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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]

Mihnea Popa works on algebraic geometry, including the birational classification of varieties and the geometry of families. His Harvard course on Kodaira dimension documents the connection with positivity and Hodge theory.^[ref]

His Kodaira dimension of algebraic varieties by Mihnea Popa explains how sections of powers of the canonical bundle measure the growth of pluricanonical maps. For a smooth projective curve, the trichotomy between genus zero, genus one and higher genus appears as Kodaira dimension $-\infty$, zero and one. This gives a concrete entry into a classification invariant that remains meaningful in higher dimensions.^[ref]