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{{short description|Curve defined as zeros of polynomials}}
[[File:Tschirnhausen cubic.svg|thumb|right|The [[Tschirnhausen cubic]] is an algebraic curve of degree three.]]

In [[mathematics]], an '''affine algebraic plane curve''' is the [[zero set]] of a [[polynomial]] in two variables. A '''projective algebraic plane curve''' is the zero set in a [[projective plane]] of a [[homogeneous polynomial]] in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by [[homogenization of a polynomial|homogenizing]] its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation {{math|1=''h''(''x'', ''y'', ''t'') = 0}} can be restricted to the affine algebraic plane curve of equation {{math|1=''h''(''x'', ''y'', 1) = 0}}. These two operations are each [[inverse function|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]] of [[dimension of an algebraic variety|dimension]] one. Equivalently, an algebraic curve is an algebraic variety that is [[birational equivalence|birationally equivalent]] to an algebraic plane curve. If the curve is contained in an [[affine space]] or a [[projective space]], one can take a [[projection (mathematics)|projection]] 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 of an algebraic variety|degree]] and [[Singular point of an algebraic variety|smoothness]]. For example, there exist smooth curves of [[genus (mathematics)|genus]] 0 and degree greater than two, but any plane projection of such curves has singular points (see [[Genus–degree formula]]).

A non-plane curve is often called a ''space curve'' or a ''[[skew curve]]''.

== In Euclidean geometry ==
An algebraic curve in the [[Euclidean plane]] is the set of the points whose [[coordinates]] are the solutions of a bivariate [[polynomial equation]] ''p''(''x'', ''y'') = 0. This equation is often called the [[implicit equation]] 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 [[arc (geometry)|arc]]s (also called ''branches'') sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points called [[acnode]]s. A ''smooth monotone arc'' is the graph of a [[smooth function]] which is defined and [[monotone function|monotone]] on an [[open interval]] 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]], there are two infinite arcs having the origin (0,0) as of endpoint. This point is the only [[mathematical singularity|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 [[Sine wave|sinusoid]] 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]]s (if any) and the way in which the arcs connect them. It is also useful to consider the [[inflection point]]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, after the section [[#Projective curves|Projective curves]].

== Plane projective curves ==

It is often desirable to consider curves in the [[projective space]]. An algebraic curve in the [[projective plane]] or '''plane projective curve''' is the set of the points in a [[projective plane]] whose [[projective coordinates]] are zeros of a [[homogeneous polynomial]] 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 <math>^hp(x,y,z)=0,</math> where

:<math>^hp(x,y,z)=z^{\deg(p)}p(\tfrac{x}{z},\tfrac{y}{z})</math>

is the result of the [[homogeneous polynomial#Homogenization|homogenization]] 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 <math>^hp(x,y,1)=p(x,y)</math> and, if ''p'' is defined by <math>p(x,y)=P(x,y,1)</math>, then <math>^hp(x,y,z)=P(x,y,z),</math>, as soon as the homogeneous polynomial ''P'' is not divisible by ''z''.

For example, the projective curve of equation ''x''<sup>2</sup> + ''y''<sup>2</sup> − ''z''<sup>2</sup> is the projective completion of the [[unit circle]] of equation ''x''<sup>2</sup> + ''y''<sup>2</sup> − 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 <math> p'_x</math> and <math> p'_y</math>, it is useful to consider the '''derivative at infinity'''

:<math> p'_\infty(x,y)={^hp'_z(x,y,1)}.</math>

For example, the equation of the tangent of the affine curve of equation ''p''(''x'', ''y'') = 0 at a point (''a'', ''b'') is

:<math>xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0.</math>

== Remarkable points of a plane curve ==
{{See also|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 <math>P(x,y,z)= {}^hp(x,y,z)</math> 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]]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]] 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]] 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]] asserts that this number is exactly ''d'', if the points are searched in the projective plane over an [[Algebraically closed field|algebraically closed]] [[Field (mathematics)|field]] (for example the [[complex number]]s), and counted with their [[multiplicity (mathematics)|multiplicity]]. 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 <math>(x-a)p'_x(a,b)+(y-b)p'_y(a,b)=0</math>, like for every [[differentiable curve]] defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:

:<math>xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0,</math>

where <math>p'_\infty(x,y)=P'_z(x,y,1)</math> is the derivative at infinity. The equivalence of the two equations results from [[Euler's homogeneous function theorem]] applied to ''P''.

