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| In [[mathematics]], '''algebraic curves''' are the simplest objects of [[Euclidean geometry]] that can not be defined by [[linear]] properties. Specifically, in Euclidean geometry, a '''plane algebraic curve''' is the [[set (mathematics)|set]] of the points of the [[Euclidean plane]] whose [[coordinates]] are [[zero of a function|zeros]] of some [[polynomial]] in two variables.
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| For example, the [[unit circle]] is an algebraic curve, being the set of zeros of the polynomial ''x''<sup>2</sup> + ''y''<sup>2</sup> − 1
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| Various technical considerations have led to consider that the [[complex number|complex]] zeros of a polynomial belong to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to any [[field (mathematics)|field]], leading to the following definition.
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| In [[algebraic geometry]], a '''plane affine algebraic curve''' defined over a field ''k'' is the set of points of ''K''<sup>2</sup> whose coordinates are zeros of some [[polynomial#Number of variables|bivariate polynomial]] with coefficients in ''k'', where ''K'' is some [[algebraically closed field|algebraically closed extension]] of ''k''. The points of the curve with coordinates in ''k'' are the ''k''-points of the curve and, all together, are the ''k'' part of the curve.
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| For example, <math>(2,\sqrt{-3})</math> is a point of the curve defined by ''x''<sup>2</sup> + ''y''<sup>2</sup> − 1 =0 and the usual unit circle is the real part of this curve. The term "unit circle" may refer to all the complex points as well to only the real points, the exact meaning being usually clear form the context. The equation ''x''<sup>2</sup> + ''y''<sup>2</sup> + 1 = 0 defines an algebraic curve, whose real part is empty.
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| More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a [[skew curve]]. The simplest example of a skew algebraic curve is the [[twisted cubic]]. One may also consider algebraic curves contained in the [[projective space]] and even algebraic curves that are defined independently to any embedding in an [[affine space|affine]] or projective space. This leads to the most general definition of an algebraic curve:
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| In [[algebraic geometry]], an '''algebraic curve''' is an [[algebraic variety]] of [[dimension of an algebraic variety|dimension]] one.
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| [[File:Tschirnhausen cubic.svg|thumb|450px|right|The [[Tschirnhausen cubic]] is an algebraic curve of degree three.]]
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| == In Euclidean geometry ==
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| 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, by opposition to the curves that are the graph of a function defining ''explicitly'' ''y'' as a function of ''x''.
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| Given a curve given by such an implicit equation, the first problems that occur is 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 to solve these problems.
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| Every algebraic curve may be uniquely decomposed into a finite numbers smooth monotone [[arc (geometry)|arc]]s (also called ''branches'') connected by some points sometimes called "remarkable points". 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 (one talk of an ''infinite arc'') or has an end point which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes.
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| For example, for the [[Tschirnhausen cubic]] of the figure, there are two infinite arcs having the origin (0,0) as end point. This point is the only singular point of the curve. There are two arcs having this singular point as one end point and having a second end point with a horizontal tangent. Finally, there two other arcs having these points with horizontal tangent as first end point and sharing the unique point with vertical tangent as second end point. On the other hand, the [[Sine wave|sinusoid]] is certainly not an algebraic curve, having an infinite number of monotone arcs.
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| To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their [[asymptote]] (if any) and the way in which the arcs connect them. It is also useful to consider also the [[inflection point]]s as remarkable points. When all this information is drawn on a paper sheet, the shape of the curve appears usually rather clearly. If not it suffices to add a few other points and their tangents to get a good description of the curve.
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| The methods for computing the remarkable points and their tangents are described below, after section [[#Projective curves|Projective curves]].
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| == Plane projective curves ==
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| 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'').
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| 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
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| :<math>^hp(x,y,z)=z^{\deg(p)}p(\tfrac{x}{z},\tfrac{y}{z})</math>
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| 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 the restriction of the projective curve to the affine plane of the points 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 <math>^hP(x,y,1)=P(x,y,z),</math> as soon as the homogeneous polynomial ''P'' is not divisible by ''z''.
