Curve - Algebraic Curve

Algebraic Curve

Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the locus of the points of coordinates x, y such that f(x, y) = 0, where f is a polynomial in two variables defined over some field F. Algebraic geometry normally looks not only on points with coordinates in F but on all the points with coordinates in an algebraically closed field K. If C is a curve defined by a polynomial f with coefficients in F, the curve is said defined over F. The points of the curve C with coordinates in a field G are said rational over G and can be denoted C(G)); thus the full curve C = C(K).

Algebraic curves can also be space curves, or curves in even higher dimension, obtained as the intersection (common solution set) of more than one polynomial equation in more than two variables. By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce singularities such as cusps or double points.

A plane curve may also may also be completed in a curve in the projective plane: if a curve is defined by a polynomial f of total degree d, then wdf(u/w, v/w) simplifies to a homogeneous polynomial g(u, v, w) of degree d. The values of u, v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such w is not zero. An example is the Fermat curve un + vn = wn, which has an affine form xn + yn = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces

Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography. Because algebraic curves in fields of characteristic zero are most often studied over the complex numbers, algebraic curves in algebraic geometry may be considered as real surfaces. In particular, the non-singular complex projective algebraic curves are called Riemann surfaces.