blow-ups and singularities

Question

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!

Answer 1

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 $\tilde{Y}$.

Answer 2

I don't see how $D$ makes any difference in the question you describe. Perhaps you forgot a condition about $D$ such as $\pi$ is ramified along $D$?

In any case, here is an example that shows that you need more conditions to have a chance here.

Let $Y=X=\mathbb A^2$ and $\pi: \mathbb A^2\to \mathbb A^2$ be defined by $(x,y)\mapsto (x^2,y^2)$. Let $Z\subset X$ be the origin. Then $\widetilde X$ is just $\mathbb A^2$ blown-up at a closed point (in particular smooth, normal, etc.), but $\widetilde Y$ is the blow-up of $\mathbb A^2$ along the fat point defined by the ideal $(x^2,y^2)$ so it is a pinch point and hence not normal and not even normal crossings.

I guess now you could say that you want $\pi$ ramified along a divisor, but it will be the same thing, if $Z$ is a point on $D$, it's pre-image will be a fat point and blowing that up will not be normal. It seems that the only chance is to require that $Z$ (or at least its generic point) is disjoint from the ramification locus of $\pi$, in other words if $\pi$ is étale in a neighbourhood of (the generic point of) $Z$.