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| 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.
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| This allows to consider that an affine curve and its projective completion are the same curve, 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.
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| 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'''
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| :<math> p'_\infty(x,y)=^hp'_z(x,y,1).</math>
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| For example, the equation of the tangent of the affine curve of equation ''p''(''x'', ''y'') = 0 at a point (''a'', ''b'') is
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| :<math>xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0.</math>
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| == Remarkable points of a plane curve ==
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| {{expand section|date=October 2012}}
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| 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''.
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| ===Intersection with a line===
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| 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 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.
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| If the polynomial defining the curve has 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 (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.
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| To compute the intersection of the curve defined by the polynomial ''p'' with the line of equation ''ax''+''by''+''c'' = 0, one solves in ''x'' (or in ''y'' if ''a'' = 0) the equation of the line. Substituting the result in ''p'', one gets a univariate equation 1 = ''q''(''y'') = 0 (or 1 = ''q''(''x'') = 0), whose roots are one coordinate of the intersection points. 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''.
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| ===Tangent at a point===
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| 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:
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| :<math>xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0,</math>
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| 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''.
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| If <math>p'_x(a,b)=p'_y(a,b)=0,</math> the tangent is not defined and the point is a '''singular point'''.
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| 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
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| :<math>xP'_x(a,b,c)+yP'_y(a,b,c)+zP'_z(a,b,c)=0,</math>
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| and the points of the curves that are singular are the points such that
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| :<math>P'_x(a,b,c)=P'_y(a,b,c)=P'_z(a,b,c)=0.</math>
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| (The condition ''P''(''a'', ''b'', ''c'') = 0 is implied by these conditions, by Euler's homogeneous function theorem.)
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| ===Asymptotes===
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| 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 belongs 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.
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| 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
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| :<math>P={^hp}=p_d+zp_{d-1}\cdots+z^dp_0</math>
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| :<math>P'_z(a,b,0) =p_{d-1}(a,b).</math>
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| 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''−''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.
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| 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
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| :<math>xq'_x(a,b)+yq'_y(a,b)+p_{d-1}(a,b)=0.</math>
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| 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
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| :<math>q'_x(a,b)=q'_y(a,b)=p_{d-1}(a,b)=0,</math>
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| 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.
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| ===Critical points===
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| {{expand section|date=October 2012}}
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| ===Singular points===
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| 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:
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| :<math>p'_x(x,y)=p'_y(x,y)=p(x,y)=0.</math>
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| In [[characteristic (algebra)|characteristic zero]], this system is equivalent with
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| :<math>p'_x(x,y)=p'_y(x,y)=p'_\infty(x,y)=0,</math>
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| where, with the notation of the preceding section, <math>p'_\infty(x,y)=P'_z(x,y,1).</math>
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| 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''.
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| 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
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| :<math>P'_x(x,y,z)=P'_y(x,y,z)=P'_z(x,y,z)=0.</math>
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| as [[homogeneous coordinates]]. (In positive characteristic, the equation <math>P(x,y,z)</math> has to be added to the system.)
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| This implies that the number of singular points is finite as soon 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 being 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).
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| The equation of the tangents at a singular point are given by the nonzero homogeneous part of 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 lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.
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| ===Inflection points===
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| {{expand section|date=October 2012}}
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| == Non plane algebraic curves ==
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| 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.
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| Let <math>f, g_0, g_3, \ldots, g_n</math> be ''n''−1 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
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| :<math>\begin{align}
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| &f(x_1,x_2)=0\\
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| &g_0(x_1,x_2)\neq 0\\
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| x_3&=\frac{g_3(x_1,x_2)}{g_0(x_1,x_2)}\\
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| \vdots &\\
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| x_n&=\frac{g_n(x_1,x_2)}{g_0(x_1,x_2)}
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| \end{align}</math>
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| 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>.
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| This representation is a [[rational 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.
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| This representation allows to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection.
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| 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>''.
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| == Algebraic function fields ==
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| The study of algebraic curves can be reduced to the study of [[irreducible component|irreducible]] algebraic curves. Up to [[birational geometry|birational]] equivalence, these are [[equivalence of categories|categorically equivalent]] to [[Function field of an algebraic variety|algebraic function field]]s. An algebraic function field is a field of algebraic functions in one variable ''K'' defined over a given field ''F''. This means there exists an element ''x'' of ''K'' 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''.
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| For example, consider the field '''C''' of complex numbers, over which we may define the field '''C'''(''x'') of rational functions in '''C'''. If
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| ''y''<sup>2</sup> = ''x''<sup>3</sup> − ''x'' − 1, then the field '''C'''(''x'', ''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'', ''y'') in '''C'''<sup>2</sup> satisfying ''y''<sup>2</sup> = ''x''<sup>3</sup> − ''x'' − 1.
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| If the field ''F'' is not [[Algebraically closed field|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. If the base field ''F'' is the field '''R''' of real numbers, then ''x''<sup>2</sup> + ''y''<sup>2</sup> = −1 defines an algebraic extension field of '''R'''(''x''), but the corresponding curve considered as a locus has no points in '''R'''. However, it does have points defined over the algebraic closure '''C''' of '''R'''.
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| == Complex curves and real surfaces ==
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| 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 likewise has topological dimension two; in other words, it is a [[surface]]. A nonsingular complex projective algebraic curve will then be a smooth orientable surface as a real manifold, embedded in a compact real manifold of dimension 2''n'' which is '''CP'''<sup>''n''</sup> regarded as a real manifold.
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| The [[Genus (mathematics)|topological genus]] of this surface, that is the number of handles or donut holes, is equal to the [[Geometric genus|genus]] of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a non singular 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'' - 1)(''d'' - 2)/2 - ''k'', where ''k'' is the number of these singularities.
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| === Compact Riemann surfaces ===
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| The theory of compact [[Riemann surface]]s consists in studying non singular complex algebraic curves through the complex analytic structure induced on this real compact surface.
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| 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.
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| There is a triple equivalence of categories between the category of smooth projective algebraic curves over the complex numbers (with [[rational map]]s as morphisms), the category of compact [[Riemann surface]]s, and the category of complex algebraic function fields, so that in studying these subjects we are in a sense studying the same thing. This 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, which is characteristic of a much wider class of problems than simply curves and Riemann surfaces.
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| See also [[Algebraic geometry and analytic geometry]], as more general theory.
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| == Singularities ==
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| Using the intrinsic concept of [[tangent space]], points ''P'' on an algebraic curve ''C'' are classified as ''smooth'' or ''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'') = 0, then the singular points are precisely the points ''P'' where the rank of the 1×(''n''+1) matrix is zero, that is, where
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| :<math>\frac{ \partial f }{ \partial x }(P)=\frac{ \partial f }{ \partial y }(P)=\frac{ \partial f }{ \partial z }(P)=0.</math>
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| 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.
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| 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
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| :<math>f(P)=\frac{ \partial f }{ \partial x }(P)=\frac{ \partial f }{ \partial y }(P)=0.</math> | |
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| 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.
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| === Classification of singularities ===
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| [[File:Cusp.svg|thumb|right|''x''<sup>3</sup> = ''y''<sup>2</sup>]]
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| 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> = ''y''<sup>2</sup> at (0,0).
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| A curve ''C'' has at most a finite number of singular points. If it has none, it can be called ''smooth'' or ''non-singular''. For this definition to be correct, we must use an [[algebraically closed field]] and a curve ''C'' in [[projective space]] (i.e., ''complete'' in the sense of algebraic geometry). If, for example, we simply look at a curve in the real affine plane there might be singular ''P'' modulo the stalk, or alternatively as the sum of ''m''(''m''−1)/2, where ''m'' is the multiplicity, over all infinitely near singular points ''Q'' lying over the singular point ''P''. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at ''P''. For an irreducible and reduced curve and a point ''P'' we can define 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. See also Hartshorne, Algebraic Geometry, IV Ex. 1.8.
