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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.
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.
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 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.
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 contrast to positively curved varieties such as del Pezzo surfaces, a complex algebraic K3 surface X is not uniruled; that is, it is not covered by a continuous family of rational curves. On the other hand, in contrast to negatively curved varieties such as surfaces of general type, X contains a large discrete set of rational curves ...
Chern numbers of minimal complex surfaces. The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.