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Concept
Iitaka dimension
Summary
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).
Official source
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