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 one-dimensional subspaces of the two-dimensional vector space *K*^{2}; it does carry other geometric stuctures. 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. ## 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 *x*-axis, *modulo* π. That is, the **real projective line** is related to the unit circle by the identication 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 potentiially *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 one-dimensional 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 3-tuple *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 cross-ratio.
## As algebraic curve From the point of view of algebraic geometry, **P**^{1}(*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 *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 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 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. |