Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Iitaka dimension
Résumé
In algebraic geometry, the Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This is 1 less than the dimension of the section ring of L
The Iitaka dimension of L is always less than or equal to the dimension of X. If L is not effective, then its Iitaka dimension is usually defined to be or simply said to be negative (some early references define it to be −1). The Iitaka dimension of L is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by .
A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If f : Y → X is a birational morphism of varieties, and if L is a big line bundle on X, then f*L is a big line bundle on Y.
All ample line bundles are big.
Big line bundles need not determine birational isomorphisms of X with its image. For example, if C is a hyperelliptic curve (such as a curve of genus two), then its canonical bundle is big, but the rational map it determines is not a birational isomorphism. Instead, it is a two-to-one cover of the canonical curve of C, which is a rational normal curve.
Kodaira dimension
The Iitaka dimension of the canonical bundle of a smooth variety is called its Kodaira dimension.
Consider on complex algebraic varieties in the following.
Let K be the canonical bundle on M. The dimension of H0(M,Km), holomorphic sections of Km, is denoted by Pm(M), called m-genus. Let
then N(M) becomes to be all of the positive integer with non-zero m-genus. When N(M) is not empty, for m-pluricanonical map is defined as the map
where are the bases of H0(M,Km). Then the image of , is defined as the submanifold of .
For certain let be the m-pluricanonical map where W is the complex manifold embedded into projective space PN.
In the case of surfaces with κ(M)=1 the above W is replaced by a curve C, which is an elliptic curve (κ(C)=0).
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.