Projective line

In algebraic geometry, a projective line is a set of points which are equivalence classes in the collection of ordered pairs in the direct product of a ring with itself. Most commonly the ring is a field K, and frequently K is the field of complex numbers. This instance, the complex projective line, is also called the Riemann sphere in commemoration of the work Bernhard Riemann did in geometry.

Equivalence relation

Two pairs (a,b) and (c,d) in A x A, where A is a ring, are related under ~ when there is a unit u in the group of units U in A, such that ua = c and ub = d. Since 1 is in U, the relation is reflexive. Since the product of two units is another unit, the relation is transitive. Any unit u has a multiplicative inverse v, uv = 1, so vc = a and vd = b follows from the relation, and hence it is symmetric. Any relation that has these three properties is an equivalence relation. A set with an equivalence relation is partitioned into equivalence classes. It is this partition that determines the "points" of the projective line P¹(A).

The ideal (ring theory) aA + bA = {ax + by : x,y in A} is required to be the whole ring A, not a "proper ideal". This criterion excludes deficient pairs (a,b) which are insufficient to generate A. In the case of a field A = K, this criterion merely means that one of a or b is not zero.

The notation for a point in homogeneous coordinates uses an element (a,b) of the equivalence class, and the symbol for the group of units: U(a,b) is the equivalence class containing (a,b). Then the definition of the projective line is

$P(A)=\{U(a,b):aA+bA=A\}/\thicksim .$

In the case of a field K, $P(K)=\{U(a,b):a\neq 0\ or\ b\neq 0\}/\thicksim .$

A projective line over a ring was described by Corrado Segre in 1912 when he took A to be dual numbers. The concept was formally presented by Walter Benz in Vorlesungen über Geometrie der Algebren (1973)

Field extended by a point at infinity

When K is a field, the projective line may be identified with an extension of K by a point at infinity. More precisely, K may be identified with the subset of P¹(K) given by

$\left\{[x:1]\in \mathbf {P} ^{1}(K)\mid x\in K\right\}.$

This subset covers all points in P¹(K) except one, which is called the point at infinity:

$\infty =[1:0].$

This allows to extend the arithmetic on K to P¹(K) by the formulas

${\frac {1}{0}}=\infty ,\qquad {\frac {1}{\infty }}=0,$

$x\cdot \infty =\infty \quad {\text{if}}\quad x\not =0$

$x+\infty =\infty \quad {\text{if}}\quad x\not =\infty$

Translating this arithmetic in term of homogeneous coordinates gives, when [0 : 0] does not occur:

$[x_{1}:x_{2}]+[y_{1}:y_{2}]=[x_{1}y_{2}+y_{1}x_{2}:x_{2}y_{2}],$

$[x_{1}:x_{2}]\cdot [y_{1}:y_{2}]=[x_{1}y_{1}:x_{2}y_{2}],$

$[x_{1}:x_{2}]^{-1}=[x_{2}:x_{1}].$

Examples

Real projective line

Main article: 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 P¹(F) has q + 1 elements.

We can write all but one of the subspaces as

$X_{a}=\{ax:x\in F\}\ ,$, a non-zero in F .

The only other subspace is given by a = 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.

Symmetry group

Quite generally, the group of homographies 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 the projective nature of these transformations. Transitivity says that any point Q may be transformed to any other point R by a homography. The point at infinity on P¹(K) is therefore an artifact 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. Indeed, a generalized converse is true: a sharply 3-transitive group action is always (isomorphic to) a generalized form of a PGL₂(K) action on a projective line, replacing "field" by "KT-field" (generalizing the inverse to a weaker kind of involution), and "PGL" by a corresponding generalization of projective linear maps.

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

[1 : t : t² : ... : tⁿ].

See twisted cubic for the first interesting case.

See also

Examples

• Real projective line
• Riemann sphere, the complex projective line

Geometry

• Cross-ratio
• Projective range
• Möbius transformations

Generalizations

• Algebraic curve
• Projective line over a ring
• Twisted cubic