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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.
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 ...
Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety. [3] 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.
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.
Rational curves. Rational curves are subdivided according to the degree of the polynomial. Degree 1. Line; Degree 2. Plane curves of degree 2 are known as conics or ...
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 ...
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 ...