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! 



blow-ups and singularities 
I think this is true. Taking a point $x\in X$, and base change your setting up to its formal neighborhood, we can assume $K(X)\subset K(Y)$ is a Galois extension with finite abelian group.


Now we can put a root stack structure $\mathcal{X}\to X$ which is branched over $D$ such that the morphism $Y\setminus\pi^{-1}(V)\to \mathcal{X}\setminus{V}$ is etale where $V\subset \mathcal{X}$ is of codimension at least 2 (See Matsuki-Olsson's paper). Then by purity, this indeed gives a DM-stack $\mathcal{Y}$ which is finite etale over $\mathcal{X}$. In particular $\mathcal{Y}$ is smooth and it has Y as its coarse moduli space. (By the way, I think once you assume $Y\to X$ branches over a snc divisor and $Y$ is normal, it comes for free that $Y$ has only abelian quotient singularities.) 


Now the condition you have just implies the preimage $\mathcal{Z}$ of $Z$ in $\mathcal{X}$ is smooth. So its preimage $\mathcal{Z}\_{\mathcal{Y}}$ in $\mathcal{Y}$ is smooth.


But then $Bl\_{\mathcal{Z}\_{\mathcal{Y}}}\mathcal{Y}$ is smooth, so its coarse moduli space only has abelian quotient singularities, which is precisely $\tilde{Y}$.


