Let $X$ be a smooth and projective variety over a field of characteristic zero. Let $Y$ be a normal variety, with finite quotient singularities (an orbifold!) and let $\pi: Y \to X$ be a finite morphism, ramified along a simple normal crossings divisor $D$. Assume that the singularities of $Y$ lie over the singular locus $D_{sing}$, i.e. $\pi(Y_{sing}) \subset D_{sing}$.

Now $Z \subset X$ is a *smooth* closed subvariety of codimension 2 in $X$, intersecting transversally all $D_J:=\bigcap_{i \in J} D_i$ (here $D=\sum_{i \in I} D_i$, with $D_i$ the irreducible components).

Consider the blow-up $\tilde{X} \to X$ of $X$ along $Z$ and form the cartesian product $\tilde{Y}=Y \times_X \tilde{X} \longrightarrow \tilde{X}$.

**My question is**: is $\tilde{Y}$ still normal and with finite quotient singularities?

Thanks for your help!