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The first key result is the cone theorem of Shigefumi Mori, describing the structure of the cone of curves of . Briefly, the theorem shows that starting with X {\displaystyle X} , one can inductively construct a sequence of varieties X i {\displaystyle X_{i}} , each of which is "closer" than the previous one to having K X i {\displaystyle K_{X ...
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]
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.
This is a rational number, the Gromov–Witten invariant for the given classes. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class A {\displaystyle A} , of genus g {\displaystyle g} , with domain in the β {\displaystyle \beta } -part of the Deligne–Mumford space) whose n {\displaystyle n} marked ...
The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Jean-Pierre Serre, [13] now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these ...
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 ...
This is a list of Wikipedia articles about curves used in different fields: mathematics ... Rational normal curve; Rose curve; Curves with genus 1. Bicuspid curve;
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 ...