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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.
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 ...
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]
The species–area relationship or species–area curve describes the relationship between the area of a habitat, or of part of a habitat, and the number of species found within that area. Larger areas tend to contain larger numbers of species, and empirically, the relative numbers seem to follow systematic mathematical relationships. [ 1 ]
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 ...
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.
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.
These sheaves have trivial non-zero cohomology, and hence they are always convex. In particular, Abelian varieties have this property since the Albanese variety of a rational curve is trivial, and every map from a variety to an Abelian variety factors through the Albanese. [4]