Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do we know anything about the number of connected components of $\bigcup D_i$?
Does the number equal to the number of $D_i$'s? (i.e. $D_i$'s do not intersect with each other.)
Thanks!