Projective Line - As Algebraic Curve

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 Pn 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

.

See twisted cubic for the first interesting case.