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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an affine algebraic plane curve is the
zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...
of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
in two variables. A projective algebraic plane curve is the zero set in a
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
of a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by
homogenizing Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, s ...
its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
one. Equivalently, an algebraic curve is an algebraic variety that is
birationally equivalent In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
to an algebraic plane curve. If the curve is contained in an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
or a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, one can take a
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
for such a birational equivalence. These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
and
smoothness In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
. For example, there exist smooth curves of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
0 and degree greater than two, but any plane projection of such curves has singular points (see
Genus–degree formula In classical algebraic geometry, the genus–degree formula relates the degree ''d'' of an irreducible plane curve C with its arithmetic genus ''g'' via the formula: :g=\frac12 (d-1)(d-2). Here "plane curve" means that C is a closed curve in the p ...
). A non-plane curve is often called a ''space curve'' or a '' skew curve''.


In Euclidean geometry

An algebraic curve in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
is the set of the points whose
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
are the solutions of a bivariate
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...
''p''(''x'', ''y'') = 0. This equation is often called the
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
of the curve, in contrast to the curves that are the graph of a function defining ''explicitly'' ''y'' as a function of ''x''. With a curve given by such an implicit equation, the first problems are to determine the shape of the curve and to draw it. These problems are not as easy to solve as in the case of the graph of a function, for which ''y'' may easily be computed for various values of ''x''. The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help in solving these problems. Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs (also called ''branches'') sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points called
acnode An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are " isolated point or hermit point". For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is ...
s. A ''smooth monotone arc'' is the graph of a
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
which is defined and
monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...
on an
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
of the ''x''-axis. In each direction, an arc is either unbounded (usually called an ''infinite arc'') or has an endpoint which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes. For example, for the
Tschirnhausen cubic In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation :r = a\sec^3 \left(\frac\right) where is the secant function. History The curve was studied by Ehrenf ...
, there are two infinite arcs having the origin (0,0) as of endpoint. This point is the only singular point of the curve. There are also two arcs having this singular point as one endpoint and having a second endpoint with a horizontal tangent. Finally, there are two other arcs each having one of these points with horizontal tangent as the first endpoint and having the unique point with vertical tangent as the second endpoint. In contrast, the
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
is certainly not an algebraic curve, having an infinite number of monotone arcs. To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
s (if any) and the way in which the arcs connect them. It is also useful to consider the
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
s as remarkable points. When all this information is drawn on a sheet of paper, the shape of the curve usually appears rather clearly. If not, it suffices to add a few other points and their tangents to get a good description of the curve. The methods for computing the remarkable points and their tangents are described below in the section Remarkable points of a plane curve.


Plane projective curves

It is often desirable to consider curves in the
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. An algebraic curve in the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
or plane projective curve is the set of the points in a
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
whose
projective coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
are zeros of a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
in three variables ''P''(''x'', ''y'', ''z''). Every affine algebraic curve of equation ''p''(''x'', ''y'') = 0 may be completed into the projective curve of equation ^hp(x,y,z)=0, where ^hp(x,y,z)=z^p(\tfrac,\tfrac) is the result of the
homogenization Homogeneity is a sameness of constituent structure. Homogeneity, homogeneous, or homogenization may also refer to: In mathematics *Transcendental law of homogeneity of Leibniz * Homogeneous space for a Lie group G, or more general transformati ...
of ''p''. Conversely, if ''P''(''x'', ''y'', ''z'') = 0 is the homogeneous equation of a projective curve, then ''P''(''x'', ''y'', 1) = 0 is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as ^hp(x,y,1)=p(x,y) and, if ''p'' is defined by p(x,y)=P(x,y,1), then ^hp(x,y,z)=P(x,y,z), as soon as the homogeneous polynomial ''P'' is not divisible by ''z''. For example, the projective curve of equation ''x''2 + ''y''2 − ''z''2 is the projective completion of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
of equation ''x''2 + ''y''2 − 1 = 0. This implies that an affine curve and its projective completion are the same curves, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part. Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if ''p''(''x'', ''y'') is the polynomial defining an affine curve, beside the partial derivatives p'_x and p'_y, it is useful to consider the derivative at infinity p'_\infty(x,y)=. For example, the equation of the tangent of the affine curve of equation ''p''(''x'', ''y'') = 0 at a point (''a'', ''b'') is xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0.


