Kodaira dimension ($\kappa (X)$)

The Kodaira dimension measures the asymptotic growth of the pluricanonical spaces $H^0(X,K_X^{\otimes m})$. For a surface it takes values $-\infty ,0,1,2$.

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

Let $B\subseteq {\mathbb{P}}^2$ be a smooth curve of degree $2d$, and let $\pi :X\to {\mathbb{P}}^2$ be the double cover branched along $B$. ThenFor $d\ge 4$ this canonical divisor is ample, so $X$ is a surface of general type. A smooth octic branch curve gives the first such example.