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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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Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety. [2] This definition differs from that of path connectedness only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones.
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} .)
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Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell , [ 1 ] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings . [ 2 ]
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 rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve defined over the field of rational numbers or more generally a number field K. Mordell's theorem (generalized to arbitrary number fields by André Weil) says the group of rational points on an elliptic curve has a finite basis. This means that ...