KodairaDimension 2006-09-01 15:08:51 2007-08-04 02:45:16 Definition Kodaira dimension bigness general type \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} 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 \emph{big}. If $\kappa(X)=\kappa(K_X)=\dim X$, then $X$ is said to be of \emph{general type}.