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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.
Upload file; Special pages ... Cite this page; Get shortened URL; Download QR code; Print/export Download as PDF; Printable version ... Rational curves are subdivided ...
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 ]
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.
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.
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 ...
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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} .)