If <math>p'_x(a,b)=p'_y(a,b)=0,</math> 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]] (''a'':''b'':''c'') of the projective curve of equation ''P''(''x'', ''y'', ''z'') = 0 is

:<math>xP'_x(a,b,c)+yP'_y(a,b,c)+zP'_z(a,b,c)=0,</math>

and the points of the curves that are singular are the points such that

:<math>P'_x(a,b,c)=P'_y(a,b,c)=P'_z(a,b,c)=0.</math>

(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]] 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 <math>p=p_d+\cdots+p_0</math> be the decomposition of the polynomial defining the curve into its homogeneous parts, where ''p<sub>i</sub>'' is the sum of the monomials of ''p'' of degree ''i''. It follows that

:<math>P={^hp}=p_d+zp_{d-1}+\cdots+z^dp_0</math>

and

:<math>P'_z(a,b,0) =p_{d-1}(a,b).</math>

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 ''p<sub>d</sub>''. The [[fundamental theorem of algebra]] implies that, over an algebraically closed field (typically, the field of complex numbers), ''p''<sub>''d''</sub> factors into a product of linear factors. Each factor defines a point at infinity on the curve: if ''bx''&nbsp;−&nbsp;''ay'' is such a factor, then it defines the point at infinity (''a'', ''b'', 0). Over the reals, ''p''<sub>''d''</sub> factors into linear and quadratic factors. The [[irreducible polynomial|irreducible]] 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''<sub>''d''</sub> the equation of the corresponding asymptote is

:<math>xq'_x(a,b)+yq'_y(a,b)+p_{d-1}(a,b)=0.</math>

If <math>q'_x(a,b)=q'_y(a,b)=0</math> and <math>p_{d-1}(a,b)\neq 0,</math> the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a [[parabola]]. In this case one says that the curve has a ''parabolic branch''. If

:<math>q'_x(a,b)=q'_y(a,b)=p_{d-1}(a,b)=0,</math>

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 point of a curve|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:
:<math>p'_x(x,y)=p'_y(x,y)=p(x,y)=0.</math>
In [[characteristic (algebra)|characteristic zero]], this system is equivalent to
:<math>p'_x(x,y)=p'_y(x,y)=p'_\infty(x,y)=0,</math>
where, with the notation of the preceding section, <math>p'_\infty(x,y)=P'_z(x,y,1).</math>
The systems are equivalent because of [[Euler's homogeneous function theorem]]. 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
:<math>P'_x(x,y,z)=P'_y(x,y,z)=P'_z(x,y,z)=0</math>
as [[homogeneous coordinates]]. (In positive characteristic, the equation <math>P(x,y,z)</math> 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 polynomial|square free]]. [[Bézout's theorem]] implies thus that the number of singular points is at most (''d''−1)<sup>2</sup>, but this bound is not sharp because the system of equations is [[overdetermined system|overdetermined]]. If [[irreducible polynomial|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]] 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 function|analytic structure]] of an algebraic curve in the [[neighborhood (topology)|neighborhood]] 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 (singularity)|cusp]] or as a smooth curve.

Near a regular point, one of the coordinates of the curve may be expressed as an [[analytic function]] of the other coordinate. This is a corollary of the analytic [[implicit function theorem]], and implies that the curve is [[smooth curve|smooth]] near the point. Near a singular point, the situation is more complicated and involves [[Puiseux series]], which provide analytic [[parametric equation]]s of the branches.

For describing a singularity, it is worth to [[translation (geometry)|translate]] the curve for having the singularity at the origin. This consists of a change of variable of the form <math>X=x-a, Y=y-b,</math> where <math>a, b</math> 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 <math>f(x,y)=0, </math> where {{math|''f''}} is a polynomial in {{math|''x''}} and {{math|''y''}}. This polynomial may be considered as a polynomial in {{math|''y''}}, with coefficients in the algebraically closed field of the [[Puiseux series]] in {{math|''x''}}. Thus {{math|''f''}} may be factored in factors of the form <math>y-P(x),</math> where {{math|''P''}} is a Puiseux series. These factors are all different if {{math|''f''}} is an [[irreducible polynomial]], because this implies that {{math|''f''}} is [[square-free polynomial|square-free]], a property which is independent of the field of coefficients.

