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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.
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 mouse lemur is a nocturnal, solitary-but-social lemur native to Madagascar. Solitary-but-social animals forage separately, but some individuals sleep in the same location or share nests. The home ranges of females usually overlap, whereas those of males do not. Males usually do not associate with other males, and male offspring are usually ...
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.
The division of labor creates specialized behavioral groups within an animal society, sometimes called castes. Eusociality is distinguished from all other social systems because individuals of at least one caste usually lose the ability to perform behaviors characteristic of individuals in another caste.
Every rational variety, including the projective spaces, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties. Unirational varieties are rationally connected, but it is not known if the converse holds.
Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K v-rational points for all places v of K. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by f is finite due to the following exact sequence. 0 → B(K)/f(A(K)) → Sel (f) (A/K) → ะจ(A/K)[f] → 0.
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 ]