Remarkable points of a plane curve

In this section, we consider a plane algebraic curve defined by a bivariate polynomial ''p''(''x'', ''y'') and its projective completion, defined by the homogenization P(x,y,z)= ^hp(x,y,z) of ''p''.


Intersection with a line

Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
s are useful to draw the curve. Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve. If an efficient
root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...
is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the ''y''-axis and passing through each
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smal ...
on the ''x''-axis. If the polynomial defining the curve has a degree ''d'', any line cuts the curve in at most ''d'' points.
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
asserts that this number is exactly ''d'', if the points are searched in the projective plane over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
(for example the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s), and counted with their
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
. The method of computation that follows proves again this theorem, in this simple case. To compute the intersection of the curve defined by the polynomial ''p'' with the line of equation ''ax''+''by''+''c'' = 0, one solves the equation of the line for ''x'' (or for ''y'' if ''a'' = 0). Substituting the result in ''p'', one gets a univariate equation ''q''(''y'') = 0 (or ''q''(''x'') = 0, if the equation of the line has been solved in ''y''), each of whose roots is one coordinate of an intersection point. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity if the degree of ''q'' is lower than the degree of ''p''; the multiplicity of such an intersection point at infinity is the difference of the degrees of ''p'' and ''q''.


Tangent at a point

The tangent at a point (''a'', ''b'') of the curve is the line of equation (x-a)p'_x(a,b)+(y-b)p'_y(a,b)=0, like for every
differentiable curve Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the ...
defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric: xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0, where p'_\infty(x,y)=P'_z(x,y,1) is the derivative at infinity. The equivalence of the two equations results from
Euler's homogeneous function theorem In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
applied to ''P''. If p'_x(a,b)=p'_y(a,b)=0, the tangent is not defined and the point is a singular point. This extends immediately to the projective case: The equation of the tangent of at the point of
projective coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
(''a'':''b'':''c'') of the projective curve of equation ''P''(''x'', ''y'', ''z'') = 0 is xP'_x(a,b,c)+yP'_y(a,b,c)+zP'_z(a,b,c)=0, and the points of the curves that are singular are the points such that P'_x(a,b,c)=P'_y(a,b,c)=P'_z(a,b,c)=0. (The condition ''P''(''a'', ''b'', ''c'') = 0 is implied by these conditions, by Euler's homogeneous function theorem.)


Asymptotes

Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belong to its affine part. The corresponding
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case. Let p=p_d+\cdots+p_0 be the decomposition of the polynomial defining the curve into its homogeneous parts, where ''pi'' is the sum of the monomials of ''p'' of degree ''i''. It follows that P

p_d+zp_+\cdots+z^dp_0
and P'_z(a,b,0) =p_(a,b). A point at infinity of the curve is a zero of ''p'' of the form (''a'', ''b'', 0). Equivalently, (''a'', ''b'') is a zero of ''pd''. The
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
implies that, over an algebraically closed field (typically, the field of complex numbers), ''p''''d'' factors into a product of linear factors. Each factor defines a point at infinity on the curve: if ''bx'' − ''ay'' is such a factor, then it defines the point at infinity (''a'', ''b'', 0). Over the reals, ''p''''d'' factors into linear and quadratic factors. The
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (''a'', ''b'', 0) is a point at infinity of the curve, one says that (''a'', ''b'') is an asymptotic direction. Setting ''q'' = ''p''''d'' the equation of the corresponding asymptote is xq'_x(a,b)+yq'_y(a,b)+p_(a,b)=0. If q'_x(a,b)=q'_y(a,b)=0 and p_(a,b)\neq 0, the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
. In this case one says that the curve has a ''parabolic branch''. If q'_x(a,b)=q'_y(a,b)=p_(a,b)=0, the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point.


