Let $X$ and $Y$ be complex projective surfaces, where $Y$ is smooth. Suppose $p:X\rightarrow Y$ is finite degree two map, i.e. $X$ is a double cover of $Y$. Let $B\subset Y$ denote the branch curve. It is known that if $B$ is smooth, then $X$ is smooth. And in general, $X$ can have singularities at most over the singular points of $B$.
In case of double covers, is the converse true? That is, if $X$ is smooth, then does it imply that $B$ is smooth?
Let $\tau \colon X \to X$ be the involution of the double cover. If $X$ is smooth then the fixed locus $R \subset X$ of $\tau$ (i.e., ramification divisor of $p$) is also smooth. The map $p$ induces an isomorphism $R \to B$, hence $B$ is also smooth.