Let $f:X \to Y$ be a finite morphism of smooth projective surfaces over a field $k$ of characteristic $0$. Is the branch divisor $B \subset Y$ a smooth curve?

The answer is negative if $\mathrm{cha}(k) = 2$. Consider the del Pezzo surface of degree $2$ given by $$X: \quad w^2 + xyw = f(x,y,z) \subset \mathbb{P}(1,1,1,2),$$ where $f$ is homogeneous of degree $4$. Then the projection $(x,y,z,w) \mapsto (x,y,z)$ is a separable double cover branched over the reducible plane curve $xy = 0 \subset \mathbb{P}^2$ when $\mathrm{cha}(k) = 2$. In particular it is singular curve, even though $X$ is smooth over $k$ when $f$ is general.

2more comments