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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.
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.
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any smooth projective surface.
An elliptic surface is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve B such that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field of B. Conversely, given a genus 1 curve over the function field of a ...
This is a rational number, the Gromov–Witten invariant for the given classes. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class A {\displaystyle A} , of genus g {\displaystyle g} , with domain in the β {\displaystyle \beta } -part of the Deligne–Mumford space) whose n {\displaystyle n} marked ...
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) → ะจ(A/K)[f] → 0.
Andrew Ogg drew the connection between the torsion conjecture for elliptic curves over the rationals and the theory of classical modular curves. [1] In the early 1970s, the work of Gérard Ligozat, Daniel Kubert , Barry Mazur , and John Tate showed that several small values of n do not occur as orders of torsion points on elliptic curves over ...
An irreducible plane algebraic curve of degree d has (d − 1)(d − 2)/2 − g singularities, when properly counted. It follows that, if a curve has (d − 1)(d − 2)/2 different singularities, it is a rational curve and, thus, admits a rational parameterization.