Surface of general type

Remarks

Equivalently, with $K_X$ the canonical divisor, there is a constant $c > 0$ with $h^0(X, nK_X) \ge c n^2$ for all large divisible $n$; that is, $K_X$ is big. Being of general type is a birational property. A minimal surface of general type satisfies $K_X^2 \ge 1$ and $\chi(\mathcal{O}_X) \ge 1$. Example: a smooth surface $X \subset \mathbb{P}^3$ of degree $d \ge 5$ has $K_X = (d - 4) H$ by adjunction, with $H$ a plane section. Since $d - 4 \ge 1$, $K_X$ is ample and $h^0(nK_X)$ grows like $n^2$, so $X$ is of general type. Non-example: a smooth quartic surface in $\mathbb{P}^3$ has $K_X = 0$, so $h^0(nK_X) = 1$ for all $n \ge 1$ and its Kodaira dimension is $0$. Example: the product $C_1 \times C_2$ of two curves of genus at least $2$ is of general type.