From Academic Kids

In mathematics, the projective line is a fundamental example of an algebraic curve. It may be defined over any field K, as the set of onedimensional subspaces of the twodimensional vector space K^{2}; it does carry other geometric structures. It may be denoted as P^{1}(K), but goes also by other names in particular areas. For the generalisation to the projective line over an associative ring, see inversive ring geometry.
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Real projective line
For example in the case that K is the real number field, such a subspace is defined by the angle in radians it makes with the xaxis, modulo π. That is, the real projective line is related to the unit circle by the identification of diametrically opposite points; in terms of group theory we can take the quotient by the subgroup {1,−1}. Topologically it is again a circle.
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 leave 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 set of points in an algebraic closure as potentially on the line.
Complex projective line: the Riemann sphere
The case of K the complex number field is the Riemann sphere (sometimes also called the Gauss sphere). It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a Riemann surface.
Symmetry group
Quite generally, the group of Möbius transformations with coefficients in K acts on the projective line P^{1}(K). This group action is transitive, so that P^{1}(K) is a homogeneous space for the group, often written PGL_{2}(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 P^{1}(K) is therefore an artefact of choice of coordinates: homogeneous coordinates [X:Y] = [tX:tY] express a onedimensional subspace by a single point (X,Y) on it, but the symmetries of the projective line can move this 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 3tuple R_{i} of distinct points (triple transitivity). This amount of specification 'uses up' the three dimensions of PGL_{2}(K). The computational aspect of this is the crossratio.
As algebraic curve
From the point of view of algebraic geometry, P^{1}(K) is a nonsingular curve of genus 0. If K is algebraically closed, it is the unique such curve over K, up to isomorphism. In general (nonsingular) 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 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 of birational geometry, this means that there will be a rational 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 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' P^{1}(K). Assuming C is nonsingular (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 is ramification.
Many curves, for example hyperelliptic curves, are best presented abstractly, as ramified covers of the projective line. According to the RiemannHurwitz formula, the genus then depends only on the type of ramification.