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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.
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 ...
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} .)
If k is the field of rational numbers (or more generally a number field), there is an algorithm to determine whether a given conic has a rational point, based on the Hasse principle: a conic over has a rational point if and only if it has a point over all completions of , that is, over and all p-adic fields .
To state Kleiman's criterion (1966), let X be a projective scheme over a field. Let N 1 ( X ) {\displaystyle N_{1}(X)} be the real vector space of 1-cycles (real linear combinations of curves in X ) modulo numerical equivalence, meaning that two 1-cycles A and B are equal in N 1 ( X ) {\displaystyle N_{1}(X)} if and only if every line bundle ...
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In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves ; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures.
The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties .