Let $\pi:Y\longrightarrow \mathbf{P}^1\_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch locus $D$. Let us suppose that $\pi$ is tamely ramified.


**Question 1.** Does this mean that for every prime number $p$ such that some vertical component of $D$ maps to $p$ in $\textrm{Spec} \mathbf{Z}$ does not divide the degree of $\pi$?


By Abhyankar's Lemma, there exists a number field $K$ with ring of integers $O\_K$ such that the branch locus $D\subset (B\_{O\_K})\_{\textrm{hor}}$ of the morphism $\pi\_{O\_K}:Y\_{O\_K}\longrightarrow \mathbf{P}^1\_{O\_K}$ is horizontal.


By embedded resolution, there exists a projective birational morphism $f:X\longrightarrow \mathbf{P}^1\_{O\_K}$ such that $f^\ast D$ is a divisor with normal crossings. 


**Question 2.** Since $D$ is horizontal, does it follow that $f^\ast D$ is horizontal. I thought so but wasn't completely sure.


**Question 3.** Does the (edit: normalization of the) fibre product $Y^\prime$ of $Y\_{O\_K}$ and $X$ over $\mathbf{P}^1\_{O\_K}$ give a morphism $Y^\prime\longrightarrow X$ with branch locus a horizontal normal crossings divisor? I would be surprised if the branch locus picked up vertical components...



Does combining Abhyankar's Lemma and embedded resolution  give  horizontal normal crossings 
**Question 1**: it is not exactly the meaning of tameness (ramification index prime to $p$ and separable residue extension). But if $p>\deg \pi$, then $\pi$ is tame at $p$. 


**Question 2**: The strict transform of $D$ is horizontal because it is finite birational to $D$, but the preimage of $D$ by $f$ is not horizontal in general. 


**Question 3**: I don't think so. Consider the following example. Let $x$ be a coordinate of $P^1$ and consider the cover induced by the equation $y^3=x^3+p^2$. Then $D$ is the divisor $V(x^3+p^2)$. To solve the singularity of $D$, you have to blowup $P^1$ along the point $(x,p)$. Then you get a vertical ramification at $p$. (In general you need to normalize the fiber product of $X$ with $Y$). 


