# Kodaira-Itaka dimension

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 $\varphi_{|mL|}$ 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 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 said to be of general type.

Title Kodaira-Itaka dimension KodairaItakaDimension 2013-03-22 16:12:43 2013-03-22 16:12:43 yark (2760) yark (2760) 17 yark (2760) Definition msc 14E05 Kodaira dimension bigness general type