Search results
Results From The WOW.Com Content Network
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 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.
The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points. [ 8 ] If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem , as it would imply that there do not exist dense subsets of the Euclidean ...
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]
Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic surfaces, for pioneering the notion of rational connectedness (i.e. extending the theory of rationally connected varieties for varieties over the complex field to varieties over local fields), and finding counterexamples to a conjecture of John Nash.
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]
The wood-pasture hypothesis (also known as the Vera hypothesis and the megaherbivore theory) is a scientific hypothesis positing that open and semi-open pastures and wood-pastures formed the predominant type of landscape in post-glacial temperate Europe, rather than the common belief of primeval forests.
Like plants, animals display a range of abilities to cope with fire, but they differ from most plants in that they must avoid the actual fire to survive. Although birds may be vulnerable when nesting, they are generally able to escape a fire; indeed they often profit from being able to take prey fleeing from a fire and to recolonize burned ...