Double plane of general type

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.