In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
projective line is a one-dimensional
projective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. The projective line over a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, denoted
P1(
K), may be defined as the set of one-dimensional
subspacesThe concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of the two-dimensional
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
K2 (it does carry other geometric structures).
For the generalisation to the
projective line over an associative ring, see
inversive ring geometryIn mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....
.
Homogeneous coordinates
An arbitrary point in the projective line
P1(
K) may be given in
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
by a pair
of points in
K which are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factor λ:
The line
K may be identified with the subset of
P1(
K) given by
This subset covers all points in
P1(
K) except one: the
point at infinity, which may be given as
Real projective line
The projective line over the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s 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 circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
and then
identifyingIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
diametrically opposite points. In terms of
group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
we can take the quotient by the
subgroupIn group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
{1,−1}.
Compare the
extended real number lineIn mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
, which distinguishes ∞ and −∞.
Complex projective line: the Riemann sphere
Adding a point at infinity to the
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
results in a space that is topologically a
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. Hence the complex projective line is also known as the
Riemann sphereIn mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
(or sometimes the
Gauss sphere). It is in constant use in
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
,
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and
complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
theory, as the simplest example of a
compact Riemann surfaceIn mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
.
For a finite field
The case of
K a
finite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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 closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
as potentially
on the line.
Symmetry group
Quite generally, the group of
Möbius transformations with
coefficientIn mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s in
K acts on the projective line
P1(
K). This
group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
is
transitive-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...
, so that
P1(
K) is a
homogeneous spaceIn mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
for the group, often written
PGL2(K) to emphasise its definition as a
projective linear groupIn mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...
.
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
artifact of choice of coordinates:
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
- [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
Qi for
i = 1,2,3 to any other 3-tuple
Ri 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-transitiveIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
. The computational aspect of this is the
cross-ratioIn geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
. 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 curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
. From the point of view of algebraic geometry,
P1(
K) is a non-singular curve of
genusIn mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
0. If
K is algebraically closed, it is the unique such curve over
K, up to
isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
. 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 fieldFunction field may refer to:*Function field of an algebraic variety*Function field...
of the projective line is the field
K(
T) of
rational functionIn mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s 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 varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V over
K, other than a single point, will have a subfield isomorphic with
K(
T). From the point of view of
birational geometryIn mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
, 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 functionIn complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s of
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, and indeed in the case of
compact Riemann surfaceIn mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
s 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
ramificationIn mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
.
Many curves, for example
hyperelliptic curves, are best presented abstractly, as ramified covers of the projective line. According to the
Riemann-Hurwitz formulaIn mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case...
, 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 varietyIn mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K...
). 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 cubicIn mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation...
for the first interesting case.
Generalizations
- Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
- Inversive ring geometry
In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....
, generalization to rings
- Twisted cubic
In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation...