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In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
Circle illustration This article is about the shape and mathematical concept of circle. ...
It has been suggested that Latus rectum be merged into this article or section. ...
Plane algebraic curves An algebraic curve defined over a field F may be considered as the locus of points in Fn determined by n−1 independent polynomial functions in n variables with coefficients in F, gi(x1, …, xn), where the curve is defined by setting each gi = 0.
Using the resultant, we can eliminate all but two of the variables and reduce the curve to a birationally equivalent plane curve, f(x,y) = 0, still with coefficients in F, but usually of higher degree, and often possessing additional singularities. For example, eliminating z between the two equations x2+y2−z2 = 0 and x+2y+3z−1 = 0, which defines an intersection of a cone and a plane in three dimensions, we obtain the conic section 8x2+5y2−4xy+2x+4y−1 = 0, which in this case is an ellipse. If we eliminate z between 4x2+y2−z2 = 1 and z = x2, we obtain y2 = x4−4x2+1, which is the equation of an hyperelliptic curve. For the technique in organ building, see Resultant (organ). ...
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...
It has been suggested that Latus rectum be merged into this article or section. ...
In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ...
Projective curves It is often desirable to consider that curves are a locus of points in projective space. In the set of equations gi = 0, we can replace each xk with xk/x0, and multiply by x0n, where n is the degree of gi. In this way we obtain n−1 homogeneous polynomial functions, which define the corresponding curve in projective space. For a plane algebraic curve we have a single equation f(x,y,z) = 0, where f is homogeneous; for example, the Fermat curve xn+yn+zn = 0 is a projective curve. In mathematics, a projective space is a fundamental construction from any vector space. ...
In mathematics, homogeneous has a variety of meanings. ...
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation Xn + Yn = Zn. ...
Algebraic function fields The study of algebraic curves can be reduced to the study of irreducible algebraic curves. Up to birational equivalence, these are categorically equivalent to algebraic function fields. 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. In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation XY = 0 is the union of the two lines X = 0 and Y = 0. ...
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...
In category theory, an abstract branch of 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 from many areas of mathematics. ...
In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ...
For example, consider the field C of complex numbers, over which we may define the field C(x) of rational functions in C. If y2 = x3−x−1, 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 y2 = x3−x−1. In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...
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. If the base field F is the field R of real numbers, then x2+y2 = −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. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ...
Complex curves and real surfaces A complex projective algebraic curve resides in n-dimensional complex projective space CPn. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve likewise has topological dimension two; in other words, it is a surface. A nonsingular n-dimensional complex projective algebraic curve will then be a smooth orientable surface as a real manifold, embedded in a compact real manifold of dimension 2n which is CPn regarded as a real manifold. The topological genus of this surface, that is the number of handles or donut holes, is the genus of the curve. By considering the complex analytic structure induced on this compact surface we are led to the theory of compact Riemann surfaces. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...
The torus is an orientable surface. ...
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
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 if it is compact as a topological space. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
There is a triple equivalence of categories between the category of irreducible projective algebraic curves over the complex numbers, the category of compact Riemann surfaces, 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. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...
Singularities Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth or non-singular, or else singular. Given n−1 homogeneous polynomial functions in n+1 variables, we may find the Jacobian matrix as the (n−1)×(n+1) matrix of partial derivatives. If the rank of this matrix at a point P on the curve has the maximal value of n−1, then the point is a smooth 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 The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...
A singular point on an curve is one where it is not smooth, for example, at a cusp. ...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...
 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.
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, 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. In mathematics, the geometric genus in algebraic geometry is a basic birational invariant pg of algebraic varieties, defined for non-singular complex projective varieties (and more generally for complex manifolds) as the Hodge number hn,0 (equal to h0,n by Serre duality). ...
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 x3 = y2 at (0,0). Image File history File links Cusp. ...
Image File history File links Cusp. ...
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. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ...
In mathematics, a projective space is a fundamental construction from any vector space. ...
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 μ = 2δ−r+1. Another singularity invariant of note is the multiplicity m, defined as the maximum integer such that
 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  where the sum is taken over all singular points P of the complex projective plane curve.
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].
Examples of curves
Rational curves A rational curve, also called a unicursal curve, is any curve which is birationally equivalent to a line, which we may take to be a projective line and identify with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivalent to a curve of genus zero; however the field R(x,y) with x2+y2 = −1 is a field of genus zero which is not a rational function field. In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
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 rational normal curve. In mathematics, the rational normal curve is a smooth, rational curve of degree n in projective n-space . ...
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. It has been suggested that Latus rectum be merged into this article or section. ...
For example, consider the ellipse x2 + xy + y2 = 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  We then have that the equation for y is  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.
Viewing rational parameterizations with rational coefficients projectively, we can view them as giving number theoretical information about homogeneous equations defined over the integers. For example from the above, we obtain
 for which  is true for integer X, Y and Z if t is an integer. Hence we obtain triangles with integer length sides, such as sides of length 3, 7, and 8, where one of the angles is 60°, from relationships such as 82−3·8+32 = 72.
Many of the curves on Wikipedia's list of curves are rational, and hence have similar rational parameterizations. This is a list of curves, by Wikipedia page. ...
Elliptic curves An elliptic curve may be defined as any curve of genus one, but often it is further required that it be a nonsingular cubic curve, which suffices to model any genus one curve. A further restriction that it have an inflection point at infinity is also often imposed; this amounts to requiring that it can be written in Tate-Weierstrass form, which in its projective version is  Elliptic curves carry the structure of an abelian group, which in terms of curves defined over the complex numbers is the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...
In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...
Curves of genus greater than one Curves of 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 curves, the Klein quartic curve, and the Fermat curve xn+yn = zn when n is greater than three. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ...
Lines through a given point P and hyperparallel to line l. ...
In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ...
The Klein quartic x3y + y3z + z3x = 0, named after Felix Klein, is a Riemann surface, and a curve of genus 3 over the complex numbers C. The Klein quartic has automorphism group isomorphic to the projective special linear group G = PSL(2,7). ...
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation Xn + Yn = Zn. ...
See also An acnode is an isolated point not on an curve, but whose coordinates satisfy the equation of the curve. ...
This article refers to Bézouts theorem in algebraic geometry. ...
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...
It has been suggested that Latus rectum be merged into this article or section. ...
A crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...
In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ...
In mathematics, in particular commutative algebra, the idea of fractional ideal is introduced in the context of Dedekind domains. ...
In algebraic geometry, the function field of an algebraic variety V is the field of fractions of the ring of regular functions on V. Since algebraic varieties are irreducible varieties by definition, the ring of regular functions on V is an integral domain, and hence has a field of fractions. ...
In algebraic geometry, the function field KX of a scheme X is a generalization of the notion of a sheaf of rational functions on a variety. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
Hilberts sixteenth problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems. ...
In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ...
For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ...
The Klein quartic x3y + y3z + z3x = 0, named after Felix Klein, is a Riemann surface, and a curve of genus 3 over the complex numbers C. The Klein quartic has automorphism group isomorphic to the projective special linear group G = PSL(2,7). ...
This is a list of curves, by Wikipedia page. ...
A quartic plane curve is a plane curve of the fourth degree. ...
In mathematics, the rational normal curve is a smooth, rational curve of degree n in projective n-space . ...
In mathematics, the Riemann-Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. ...
In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ...
Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...
References - Egbert Brieskorn and Horst Knörrer, Plane Algebraic Curves, John Stillwell, trans., 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
- 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
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