Kodaira dimension

For a smooth projective variety $X$ over the complex numbers, its plurigenera are $P_m(X)=\dim H^0(X,K_X^{\otimes m})$, where $K_X$ is the canonical line bundle of a smooth variety. If every $P_m$ vanishes for $m>0$, define $\kappa (X)=-\infty$. Otherwise, $\kappa (X)$ is the largest dimension of the image of a rational map defined by a complete linear system of a divisor. Equivalently, there are positive constants $a,b$ such thatfor all sufficiently large, sufficiently divisible $m$. Restricting to divisible $m$ matters because some plurigenera can vanish periodically.^[ref]

For a smooth projective curve, genus zero, one and at least two give $\kappa =-\infty ,0,1$, respectively. For smooth projective surfaces, ${\mathbb{P}}^2$, a product of two elliptic curves, the product of an elliptic curve with a curve of genus at least two, and the product of two such higher-genus curves realize $-\infty ,0,1,2$. The plurigenera, hence $\kappa$, are invariant under birational between smooth projective varieties, making this growth rate useful for classification.^[ref]