= 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 $L^{\otimes2}$, 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 $\mathcal O_X(R)=\pi^*L$. The <determinant> of the <differential> gives
$$
K_X=\pi^*K_{\mathbb P^2}+R=\pi^*((d-3)H),
$$
where $H$ is the <hyperplane class>. Thus $K_X$ is <ample> for $d\geq4$, 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.
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