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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 ...
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.
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 ...
Current tomato Indeterminate Regular leaf Ostensibly from the original wild tomato from Mexico. They are smaller than most cherry tomato types. [88] Micro Tom Red 50–60 1 oz Cherry Micro Determinate Regular Leaf Considered world's smallest tomato, Micro Tom is a cultivar used mainly in laboratory experiments [89] Millionaire Pink 80–85 Heirloom
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.
An elliptic curve is a smooth projective curve of genus one. In algebraic geometry , a projective variety is an algebraic variety that is a closed subvariety of a projective space . That is, it is the zero-locus in P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal , the defining ...
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If X is a smooth complete curve (for example, P 1) and if f is a rational map from X to a projective space P m, then f is a regular map X → P m. [5] In particular, when X is a smooth complete curve, any rational function on X may be viewed as a morphism X → P 1 and, conversely, such a morphism as a rational function on X.