Singular points

The singular points of a curve of degree ''d'' defined by a polynomial ''p''(''x'',''y'') of degree ''d'' are the solutions of the system of equations: p'_x(x,y)=p'_y(x,y)=p(x,y)=0. In
characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
, this system is equivalent to p'_x(x,y)=p'_y(x,y)=p'_\infty(x,y)=0, where, with the notation of the preceding section, p'_\infty(x,y)=P'_z(x,y,1). The systems are equivalent because of
Euler's homogeneous function theorem In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
. The latter system has the advantage of having its third polynomial of degree ''d''-1 instead of ''d''. Similarly, for a projective curve defined by a homogeneous polynomial ''P''(''x'',''y'',''z'') of degree ''d'', the singular points have the solutions of the system P'_x(x,y,z)=P'_y(x,y,z)=P'_z(x,y,z)=0 as
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
. (In positive characteristic, the equation P(x,y,z) has to be added to the system.) This implies that the number of singular points is finite as long as ''p''(''x'',''y'') or ''P''(''x'',''y'',''z'') is square free.
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
implies thus that the number of singular points is at most (''d''−1)2, but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is ''d''(''d''−1)/2, this value is reached when the polynomial factors in linear factors, that is if the curve is the union of ''d'' lines. For irreducible curves and polynomials, the number of singular points is at most (''d''−1)(''d''−2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below). The equation of the tangents at a singular point is given by the nonzero homogeneous part of the lowest degree in the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.


Analytic structure

The study of the analytic structure of an algebraic curve in the
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
or as a smooth curve. Near a regular point, one of the coordinates of the curve may be expressed as an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
of the other coordinate. This is a corollary of the analytic implicit function theorem, and implies that the curve is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
near the point. Near a singular point, the situation is more complicated and involves
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
, which provide analytic
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s of the branches. For describing a singularity, it is worth to
translate Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
the curve for having the singularity at the origin. This consists of a change of variable of the form X=x-a, Y=y-b, where a, b are the coordinates of the singular point. In the following, the singular point under consideration is always supposed to be at the origin. The equation of an algebraic curve is f(x,y)=0, where is a polynomial in and . This polynomial may be considered as a polynomial in , with coefficients in the algebraically closed field of the
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
in . Thus may be factored in factors of the form y-P(x), where is a Puiseux series. These factors are all different if is an
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
, because this implies that is
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
, a property which is independent of the field of coefficients. The Puiseux series that occur here have the form P(x)=\sum_^\infty a_nx^, where is a positive integer, and is an integer that may also be supposed to be positive, because we consider only the branches of the curve that pass through the origin. Without loss of generality, we may suppose that is
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
with the greatest common divisor of the such that (otherwise, one could choose a smaller common denominator for the exponents). Let be a primitive th root of unity. If the above Puiseux series occurs in the factorization of , then the series P_i(x)=\sum_^\infty a_n\omega_d^i x^ occur also in the factorization (a consequence of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
). These series are said conjugate, and are considered as a single branch of the curve, of ''ramification'' index . In the case of a real curve, that is a curve defined by a polynomial with real coefficients, three cases may occur. If none has real coefficients, then one has a non-real branch. If some has real coefficients, then one may choose it as . If is odd, then every real value of provides a real value of , and one has a real branch that looks regular, although it is singular if . If is even, then and have real values, but only for . In this case, the real branch looks as a
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
(or is a cusp, depending on the definition of a cusp that is used). For example, the ordinary cusp has only one branch. If it is defined by the equation y^2-x^3=0, then the factorization is (y-x^)(y+x^); the ramification index is 2, and the two factors are real and define each a half branch. If the cusp is rotated, it equation becomes y^3-x^2=0, and the factorization is (y-x^)(y-j^2x^)(y-(j^2)^2x^), with j=(1+\sqrt)/2 (the coefficient has not been simplified to for showing how the above definition of is specialized). Here the ramification index is 3, and only one factor is real; this shows that, in the first case, the two factors must be considered as defining the same branch.


