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Kodaira-Itaka dimension (Definition)

Given a projective algebraic variety $ X$ and a line bundle $ L\to X$, the Kodaira-Itaka dimension of $ L$ is defined to be the supremum of the dimensions of the image of $ X$ by the map $ \varphi_{\vert mL\vert}$ associated to the linear system $ \vert mL\vert$, when $ m$ is a positive integer, namely

$\displaystyle \kappa(L)=\sup_{m\in\mathbb{N}}\{\dim\varphi_{\vert mL\vert}(X)\}. $

It is a standard fact that if we consider the graded ring

$\displaystyle R(X,L)=\bigoplus_{m\in\mathbb{N}}H^0(X,mL), $
then 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 Kodaira dimension of $ X$.

In paticular, if for some $ m$ we have $ \dim\varphi_{\vert mL\vert}(X)=\dim X$ then $ \kappa(L)=\dim X$ and $ L$ is called big.

If $ \kappa(X)=\kappa(K_X)=\dim X$, then $ X$ is said to be of general type.



"Kodaira-Itaka dimension" is owned by yark. [ full author list (6) | owner history (6) ]
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Also defines:  Kodaira dimension, bigness, general type
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Cross-references: canonical, graded ring, integer, positive, linear system, map, image, dimensions, supremum, line bundle, variety, algebraic
There are 4 references to this entry.

This is version 14 of Kodaira-Itaka dimension, born on 2006-09-01, modified 2007-08-04.
Object id is 8308, canonical name is KodairaDimension.
Accessed 2315 times total.

Classification:
AMS MSC: 14E05 (Algebraic geometry :: Birational geometry :: Rational and birational maps)
Pending Errata and Addenda
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Very minor detail by Mravinci on 2006-11-16 20:48:25
Minorone correzione: First instance of "Kodaira-Itaka dimension" ought to be italicized, or preferably emphasized, and "Kodaira dimension" ought to be put in the synonyms box.
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