The Puiseux series that occur here have the form 
:<math>P(x)=\sum_{n=n_0}^\infty a_nx^{n/d},</math>
where {{mvar|d}} is a positive integer, and {{tmath|n_0}} 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 {{mvar|d}} is [[coprime integers|coprime]] with the greatest common divisor of the {{mvar|n}} such that {{tmath|a_n \ne 0}} (otherwise, one could choose a smaller common denominator for the exponents).

Let {{tmath|\omega_d}} be a [[primitive root of unity|primitive {{mvar|d}}th root of unity]]. If the above Puiseux series occurs in the factorization of {{tmath|1=f(x,y)=0}}, then the {{mvar|d}} series 
:<math>P_i(x)=\sum_{n=n_0}^\infty a_n\omega_d^i x^{n/d}</math>
occur also in the factorization (a consequence of [[Galois theory]]). These {{mvar|d}} series are said [[algebraic conjugate|conjugate]], and are considered as a single branch of the curve, of ''ramification'' index {{mvar|d}}.

In the case of a real curve, that is a curve defined by a polynomial with real coefficients, three cases may occur. If none {{tmath|P_i(x)}} has real coefficients, then one has a non-real branch. If some {{tmath|P_i(x)}} has real coefficients, then one may choose it as {{tmath|P_0(x)}}. If {{mvar|d}} is odd, then every real value of {{mvar|x}} provides a real value of {{tmath|P_0(x)}}, and one has a real branch that looks regular, although  it is singular if {{math|''d'' > 1}}. If {{mvar|d}} is even, then {{tmath|P_0(x)}} and {{tmath|P_{d/2}(x)}} have real values, but only for {{math|''x'' ≥ 0}}. In this case, the real branch looks as a [[cusp (singularity)|cusp]] (or is a cusp, depending of the definition of a cusp that is used).

For example, the ordinary cusp has only one branch. If it is defined by the equation <math>y^2-x^3=0,</math> then the factorization is <math>(y-x^{3/2})(y+x^{3/2});</math> the ramification index is 2, and the two factors are real and define each an half branch. If the cusp is rotated, it equation becomes <math>y^3-x^2=0,</math> and the factorization is <math>(y-x^{2/3})(y-j^2x^{2/3})(y-(j^2)^2x^{2/3}),</math> with <math>j=(1+\sqrt{-3})/2</math> (the coefficient {{tmath|(j^2)^2}} has not been simplified to {{mvar|j}} for showing how the above definition of {{tmath|P_i(x)}} 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]] of [[dimension of an algebraic variety|dimension]] one. This implies that an '''affine curve''' in an [[affine space]] 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]] of [[Krull dimension]] 1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred.

Let <math>f, g_0, g_3, \ldots, g_n</math> be ''n'' polynomials in two variables ''x''<sub>1</sub> and ''x''<sub>2</sub> such that ''f'' is irreducible. The points in the affine space of dimension ''n'' such whose coordinates satisfy the equations and inequations

:<math>\begin{align}
&f(x_1,x_2)=0\\
&g_0(x_1,x_2)\neq 0\\
x_3&=\frac{g_3(x_1,x_2)}{g_0(x_1,x_2)}\\
& {}\ \vdots \\
x_n&=\frac{g_n(x_1,x_2)}{g_0(x_1,x_2)}
\end{align}</math>

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 <math>g_0^kh</math> belongs to the ideal generated by <math>f, x_3g_0-g_3, \ldots, x_ng_0-g_n</math>.
This representation is a [[birational equivalence]] 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 (mathematics)|projection]] 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]] for a [[Monomial order#Block order|block ordering]] such that the block of the smaller variables is (''x''<sub>1</sub>, ''x''<sub>2</sub>). The polynomial ''f'' is the unique polynomial in the base that depends only of ''x''<sub>1</sub> and ''x''<sub>2</sub>. The fractions ''g<sub>i</sub>''/''g''<sub>0</sub> are obtained by choosing, for ''i'' = 3, ..., ''n'', a polynomial in the basis that is linear in ''x<sub>i</sub>'' and depends only on ''x''<sub>1</sub>, ''x''<sub>2</sub> and ''x<sub>i</sub>''. If these choices are not possible, this means either that the equations define an [[algebraic set]] 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''<sub>1</sub>, ''x''<sub>2</sub> and ''x<sub>i</sub>''.