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| The [[Milnor number]] μ of the singularity is the degree of the mapping grad ''f''(''x'',''y'')/|grad ''f''(''x'',''y'')| on the small sphere of radius ε, in the sense of the topological [[degree of a continuous mapping]], where grad ''f'' is the (complex) gradient vector field of ''f''. It is related to δ and ''r'' by the [[Milnor-Jung formula]],
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| :μ = 2δ − ''r'' + 1.
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| Another singularity invariant of note is the multiplicity ''m'', defined as the maximum integer such that the derivatives of ''f'' to all orders up to ''m'' vanish.
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| Computing the delta invariants of all of the singularities allows the genus ''g'' of the curve to be determined; if ''d'' is the degree, then
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| :<math>g = \frac{1}{2}(d-1)(d-2) - \sum_P \delta_P,</math>
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| where the sum is taken over all singular points ''P'' of the complex projective plane curve. It is called the [[genus formula]].
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| Singularities may be classified by the triple [''m'', δ, ''r''], where ''m'' is the multiplicity, δ is the delta-invariant, and ''r'' is the branching number. In these terms, an ''ordinary cusp'' is a point with invariants [2,1,1] and an ''ordinary double point'' is a point with invariants [2,1,2]. An ordinary n-multiple point may be defined as one having invariants [''n'', ''n''(''n''−1)/2, ''n''].
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| == Examples of curves ==
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| === Rational curves ===
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| 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> = −1 is a field of genus zero which is not a rational function field.
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| Concretely, a rational curve of dimension ''n'' over ''F'' can be parameterized (except for isolated exceptional points) by means of ''n'' rational functions defined in terms of a single parameter ''t''; by clearing denominators we can turn this into ''n''+1 polynomial functions in projective space. An example would be the
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| [[rational normal curve]].
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| 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 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.
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| [[File:Rotated ellipse.svg|thumb|right|''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup> = 1]]
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| For example, consider the ellipse ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup> = 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
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| :<math>x = \frac{1-t^2}{1+t+t^2}.</math>
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| We then have that the equation for ''y'' is
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| :<math>y=t(x+1)=\frac{t(t+2)}{1+t+t^2}\,,</math>
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| 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.
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| 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 above ellipse is
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| :<math>X=U^2-T^2,\quad Y=T\,(T+2\,U),\quad Z=T^2+TU+U^2.</math>
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| [[Elimination theory|Eliminating]] ''T'' and ''U'' between these equations we get again the projective equation of the ellipse
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| :<math>X^2+X\,Y+Y^2=Z^2,</math>
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| which may be easily obtained directly by homogenizing above equation.
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| Many of the curves on Wikipedia's [[list of curves]] are rational, and hence have similar rational parameterizations.
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| === Elliptic curves ===
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| 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
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| :<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>
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| 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.
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| 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 a line, or two conics, or a conic and two lines, or four lines).
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| === Curves of genus greater than one ===
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| 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' 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>+''y''<sup>''n''</sup> = ''z''<sup>''n''</sup> when ''n'' is greater than three.
