Kodaira dimension ($\kappa (X)$)

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]

A smooth projective surface is of general type when its Kodaira dimension is two, equivalently when its canonical divisor is big.

Work over an algebraically closed field of characteristic different from two. A smooth projective plane curve $B\subset {\mathbb{P}}^2$ of degree $2d$ is the zero set of a global section $s$ of $L^{\otimes 2}$, where $L={\mathcal{O}}_{{\mathbb{P}}^2}(d)$. The finite morphism $\pi :X\to {\mathbb{P}}^2$ defined locally by $w^2=s$ is a double cover, branched along $B$. The Jacobian criterion shows that $X$ is smooth: away from $B$ the derivative in $w$ is nonzero, while on $B$ some derivative of $s$ is nonzero.

The ramification divisor $R$ satisfies ${\mathcal{O}}_X(R)={\pi }^{\ast }L$. The determinant of the differential giveswhere $H$ is the hyperplane class. Thus $K_X$ is ample for $d\ge 4$, so $X$ is a surface of general type; its self-intersection number is $K_X^2=2(d-3{)}^2$. A smooth octic branch curve gives $K_X^2=2$. For $d=3$, the same formula instead gives a trivial canonical line bundle of a smooth variety, illustrating why the branch degree changes the classification. The assumption on characteristic is needed for this local double-cover construction and its smoothness argument.