Surface of general type

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.