== Algebraic function fields ==
The study of algebraic curves can be reduced to the study of [[irreducible component|irreducible]] algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to [[birational geometry|birational]] equivalence, the irreducible curves over a field ''F'' are [[equivalence of categories|categorically equivalent]] to [[Function field of an algebraic variety|algebraic function field]]s in one variable over ''F''. Such an algebraic function field is a [[field extension]] ''K'' of ''F'' that contains an element ''x'' which is [[algebraic element|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&nbsp;''F''.

For example, consider the field '''C''' of complex numbers, over which we may define the field '''C'''(''x'') of rational functions in&nbsp;'''C'''. If
''y''<sup>2</sup>&nbsp;=&nbsp;''x''<sup>3</sup>&nbsp;−&nbsp;''x''&nbsp;−&nbsp;1, then the field '''C'''(''x'',&nbsp;''y'') is an [[elliptic function|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'',&nbsp;''y'') in '''C'''<sup>2</sup> satisfying ''y''<sup>2</sup>&nbsp;=&nbsp;''x''<sup>3</sup>&nbsp;−&nbsp;''x''&nbsp;−&nbsp;1.

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 ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;−1 defines an algebraic extension field of '''R'''(''x''), but the corresponding curve considered as a subset of '''R'''<sup>2</sup> has no points. The equation ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;−1 does define an irreducible algebraic curve over '''R''' in the [[scheme (mathematics)|scheme]] sense (an [[Glossary of scheme theory#integral|integral]], [[Glossary of scheme theory#Separated and proper morphisms|separated]] [[Glossary of scheme theory#dimension|one-dimensional]] [[Scheme (mathematics)|scheme]]s of [[Glossary of scheme theory#Finite, quasi-finite, and finite type morphisms|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 [[Isomorphism|isomorphic]] 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]] 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'''<sup>''n''</sup>. This has complex dimension ''n'', but topological dimension, as a real [[manifold]], 2''n'', and is [[Compact space|compact]], [[connected space|connected]], and [[Orientability|orientable]]. An algebraic curve over '''C''' likewise has topological dimension two; in other words, it is a [[Surface (topology)|surface]].

The [[Genus (mathematics)|topological genus]] of this surface, that is the number of handles or donut holes, is equal to the [[geometric genus]] 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 of a polynomial|degree]] ''d'' and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is (''d''&nbsp;&minus;&nbsp;1)(''d''&nbsp;&minus;&nbsp;2)/2&nbsp;&minus;&nbsp;''k'', where ''k'' is the number of these singularities.

=== Compact Riemann surfaces ===
A Riemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is [[Compact space|compact]] if it is compact as a topological space.

There is a triple [[equivalence of categories]] between the category of smooth irreducible projective algebraic curves over '''C''' (with non-constant [[morphism of varieties|regular maps]] as morphisms), the category of compact Riemann surfaces (with non-constant [[holomorphic map]]s as morphisms), and the [[opposite category|opposite]] of the category of [[algebraic function field]]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]], points ''P'' on an algebraic curve ''C'' are classified as ''smooth'' (synonymous: ''non-singular''), or else ''[[Singular point of a curve|singular]]''. Given ''n''−1 homogeneous polynomials in ''n''+1 variables, we may find the [[Jacobian matrix and determinant#Jacobian matrix|Jacobian matrix]] as the (''n''−1)×(''n''+1) matrix of the partial derivatives. If the [[rank (linear algebra)|rank]] 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'')&nbsp;= 0, then the singular points are precisely the points ''P'' where the rank of the 1×(''n''+1) matrix is zero, that is, where

:<math>\frac{ \partial f }{ \partial x }(P)=\frac{ \partial f }{ \partial y }(P)=\frac{ \partial f }{ \partial z }(P)=0.</math>

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'')&nbsp;= 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

:<math>f(P)=\frac{ \partial f }{ \partial x }(P)=\frac{ \partial f }{ \partial y }(P)=0.</math>

The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the [[geometric genus|genus]], 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 ===
[[File:Cusp.svg|thumb|right|''x''<sup>3</sup>&nbsp;= ''y''<sup>2</sup>]]
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''<sup>3</sup>&nbsp;= ''y''<sup>2</sup> 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]] (i.e., ''complete'' in the sense of algebraic geometry). For example, the plane curve of equation <math>y-x^3=0</math> is considered as singular, as having a singular point (a cusp) at infinity.