Non-plane algebraic curves

An algebraic curve is an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
one. This implies that an affine curve in an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
of dimension ''n'' is defined by, at least, ''n''−1 polynomials in ''n'' variables. To define a curve, these polynomials must generate a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
of
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred. Let f, g_0, g_3, \ldots, g_n be ''n'' polynomials in two variables ''x''1 and ''x''2 such that ''f'' is irreducible. The points in the affine space of dimension ''n'' such whose coordinates satisfy the equations and inequations \begin &f(x_1,x_2)=0\\ &g_0(x_1,x_2)\neq 0\\ x_3&=\frac\\ & \ \vdots \\ x_n&=\frac \end are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials ''h'' such that it exists an integer ''k'' such g_0^kh belongs to the ideal generated by f, x_3g_0-g_3, \ldots, x_ng_0-g_n. This representation is a
birational equivalence In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...
between the curve and the plane curve defined by ''f''. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field. This representation allows us to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection. For a curve defined by its implicit equations, above representation of the curve may easily deduced from a
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbn ...
for a block ordering such that the block of the smaller variables is (''x''1, ''x''2). The polynomial ''f'' is the unique polynomial in the base that depends only of ''x''1 and ''x''2. The fractions ''gi''/''g''0 are obtained by choosing, for ''i'' = 3, ..., ''n'', a polynomial in the basis that is linear in ''xi'' and depends only on ''x''1, ''x''2 and ''xi''. If these choices are not possible, this means either that the equations define an
algebraic set Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when ''f'' exists and is unique, and, for ''i'' = 3, …, ''n'', there exist polynomials whose leading monomial depends only on ''x''1, ''x''2 and ''xi''.


Algebraic function fields

The study of algebraic curves can be reduced to the study of
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to
birational In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
equivalence, the irreducible curves over a field ''F'' are categorically equivalent to
algebraic function field In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...
s in one variable over ''F''. Such an algebraic function field is a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''K'' of ''F'' that contains an element ''x'' which is transcendental over ''F'', and such that ''K'' is a finite algebraic extension of ''F''(''x''), which is the field of rational functions in the indeterminate ''x'' over ''F''. For example, consider the field C of complex numbers, over which we may define the field C(''x'') of rational functions in C. If , then the field C(''x'', ''y'') is an elliptic function field. The element ''x'' is not uniquely determined; the field can also be regarded, for instance, as an extension of C(''y''). The algebraic curve corresponding to the function field is simply the set of points (''x'', ''y'') in C2 satisfying . If the field ''F'' is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. For example, if the base field ''F'' is the field R of real numbers, then defines an algebraic extension field of R(''x''), but the corresponding curve considered as a subset of R2 has no points. The equation does define an irreducible algebraic curve over R in the
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
sense (an
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
, separated
one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
s of finite type over R). In this sense, the one-to-one correspondence between irreducible algebraic curves over ''F'' (up to birational equivalence) and algebraic function fields in one variable over ''F'' holds in general. Two curves can be birationally equivalent (i.e. have
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
function fields) without being isomorphic as curves. The situation becomes easier when dealing with ''nonsingular'' curves, i.e. those that lack any singularities. Two nonsingular projective curves over a field are isomorphic if and only if their function fields are isomorphic.
Tsen's theorem In mathematics, Tsen's theorem states that a function field ''K'' of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally t ...
is about the function field of an algebraic curve over an algebraically closed field.


Complex curves and real surfaces

A complex projective algebraic curve resides in ''n''-dimensional complex projective space CP''n''. This has complex dimension ''n'', but topological dimension, as a real
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, 2''n'', and is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, and
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
. An algebraic curve over C likewise has topological dimension two; in other words, it is a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. The topological genus of this surface, that is the number of handles or donut holes, is equal to the
geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex m ...
of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a nonsingular curve that has
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
''d'' and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is , where ''k'' is the number of these singularities.


