Iitaka dimension

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).