Search results
Results From The WOW.Com Content Network
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.
This is a list of Wikipedia articles about curves used in different fields: ... Rational curves are subdivided according to the degree of the polynomial. Degree 1
Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red) In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, [1] [2] [3] are a system of coordinates used in projective geometry, just as Cartesian coordinates are used ...
Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set (hence compact), these have well-defined Donaldson–Thomas invariants (the "virtual number of points"); at least for degree 1 and 2, these agree with the actual number of points.
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 .
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]
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply contract all the −1-curves on the surface, and the resulting variety Y is either a (unique) minimal model with K nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface ...
The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the geometric genus. [2] This method is non-elementary, but several short proofs using only the basics of field theory have long been known, mainly using the concept of transcendence degree. [3]