Category: Algebraic curves

In geometry, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction as C at these points. That is, L is a tangent line at P and at Q.
Weierstrass point
In mathematics, a Weierstrass point on a nonsingular algebraic curve defined over the complex numbers is a point such that there are more functions on , with their poles restricted to only, than would
Superelliptic curve
In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form where is an integer and f is a polynomial of degree with coefficients in a field ; more precisely, it is
Vector bundles on algebraic curves
In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic cu
Smooth completion
In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset. Smooth completio
Abhyankar's conjecture
In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1
Polynomial lemniscate
In mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients of degree n. For any such polynomia
Weber's theorem
In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. Consider two non-singular curves C and C′ having the same genus g > 1. If
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-f
A squircle is a shape intermediate between a square and a circle. There are at least two definitions of "squircle" in use, the most common of which is based on the superellipse. The word "squircle" is
Hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinc
Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is a fixed non-zero integer. These curves were closely studied by Louis Mordell, from the point of view of determining
Genus–degree formula
In classical algebraic geometry, the genus–degree formula relates the degree d of an irreducible plane curve with its arithmetic genus g via the formula: Here "plane curve" means that is a closed curv
Projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by
Real plane curve
In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane.
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientat
Global field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A
Sectrix of Maclaurin
In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivale
Prym variety
In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it wa
Moment curve
In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form In the Euclidean plane, the moment curve is a par
In geometry, a cissoid ((from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') is a plane curve generated from two given curves C1, C2 and a point O (the pole). Let L be a variable line passing thr
Deltoid curve
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle
Symmetric product of an algebraic curve
In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product C × C × ... × C or Cn by the group action of the symmetric group Sn on n lett
Algebraic curve
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 i
Schottky problem
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.
Fermat curve
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 Therefore, in terms of the affine plane its eq
Weil reciprocity law
In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e
Enriques–Babbage theorem
In algebraic geometry, the Enriques–Babbage theorem states that a canonical curve is either a set-theoretic intersection of quadrics, or trigonal, or a plane quintic. It was proved by Babbage and Enri
In geometry, a superegg is a solid of revolution obtained by rotating an elongated superellipse with exponent greater than 2 around its longest axis. It is a special case of superellipsoid. Unlike an
Stable curve
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a complete connecte
In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis.
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a grou
Bicuspid curve
No description available.
Reiss relation
In algebraic geometry, the Reiss relation, introduced by Reiss, is a condition on the second-order elements of the points of a plane algebraic curve meeting a given line.
Lüroth quartic
In mathematics, a Lüroth quartic is a nonsingular quartic plane curve containing the 10 vertices of a complete pentalateral. They were introduced by Jacob Lüroth. Morley showed that the Lüroth quartic
Polar curve
In algebraic geometry, the first polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point of C whose ta
Torelli theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (co
Gonality of an algebraic curve
In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C is defined over the fiel
Rational normal curve
In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when
Cruciform curve
No description available.
Quartic plane curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: with at least one of A, B, C, D, E not equal to zero. T
Sum of residues formula
In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes.
A bifolium is a quartic plane curve with equation in Cartesian coordinates:
Cassini oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted w
Generalized conic
In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of sums defining property of the classical conic. For example, in elementary geometry, an el
Delta invariant
In mathematics, in the theory of algebraic curves, a delta invariant measures the number of double points concentrated at a point. It is a non-negative integer. Delta invariants are discussed in the "
In geometry, a hippopede (from Ancient Greek ἱπποπέδη (hippopédē) 'horse fetter') is a plane curve determined by an equation of the form where it is assumed that c > 0 and c > d since the remaining ca
ELSV formula
In mathematics, the ELSV formula, named after its four authors , , , , is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stab
Castelnuovo curve
In algebraic geometry, a Castelnuovo curve, studied by Castelnuovo, is a curve in projective space Pn of maximal genus g among irreducible non-degenerate curves of given degree d. Castelnuovo showed t
Conchoid of de Sluze
In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze. The curves are defined by the polar equatio
Goppa code
In mathematics, an algebraic geometric code (AG-code), otherwise known as a Goppa code, is a general type of linear code constructed by using an algebraic curve over a finite field . Such codes were i
In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to
Generalized Jacobian
In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by Maxwe
Ordinary singularity
In mathematics, an ordinary singularity of an algebraic curve is a singular point of multiplicity r where the r tangents at the point are distinct .In higher dimensions the literature on algebraic geo
S-equivalence is an equivalence relation on the families of semistable vector bundles on an algebraic curve.
Ampersand curve
No description available.
Singular point of a curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied
Lange's conjecture
In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by and proved by Montserrat Teixidor i Bigas and in 1999.
