Kodaira dimension

Remarks

Always $\kappa(X) \in \set{-\infty, 0, 1, \dots, n}$. Equivalently, when some $P_m(X) > 0$, $\kappa(X)$ is the largest dimension of the image of the rational map $X \dashrightarrow \mathbb{P}(H^0(X, \omega_X^{\otimes m}))$ over all $m$ with $P_m(X) > 0$. The variety $X$ is of general type if $\kappa(X) = n$.

Example: for $X = \mathbb{P}^n_k$ with $n \ge 1$, $\omega_X \cong \mathcal{O}(-n-1)$, so $\omega_X^{\otimes m} \cong \mathcal{O}(-m(n+1))$ has no nonzero sections and $\kappa(X) = -\infty$. For a smooth projective curve of genus $g \ge 2$, $\deg \omega_X^{\otimes m} = m(2g-2)$ and Riemann–Roch gives $P_m = (2m-1)(g-1)$ for $m \ge 2$, which grows like $m^1$, so $\kappa(X) = 1$.

Non-example of clause 2 with $\kappa = 0$: the curve of genus $2$ has $P_m / m^0 = P_m$ unbounded, so its Kodaira dimension is not $0$.