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2-dimensional section of Reeb foliation 3-dimensional model of Reeb foliation. In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space R n into the cosets x + R p of the standardly embedded ...
Every rational variety, including the projective spaces, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties. Unirational varieties are rationally connected, but it is not known if the converse holds.
A variety X over an uncountable algebraically closed field k is uniruled if and only if there is a rational curve passing through every k-point of X. By contrast, there are varieties over the algebraic closure k of a finite field which are not uniruled but have a rational curve through every k-point.
A birational map from X to Y is a rational map f : X ⇢ Y such that there is a rational map Y ⇢ X inverse to f.A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open subsets of X, Y by definition gives a birational map f : X ⇢ Y.
Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K v-rational points for all places v of K. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by f is finite due to the following exact sequence. 0 → B(K)/f(A(K)) → Sel (f) (A/K) → Ш(A/K)[f] → 0.
In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K.The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a rational point over K. [1]
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.
In dimension 3, there are smooth complex Fano varieties which are not rational, for example cubic 3-folds in P 4 (by Clemens - Griffiths) and quartic 3-folds in P 4 (by Iskovskikh - Manin). Iskovskih ( 1977 , 1978 , 1979 ) classified the smooth Fano 3-folds with second Betti number 1 into 17 classes, and Mori & Mukai (1981) classified the ...