''In the remainder of this section, one considers a plane curve {{mvar|C}} defined as the zero set of a bivariate polynomial'' {{math|''f''(''x'', ''y'')}}. 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 {{math|''m''}} is defined as the maximum integer such that the derivatives of {{math|''f''}} to all orders up to {{math|''m'' – 1}} vanish (also the minimal [[intersection number]]  between the curve and a straight line at {{math|''P''}}).
Intuitively, a singular point has delta invariant {{mvar|δ}} if it concentrates {{mvar|δ}} ordinary double points at {{math|''P''}}. To make this precise, the [[Blowing up|blow up]] process produces so-called [[infinitely near point]]s, and summing {{math|''m''(''m''−1)/2}} over the infinitely near points, where ''m'' is their multiplicity, produces {{mvar|δ}}.
For an irreducible and reduced curve and a point {{math|''P''}} we can define {{mvar|δ}} algebraically as the length of <math>\widetilde{\mathcal{O}_P} / \mathcal{O}_P</math> where <math>\mathcal{O}_P</math> is the local ring at ''P'' and <math>\widetilde{\mathcal{O}_P}</math> is its integral closure.<ref>Hartshorne, Algebraic Geometry, IV Ex. 1.8.</ref>

The [[Milnor number]] {{mvar|μ}} of a singularity is the degree of the mapping {{math|{{sfrac|grad ''f''(''x'',''y'')|{{!}}grad&nbsp;''f''(''x'',''y''){{!}}}}}} on the small sphere of radius ε, in the sense of the topological [[degree of a continuous mapping]], where {{math|grad&nbsp;''f''}} is the (complex) gradient vector field of ''f''. It is related to δ and ''r'' by the [[Milnor–Jung formula]],

:μ = 2δ − ''r'' + 1.

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 (mathematics)|genus]] ''g'' of the curve to be determined; if ''d'' is the degree, then

:<math>g = \frac{1}{2}(d-1)(d-2) - \sum_P \delta_P,</math>

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 [2,1,1] and an ''ordinary double point'' is a point with invariants [2,1,2], 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 [[birational geometry|birationally equivalent]] 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 (mathematics)|genus]] zero; however, the field of all real algebraic functions defined on the real algebraic variety ''x''<sup>2</sup>+''y''<sup>2</sup>&nbsp;= −1 is a field of genus zero which is not a rational function field.

Concretely, a rational curve embedded in an [[affine space]] of dimension ''n'' over ''F'' can be parameterized (except for isolated exceptional points) by means of ''n'' [[rational function]]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]] of the curve in the  projective space. An example is the
[[rational normal curve]], where all these polynomials are [[monomial]]s.

Any [[conic section]] defined over ''F'' with a [[rational point]] 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.

[[File:Rotated ellipse.svg|thumb|right|''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup> = 1]]
For example, consider the ellipse ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup>&nbsp;= 1, where (−1, 0) is a rational point. Drawing a line with slope ''t'' from (−1,0), ''y''&nbsp;= ''t''(''x''+1), substituting it in the equation of the ellipse, factoring, and solving for ''x'', we obtain

:<math>x = \frac{1-t^2}{1+t+t^2}.</math>

Then the equation for ''y'' is

:<math>y=t(x+1)=\frac{t(t+2)}{1+t+t^2}\,,</math>

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''&nbsp;= ∞; the entire curve is parameterized therefore by the real projective line.

