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.