Ad
related to: properties of algebraic curves
Search results
Results from the WOW.Com Content Network
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K. If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form
Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as ...
Taking the function field k(V) of an irreducible algebraic curve V, the functions F in the function field may all be realised as morphisms from V to the projective line over k. [clarification needed] (cf. #Properties) The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective ...
One of the most important properties of stable curves is the fact that they are local complete intersections. This implies that standard Serre-duality theory can be used. In particular, it can be shown that for every stable curve ω C / S ⊗ 3 {\displaystyle \omega _{C/S}^{\otimes 3}} is a relatively very-ample sheaf; it can be used to embed ...
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} applied to homogeneous coordinates ( x : y : z ) {\displaystyle (x:y:z)} for the projective plane ; or the inhomogeneous version for the affine space determined by setting z = 1 in such an ...
Ad
related to: properties of algebraic curves