Compact Riemann surfaces

A
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
if it is compact as a topological space. There is a triple
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences f ...
between the category of smooth irreducible projective algebraic curves over C (with non-constant regular maps as morphisms), the category of compact Riemann surfaces (with non-constant
holomorphic map In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivat ...
s as morphisms), and the opposite of the category of
algebraic function field In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...
s in one variable over C (with field homomorphisms that fix C as morphisms). This means that in studying these three subjects we are in a sense studying one and the same thing. It allows complex analytic methods to be used in algebraic geometry, and algebraic-geometric methods in complex analysis and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry. See also algebraic geometry and analytic geometry for a more general theory.


Singularities

Using the intrinsic concept of
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
, points ''P'' on an algebraic curve ''C'' are classified as ''smooth'' (synonymous: ''non-singular''), or else ''
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
''. Given ''n''−1 homogeneous polynomials in ''n''+1 variables, we may find the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
as the (''n''−1)×(''n''+1) matrix of the partial derivatives. If the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of this matrix is ''n''−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains ''n''−1 when the Jacobian matrix is evaluated at a point ''P'' on the curve, then the point is a smooth or regular point; otherwise it is a ''singular point''. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation ''f''(''x'',''y'',''z'') = 0, then the singular points are precisely the points ''P'' where the rank of the 1×(''n''+1) matrix is zero, that is, where \frac(P)=\frac(P)=\frac(P)=0. Since ''f'' is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field ''F'', which in particular need not be the real or complex numbers. It should, of course, be recalled that (0,0,0) is not a point of the curve and hence not a singular point. Similarly, for an affine algebraic curve defined by a single polynomial equation ''f''(''x'',''y'') = 0, then the singular points are precisely the points ''P'' ''of the curve'' where the rank of the 1×''n'' Jacobian matrix is zero, that is, where f(P)=\frac(P)=\frac(P)=0. The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
, which is a birational invariant. For this to work, we should consider the curve projectively and require ''F'' to be algebraically closed, so that all the singularities which belong to the curve are considered.


Classification of singularities

Singular points include multiple points where the curve crosses over itself, and also various types of ''cusp'', for example that shown by the curve with equation ''x''3 = ''y''2 at (0,0). A curve ''C'' has at most a finite number of singular points. If it has none, it can be called ''smooth'' or ''non-singular''. Commonly, this definition is understood over an algebraically closed field and for a curve ''C'' in a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
(i.e., ''complete'' in the sense of algebraic geometry). For example, the plane curve of equation y-x^3=0 is considered as singular, as having a singular point (a cusp) at infinity. ''In the remainder of this section, one considers a plane curve defined as the zero set of a bivariate polynomial'' . Some of the results, but not all, may be generalized to non-plane curves. The singular points are classified by means of several invariants. The multiplicity is defined as the maximum integer such that the derivatives of to all orders up to vanish (also the minimal
intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
between the curve and a straight line at ). Intuitively, a singular point has delta invariant if it concentrates ordinary double points at . To make this precise, the blow up process produces so-called
infinitely near point In algebraic geometry, an infinitely near point of an algebraic surface ''S'' is a point on a surface obtained from ''S'' by repeatedly blowing up points. Infinitely near points of algebraic surfaces were introduced by . There are some other mean ...
s, and summing over the infinitely near points, where ''m'' is their multiplicity, produces . For an irreducible and reduced curve and a point we can define algebraically as the length of \widetilde / \mathcal_P where \mathcal_P is the local ring at ''P'' and \widetilde is its integral closure. The
Milnor number In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If ''f'' is a complex-valued holomorphic function germ then the Milnor number of ''f'', denoted ''μ''(''f''), is ...
of a singularity is the degree of the mapping on the small sphere of radius ε, in the sense of the topological
degree of a continuous mapping In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree ...
, where is the (complex) gradient vector field of ''f''. It is related to δ and ''r'' by the Milnor–Jung formula, Here, the branching number ''r'' of ''P'' is the number of locally irreducible branches at ''P''. For example, ''r'' = 1 at an ordinary cusp, and ''r'' = 2 at an ordinary double point. The multiplicity ''m'' is at least ''r'', and that ''P'' is singular if and only if ''m'' is at least 2. Moreover, δ is at least ''m''(''m''-1)/2. Computing the delta invariants of all of the singularities allows the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
''g'' of the curve to be determined; if ''d'' is the degree, then g = \frac(d-1)(d-2) - \sum_P \delta_P, where the sum is taken over all singular points ''P'' of the complex projective plane curve. It is called the genus formula. Assign the invariants 'm'', δ, ''r''to a singularity, where ''m'' is the multiplicity, δ is the delta-invariant, and ''r'' is the branching number. Then an ''ordinary cusp'' is a point with invariants ,1,1and an ''ordinary double point'' is a point with invariants ,1,2 and an ordinary ''m''-multiple point is a point with invariants 'm'', ''m''(''m''−1)/2, ''m''


