Suppose $X$ and $Y$ are two smooth surfaces (over the complex numbers), and $f: X \to Y$ is a finite flat morphism of degree two. Is it necessarily true that the locus where $f$ is not a smooth morphism (i.e. the ramification locus) always one dimensional (if not empty)?



Is the branching locus of the double cover of surfaces always one dimensional?
Yes. Otherwise there would be an isolated ramification point in $Y$. A link of that point is $S^3$, which does not have a nontrivial double cover. So upstairs, $X$ looks near that point like two ${\mathbb C}^2$s glued at a point, which isn't smooth.


