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 ${\phi }_{|mL|}$ associated to the linear system $|mL|$,when $m$ is a positive integer, namely

$$\kappa (L)=\underset{m\in \mathbb{N}}{sup}\{dim{\phi }_{|mL|}(X)\}.$$

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

$$R(X,L)=\underset{m\in \mathbb{N}}{\oplus }{H}^0(X,mL),$$

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 $dim{\phi }_{|mL|}(X)=dimX$then $\kappa (L)=dimX$ and $L$ is called big.

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