Kodaira dimension

Given an $n$-dimensional smooth projective variety $X$, we can study the canonical line bundle $K_X$ of holomorphic $n$-forms. The dimensions of the spaces of global sections of $K_X^{\otimes r}$ are useful birational invariants of $X$ which aid in the classification of varieties (birational means they only depend on a Zariski-open subset of $X$). As $r\to \infty$, these numbers either behave asymptotically like $C\cdot r^k$ for a unique integer $k$ or are eventually zero. We define the Kodaira dimension to be this integer $k$ in the first case and $-\infty$ in the second case.

Another interpretation is as follows. For each $r$, we have a rational map of $X$ into projective space given by

$$[s_0,\dots ,s_m]:X⇢{\mathbb{P}}^n$$

where $s_0,\dots ,s_n$ are the global sections of $K_X^{\otimes r}$. The Kodaira dimension is the supremum, as $r\to \infty$, of the dimension of the image of $X$ under these maps. Hence the Kodaira dimension of $X$ takes values in $\{-\infty ,0,\dots ,n\}$.