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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, 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 ).
Graphs of curves y 2 = x 3 − x and y 2 = x 3 − x + 1. Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry.
For example, if X is the curve in the projective plane P 2 R over the real numbers R defined by the equation xy 2 = 7z 3, then X C is the complex curve in P 2 C defined by the same equation. Many properties of an algebraic variety over a field k can be defined in terms of its base change to the algebraic closure of k, which makes the situation ...
Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces. In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.
The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry , a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries , a line is a 2-dimensional vector space ...
The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel–Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be ...
An elliptic curve is a smooth projective curve of genus one. In algebraic geometry , a projective variety is an algebraic variety that is a closed subvariety of a projective space . That is, it is the zero-locus in P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal , the defining ...