Projective line
Encyclopedia
In mathematics
, a projective line is a one-dimensional projective space
. The projective line over a field
K, denoted P1(K), may be defined as the set of one-dimensional subspaces
of the two-dimensional vector space
K2 (it does carry other geometric structures).
For the generalisation to the projective line over an associative ring, see inversive ring geometry
.
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
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 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 −∞.
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
.
F is also simple to understand. In this case if F has q elements, the projective line has
elements. We can write all but one of the subspaces as
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 closure
as potentially on the line.
s 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 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 P1(K) is therefore an artifact of choice of coordinates: homogeneous coordinates
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-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.
. 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 function
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 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 function
s of complex analysis
, and indeed in the case of compact Riemann surface
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 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.
Mathematics
Mathematics 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 space
Projective space
In 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 field
Field (mathematics)
In 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 subspaces
Linear subspace
The 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 space
Vector space
A 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 geometry
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....
.
Homogeneous coordinates
An arbitrary point in the projective line P1(K) may be given in homogeneous coordinatesHomogeneous coordinates
In 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 numberReal number
In 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 circle
Unit circle
In 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 identifying
Quotient space
In 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 theory
Group theory
In 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 subgroup
Subgroup
In 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 line
Extended real number line
In 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 planeComplex plane
In 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 sphere
Sphere
A 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 sphere
Riemann sphere
In 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 analysis
Complex analysis
Complex 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 geometry
Algebraic geometry
Algebraic 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 manifold
Complex manifold
In 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 surface
Compact Riemann surface
In 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 fieldFinite field
In 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 closure
Algebraic closure
In 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 coefficientCoefficient
In 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 action
Group action
In 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
Transitivity (mathematics)
-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 space
Homogeneous space
In 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 group
Projective linear group
In 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 coordinates
Homogeneous coordinates
In 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-transitive
Group action
In 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-ratio
Cross-ratio
In 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 curveAlgebraic 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...
. From the point of view of algebraic geometry, P1(K) is a non-singular curve of genus
Genus (mathematics)
In 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 isomorphism
Isomorphism
In 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 field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
of the projective line is the field K(T) of rational function
Rational function
In 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 variety
Algebraic variety
In 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 geometry
Birational geometry
In 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 function
Meromorphic function
In 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 analysis
Complex analysis
Complex 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 surface
Compact Riemann surface
In 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 ramification
Ramification
In 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 formula
Riemann-Hurwitz formula
In 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 variety
Rational variety
In 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 cubic
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...
for the first interesting case.
Generalizations
- Algebraic curveAlgebraic 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...
- Inversive ring geometryInversive 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....
, generalization to rings - Twisted cubicTwisted 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...