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1.1 Rational curves. 1.1.1 Degree 1. 1. ... Download as PDF; Printable version ... move to sidebar hide. This is a gallery of curves used in mathematics, by Wikipedia ...
In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space P n. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the plane conic Z 0 Z 2 = Z 2 1, and for n = 3 it is the twisted cubic.
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Lüroth's problem concerns subextensions L of K(X), the rational functions in the single indeterminate X. Any such field is either equal to K or is also rational, i.e. L = K(F) for some rational function F. In geometrical terms this states that a non-constant rational map from the projective line to a curve C can only occur when C also has genus 0.
If X is a curve of genus 1 with a k-rational point p 0, then X is called an elliptic curve over k. In this case, X has the structure of a commutative algebraic group (with p 0 as the zero element), and so the set X ( k ) of k -rational points is an abelian group .
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.
A variety is called convex if the pullback of the tangent bundle to a stable rational curve: has globally generated sections. [2] Geometrically this implies the curve is free to move around infinitesimally without any obstruction. Convexity is generally phrased as the technical condition
A variety is uniruled if it is covered by a family of rational curves. (More precisely, a variety X {\displaystyle X} is uniruled if there is a variety Y {\displaystyle Y} and a dominant rational map Y × P 1 → X {\displaystyle Y\times \mathbf {P} ^{1}\to X} which does not factor through the projection to Y {\displaystyle Y} .)