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| == See also ==
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| <div style="-moz-column-count:3; column-count:3;">
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| ===Classical algebraic geometry===
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| * [[Acnode]]
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| * [[Bézout's theorem]]
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| * [[Crunode]]
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| * [[Curve]]
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| * [[Curve sketching]]
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| * [[Jacobian variety]]
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| * [[Klein quartic]]
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| * [[List of curves]]
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| * [[Hilbert's sixteenth problem]]
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| * [[Cubic plane curve]]
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| * [[Hyperelliptic curve]]
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| ===Modern algebraic geometry===
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| * [[Birational geometry]]
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| * [[Conic section]]
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| * [[Elliptic curve]]
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| * [[Fractional ideal]]
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| * [[Function field of an algebraic variety]]
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| * [[Function field (scheme theory)]]
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| * [[Genus (mathematics)]]
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| * [[Riemann–Roch theorem for algebraic curves]]
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| * [[Quartic plane curve]]
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| * [[Rational normal curve]]
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| * [[Weber's theorem]]
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| ===Geometry of Riemann surfaces===
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| * [[Riemann–Hurwitz formula]]
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| * [[Riemann-Roch theorem for Riemann surfaces]]
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| * [[Riemann surface]]
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| </div>
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| == References ==
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| {{Commons category|Algebraic curves}}
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| * Egbert Brieskorn and [[Horst Knörrer]], ''Plane Algebraic Curves'', [[John Stillwell]], translator, [[Birkhäuser]], 1986
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| * [[Claude Chevalley]], ''Introduction to the Theory of Algebraic Functions of One Variable'', [[American Mathematical Society]], Mathematical Surveys Number VI, 1951
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| * Hershel M. Farkas and Irwin Kra, ''Riemann Surfaces'', Springer, 1980
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| * W. Fulton, ''Algebraic Curves: an introduction to algebraic geometry'' available at <ref>[http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Algebraic Curves: an introduction to algebraic geometry]</ref>
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| * C.G. Gibson, ''Elementary Geometry of Algebraic Curves: An Undergraduate Introduction'', [[Cambridge University Press]], 1998.
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| * [[Phillip A. Griffiths]], ''Introduction to Algebraic Curves'', Kuniko Weltin, trans., American Mathematical Society, Translation of Mathematical Monographs volume 70, 1985 revision
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| * [[Robin Hartshorne]], ''Algebraic Geometry'', Springer, 1977
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| * [[Shigeru Iitaka]], ''Algebraic Geometry: An Introduction to the Birational Geometry of Algebraic Varieties'', Springer, 1982
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| * [[John Milnor]], ''Singular Points of Complex Hypersurfaces'', [[Princeton University Press]], 1968
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| * [[George Salmon]], ''Higher Plane Curves'', Third Edition, G. E. Stechert & Co., 1934
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| * [[Jean-Pierre Serre]], ''Algebraic Groups and Class Fields'', Springer, 1988
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| * [[Claire Voisin]] ''LECTURES ON THE HODGE AND GROTHENDIECK–HODGE CONJECTURES'';<ref>[http://seminariomatematico.dm.unito.it/rendiconti/69-2/149.pdf LECTURES ON THE HODGE AND GROTHENDIECK–HODGE CONJECTURES]</ref> ''ANTICANONICAL DIVISORS AND CURVE CLASSES ON FANO MANIFOLDS'';<ref>[http://www.math.polytechnique.fr/~voisin/Articlesweb/fano-hodge.pdf ANTICANONICAL DIVISORS AND CURVE CLASSES ON FANO MANIFOLDS]</ref> Voisin C. ''Hodge theory and complex algebraic geometry 1'';<ref>[http://librarum.org/book/1074/134 Hodge theory and complex algebraic geometry 1]</ref> ''Green's canonical syzygy conjecture for generic curves of odd genus'';<ref>[http://www.math.polytechnique.fr/~voisin/Articlesweb/syzod.pdf Green's canonical syzygy conjecture for generic curves of odd genus]</ref> ''Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface'' <ref>[http://www.math.polytechnique.fr/~voisin/Articlesweb/syzy.pdf Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface]</ref>
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| * [[Montserrat Teixidor i Bigas]] ''ON A CONJECTURE OF LANGE'';<ref>[http://www.academia.edu/419227/On_a_Conjecture_of_Lange ON A CONJECTURE OF LANGE]</ref> ''Moduli spaces of vector bundles on reducible curves'';<ref>[http://www.math.ucdavis.edu/~osserman/math/montserrat-Clay.pdf Moduli spaces of vector bundles on reducible curves]</ref> ''Green’s Conjecture for the generic r-gonal curve of genus g ¸ 3r ¡ 7'' <ref>[http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1087575039 Green’s Conjecture for the generic r-gonal curve of genus g ¸ 3r ¡ 7]</ref>
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| <br>
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| {{Reflist}}
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| {{Algebraic curves navbox}}
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| [[Category:Algebraic curves| ]]
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