Examples of curves


Rational curves

A rational curve, also called a unicursal curve, is any curve which is
birationally equivalent In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
to a line, which we may take to be a projective line; accordingly, we may identify the function field of the curve with the field of rational functions in one indeterminate ''F''(''x''). If ''F'' is algebraically closed, this is equivalent to a curve of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
zero; however, the field of all real algebraic functions defined on the real algebraic variety ''x''2+''y''2 = −1 is a field of genus zero which is not a rational function field. Concretely, a rational curve embedded in an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
of dimension ''n'' over ''F'' can be parameterized (except for isolated exceptional points) by means of ''n''
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s of a single parameter ''t''; by reducing these rational functions to the same denominator, the ''n''+1 resulting polynomials define a ''polynomial parametrization'' of the
projective completion In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
of the curve in the projective space. An example is the
rational normal curve In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the ...
, where all these polynomials are
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
s. Any
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
defined over ''F'' with a
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
in ''F'' is a rational curve. It can be parameterized by drawing a line with slope ''t'' through the rational point, and an intersection with the plane quadratic curve; this gives a polynomial with ''F''-rational coefficients and one ''F''-rational root, hence the other root is ''F''-rational (i.e., belongs to ''F'') also. For example, consider the ellipse ''x''2 + ''xy'' + ''y''2 = 1, where (−1, 0) is a rational point. Drawing a line with slope ''t'' from (−1,0), ''y'' = ''t''(''x''+1), substituting it in the equation of the ellipse, factoring, and solving for ''x'', we obtain x = \frac. Then the equation for ''y'' is y=t(x+1)=\frac\,, which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve. All points of the ellipse are given, except for (−1,1), which corresponds to ''t'' = ∞; the entire curve is parameterized therefore by the real projective line. Such a rational parameterization may be considered in the
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator. As the parameter is defined in a projective line, the polynomials in the parameter should be
homogenized Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
. For example, the projective parameterization of the above ellipse is X=U^2-T^2,\quad Y=T\,(T+2\,U),\quad Z=T^2+TU+U^2. Eliminating ''T'' and ''U'' between these equations we get again the projective equation of the ellipse X^2+X\,Y+Y^2=Z^2, which may be easily obtained directly by homogenizing the above equation. Many of the curves on Wikipedia's
list of curves This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc. Mathematics (Geometry) ...
are rational and hence have similar rational parameterizations.


Rational plane curves

Rational plane curves are rational curves embedded into \mathbb^2. Given generic sections s_1,s_2,s_3 \in \Gamma(\mathbb^1, \mathcal(d)) of degree d homogeneous polynomials in two coordinates, x,y, there is a maps:\mathbb^1 \to \mathbb^2 given by s( :y = [s_1( :y:s_2( :y:s_3( :y]defining a rational plane curve of degree d. There is an associated moduli space \mathcal = \overline_(\mathbb^2, d\cdot [H]) (where [H] is the hyperplane class) parametrizing all such stable curves. A dimension count can be made to determine the moduli spaces dimension: There are d+1 parameters in \Gamma(\mathbb^1, \mathcal(d)) giving 3d+3 parameters total for each of the sections. Then, since they are considered up to a projective quotient in \mathbb^2 there is 1 less parameter in \mathcal. Furthermore, there is a three dimensional group of automorphisms of \mathbb^1, hence \mathcal has dimension 3d + 3 - 1 - 3 = 3d - 1. This moduli space can be used to count the number N_d of degree d rational plane curves intersecting 3d-1 points using Gromov–Witten theory. It is given by the recursive relationN_d = \sum_ N_ N_ d_A^2 d_B\left( d_B\binom - d_A\binom \right)where N_1 = N_2 = 1.