Ribet's theorem
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jea
Cissoid of Diocles
In geometry, the cissoid of Diocles (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped'; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two me
Artin–Schreier curve
In mathematics, an Artin–Schreier curve is a plane curve defined over an algebraically closed field of characteristic by an equation for some rational function over that field. One of the most importa
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle
In geometry, a limaçon or limacon /ˈlɪməsɒn/, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls
Conchoid of Dürer
In geometry, the conchoid of Dürer, also called Dürer's shell curve, is a plane, algebraic curve, named after Albrecht Dürer and introduced in 1525. It is not a true conchoid.
Cayley–Bacharach theorem
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states: Assume that two cubics C1 and C2 in
Cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine sp
Spiric section
In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form Equivalently, spiric sections can be defined as bicircular quartic curves
Theta characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riema
Elkies trinomial curves
In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials givin
Lemniscate of Gerono
In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an symb
Real hyperelliptic curve
There are two types of hyperelliptic curves, a class of algebraic curves: real hyperelliptic curves and imaginary hyperelliptic curves which differ by the number of points at infinity. Hyperelliptic c
Brill–Noether theory
In algebraic geometry, Brill–Noether theory, introduced by Alexander von Brill and Max Noether, is the study of special divisors, certain divisors on a curve C that determine more compatible functions
Bring's curve
In mathematics, Bring's curve (also called Bring's surface) is the curve given by the equations It was named by , p.157) after Erland Samuel Bring who studied a similar construction in 1786 in a Promo
Modular curve
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a
Kempe's universality theorem
In 1876 Alfred B. Kempe published his article On a General Method of describing Plane Curves of the nth degree by Linkwork, which showed that for an arbitrary algebraic plane curve a linkage can be co
Folium of Descartes
In geometry, the folium of Descartes (from Latin folium 'leaf'; named for René Decartes) is an algebraic curve defined by the implicit equation
Plücker formula
In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic cu
Three-leaved clover
No description available.
Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series is a Puiseux series in the indetermin
Classical modular curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the
Jacobian variety
In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group
Hodge bundle
In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has a
Trident curve
In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: Trident curves are cubic plane curves with an ordinary do
Cramer's paradox
In mathematics, Cramer's paradox or the Cramer–Euler paradox is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitr
Theta divisor
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the asso
In mathematics, a hyperbola (/haɪˈpɜːrbələ/; pl. hyperbolas or hyperbolae /-liː/; adj. hyperbolic /ˌhaɪpərˈbɒlɪk/) is a type of smooth curve lying in a plane, defined by its geometric properties or by
Nagata's conjecture on curves
In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with
Algebraic stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques spec
Twisted cubic
In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transfor
Abel–Jacobi map
In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping
Bullet-nose curve
In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation The bullet curve has three double points in the real projective plane, at x = 0 and
Sinusoidal spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates where a is a nonzero constant and n is a rational number other than 0. With a rotation
Conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabol
Normal degree
In algebraic geometry, the normal degree of a rational curve C on a surface is defined to be –K.C–2 where K is the canonical divisor of the surface.
De Franchis theorem
In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1
Witch of Agnesi
In mathematics, the witch of Agnesi (Italian pronunciation: [aɲˈɲeːzi, -eːsi; -ɛːzi]) is a cubic plane curve defined from two diametrically opposite points of a circle. It gets its name from Italian m
Spherical conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, para
Zariski geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve,
In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also kn
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is
Bow curve
No description available.
Chiral Potts curve
The chiral Potts curve is an algebraic curve defined over the complex numbers that occurs in the study of the chiral Potts model of statistical mechanics. For an integer N, the parameters in the Boltz
Division polynomials
In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to def
Toric section
A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Special cases have been known since antiquity, and the general case was
Belyi's theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified c
Hasse–Witt matrix
In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the pr
Lemniscate of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2c from each other as the locus of points P so that PF1·PF2 = c2. The curv
Limaçon trisectrix
In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves part
Cartesian oval
In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These curves are named after French mathematician R
In geometry, the n-ellipse is a generalization of the ellipse allowing more than two foci. n-ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, k-ellipse, and Ts
Cusp (singularity)
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of sin
Riemann–Hurwitz formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of
Bean curve
No description available.
Mumford measure
In mathematics, a Mumford measure is a measure on a supermanifold constructed from a bundle of relative dimension 1|1. It is named for David Mumford.
Bitangents of a quartic
In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but
Hilbert's twenty-first problem
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specif
Imaginary hyperelliptic curve
A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus . If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic cu
Chasles–Cayley–Brill formula
In algebraic geometry, the Chasles–Cayley–Brill formula, also known as the Cayley–Brill formula, states that a correspondence T of valence k from an algebraic curve C of genus g to itself has d + e +
Clifford's theorem on special divisors
In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford on algebraic curves, showing the constraints on special linear systems on a curve C.
List of curves
This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, p
Lambda g conjecture
In algebraic geometry, the -conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification of the moduli space of curves with marked points. It was first
Modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational n
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are "isolated point or hermit point". For example the equation has an acnode at the
Abelian integral
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form where is an arbitrary rational function of the two varia