Such a rational parameterization may be considered in the [[projective space]] 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 [[Homogeneous polynomial#Homogenization|homogenized]]. For example, the projective parameterization of the above ellipse is

:<math>X=U^2-T^2,\quad Y=T\,(T+2\,U),\quad Z=T^2+TU+U^2.</math>

[[Elimination theory|Eliminating]] ''T'' and ''U'' between these equations we get again the projective equation of the ellipse

:<math>X^2+X\,Y+Y^2=Z^2,</math>

which may be easily obtained directly by homogenizing the above equation.

Many of the curves on Wikipedia's [[list of curves]] are rational and hence have similar rational parameterizations.

=== Elliptic curves ===
An [[elliptic curve]] may be defined as any curve of [[genus (mathematics)|genus]] one with a [[rational point]]: 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

:<math>y^2z + a_1 xyz + a_3 yz^2 = x^3 + a_2 x^2z + a_4 xz^2 + a_6 z^3.</math>

If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting <math>a_1=a_2=a_3=0,</math> which gives the classical Weierstrass form
:<math>y^2 = x^3 + p x + q.</math>

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 function]]s.

The intersection of two [[quadric surface]]s 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 (mathematics)|genus]] 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 curve]]s, the [[Klein quartic|Klein quartic curve]], and the [[Fermat curve]] ''x''<sup>''n''</sup>&nbsp;+&nbsp;''y''<sup>''n''</sup>&nbsp;= ''z''<sup>''n''</sup> when ''n'' is greater than three.

== See also ==

===Classical algebraic geometry===
{{div col|colwidth=22em}}
* [[Acnode]]
* [[Bézout's theorem]]
* [[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]]
{{div col end}}

===Modern algebraic geometry===
{{div col|colwidth=22em}}
* [[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]]
{{div col end}}

===Geometry of Riemann surfaces===
{{div col|colwidth=22em}}
* [[Riemann–Hurwitz formula]]
* [[Riemann–Roch theorem for Riemann surfaces]]
* [[Riemann surface]]
{{div col end}}

== Notes ==
{{Reflist}}

== References ==
{{Commons category|Algebraic curves}}
* Egbert Brieskorn and [[Horst Knörrer]], ''Plane Algebraic Curves'', [[John Stillwell]], translator, [[Birkhäuser]], 1986
* [[Claude Chevalley]], ''Introduction to the Theory of Algebraic Functions of One Variable'', [[American Mathematical Society]], Mathematical Surveys Number VI, 1951
* Hershel M. Farkas and Irwin Kra, ''Riemann Surfaces'', Springer, 1980
* W. Fulton, [http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf ''Algebraic Curves: an introduction to algebraic geometry''].
* C.G. Gibson, ''Elementary Geometry of Algebraic Curves: An Undergraduate Introduction'', [[Cambridge University Press]], 1998.
* [[Phillip A. Griffiths]], ''Introduction to Algebraic Curves'', Kuniko Weltin, trans., American Mathematical Society, Translation of Mathematical Monographs volume 70, 1985 revision
* [[Robin Hartshorne]], ''Algebraic Geometry'', Springer, 1977
* [[Shigeru Iitaka]], ''Algebraic Geometry: An Introduction to the Birational Geometry of Algebraic Varieties'', Springer, 1982
* [[John Milnor]], ''Singular Points of Complex Hypersurfaces'', [[Princeton University Press]], 1968
* [[George Salmon]], ''Higher Plane Curves'', Third Edition, G. E. Stechert & Co., 1934
* [[Jean-Pierre Serre]], ''Algebraic Groups and Class Fields'', Springer, 1988
* {{cite journal| author=[[Ernst Kötter]]| title=Grundzüge einer rein geometrischen Theorie der algebraischen ebenen Kurven (Fundamentals of a purely geometrical theory of algebraic plane curves)| journal=Transactions of the [[Royal Academy of Berlin]]| year=1887}} &mdash; gained the 1886 Academy prize<ref>{{cite journal| author=Norman Fraser| title=Kötter's synthetic geometry of algebraic curves| journal=Proceedings of the Edinburgh Mathematical Society|date=Feb 1888| volume=7| pages=46–61| url=http://journals.cambridge.org/action/displayFulltext?type=1&fid=3146652&jid=PEM&volumeId=7&issueId=-1&aid=3131388&bodyId=&membershipNumber=&societyETOCSession=}} Here: p.46</ref>

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