Elliptic curves

An elliptic curve may be defined as any curve of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
one with a
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
: a common model is a nonsingular Cubic plane curve, cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate-Weierstrass form, which in its projective version is y^2z + a_1 xyz + a_3 yz^2 = x^3 + a_2 x^2z + a_4 xz^2 + a_6 z^3. If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting a_1=a_2=a_3=0, which gives the classical Weierstrass form y^2 = x^3 + p x + q. Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law. In a plane cubic model three points sum to zero in the group if and only if they are Line (geometry), collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the Fundamental pair of periods, period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).


Curves of genus greater than one

Curves of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings's theorem, can have only a finite number of rational points, and they may be viewed as having a hyperbolic geometry structure. Examples are the hyperelliptic curves, the Klein quartic, Klein quartic curve, and the Fermat curve when is greater than three. Also projective plane curves in \mathbb^2 and curves in \mathbb^1\times\mathbb^1 provide many useful examples.


Projective plane curves

Plane curves C \subset \mathbb^2 of degree k, which can be constructed as the vanishing locus of a generic section s \in \Gamma(\mathbb^2, \mathcal(k)), has genus \frac which can be computed using Coherent sheaf cohomology. Here's a brief summary of the curves genera relative to their degree For example, the curve x^4 + y^4 + z^4 defines a curve of genus 3 which is Smooth scheme, smooth since the differentials 4x^3, 4y^3, 4z^3 have no common zeros with the curve.. A non-example of a generic section is the curve x(x^2 + y^2 + z^2) which, by Bezouts theorem, should intersect at most 2 points, is the union of two rational curves C_1 \cup C_2 intersecting at two points. Note C_1 is given by the vanishing locus of x and C_2 is given by the vanishing locus of x^2 + y^2 + z^2. These can be found explicitly: a point lies in both if x = 0. So the two solutions are the points [0:y:z] such that y^2 + z^2 = 0, which are [0:1:-\sqrt] and [0: 1: \sqrt].


Curves in product of projective lines

Curve C \subset \mathbb^1\times\mathbb^1 given by the vanishing locus of s \in \Gamma(\mathbb^1\times\mathbb^1, \mathcal(a,b)), for a,b \geq 2, give curves of genusab - a -b + 1which can be checked using Coherent sheaf cohomology. If a = 2, then they define curves of genus 2b -2 -b + 1 = b-1, hence a curve of any genus can be constructed as a curve in \mathbb^1\times\mathbb^1. Their genera can be summarized in the table and for a = 3, this is


See also


Classical algebraic geometry

* Acnode *
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
* Cramer's theorem (algebraic curves) * Crunode * Curve * Curve sketching * Jacobian variety * Klein quartic * List of curves * Hilbert's sixteenth problem * Cubic plane curve * Hyperelliptic curve


Modern algebraic geometry

* Birational geometry * Conic section * Elliptic curve * Fractional ideal * Function field of an algebraic variety * Function field (scheme theory) * Genus (mathematics) * Polynomial lemniscate * Quartic plane curve * Rational normal curve * Riemann–Roch theorem for algebraic curves * Weber's theorem


Geometry of Riemann surfaces

* Riemann–Hurwitz formula * Riemann–Roch theorem for Riemann surfaces *
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...


Notes


References

* * * * * * * * * * * * — gained the 1886 Academy prize {{Authority control Algebraic curves,