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'Kodaira-Itaka dimension'
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| Title of object: |
Kodaira-Itaka dimension |
| Canonical Name: |
KodairaDimension |
| Type: |
Definition |
| Created on: |
2006-09-01 15:08:51 |
| Modified on: |
2006-11-17 03:22:26 |
| Classification: |
msc:14E05 |
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Content:
Given a projective algebraic variety $X$ and a line bundle $L\to X$, the \emph{Kodaira-Itaka dimension} of $L$, is defined to be the supremum of the dimensions of the image of $X$ by the map $\varphi_{|mL|}$ associated to the linear system $|mL|$, when $m$ is a positive integer, namely $$\kappa(L)=\sup_{m\in\mathbb N}\{\dim\varphi_{|mL|}(X)\}.$$
It is a standard fact that if we consider the graded ring $$R(X,L)=\bigoplus_{m\in\mathbb N}H^0(X,mL),
$$
then $\text{tr.deg} R(X,L)=\kappa(L)+1$.
When the line bundle we have is the canonical bundle $K_X$ of $X$, then its Kodaira-Itaka dimension is called \emph{Kodaira dimension} of $X$.
In paticular, if for some $m$ we have $\dim\varphi_{|mL|}(X)=\dim X$ then $\kappa(L)=\dim X$ and $L$ is called big.
If $\kappa(X)=\kappa(K_X)=\dim X$, then $X$ is called of general type. |
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