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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.
Gneiss, a foliated metamorphic rock. Quartzite, a non-foliated metamorphic rock. Foliation in geology refers to repetitive layering in metamorphic rocks. [1] Each layer can be as thin as a sheet of paper, or over a meter in thickness. [1] The word comes from the Latin folium, meaning "leaf", and refers to the sheet-like planar structure. [1]
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.
Bergmann's rule - Penguins on the Earth (mass m, height h) [1] Bergmann's rule is an ecogeographical rule that states that, within a broadly distributed taxonomic clade, populations and species of larger size are found in colder environments, while populations and species of smaller size are found in warmer regions.
The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ, from the modular curve X 0 (N) to E. In particular, the points of E can be parametrized by modular functions. For example, a modular parametrization of the curve y 2 − y = x 3 − x is given by [18]
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell , [ 1 ] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings . [ 2 ]
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 ...