This question is the two-dimensional analogue of [Etale coverings of certain open subschemes in Spec O\_K](https://mathoverflow.net/questions/22883/etale-coverings-of-certain-open-subschemes-in-spec-o-k)


There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O\_K$. As Cam Mcleman answered, this is basically done by the Galois group of the maximal extension unramified outside $D$. A covering of $U$ is of the form $O\_L[\frac{1}{D}]$, where $L$ is any extension of $K$. 


Here I would like to ask the same question, only now for $X=\mathbf{P}^1\_{\mathbf{Z}}$. 


Let $D$ be a normal crossings divisor on $\mathbf{P}^1\_{\mathbf{Z}}$ and let $U$ be the complement of its support.


**Q1**. Is there an "equivalence of categories" as Georges Elencwajg mentions in his answer for the analytic case. (See above link.) Basically, is there an arithmetic Grauert-Remmert theorem? 


**Q2**. What is known about the etale fundamental group in this case? Is it "finitely generated"? Has anybody studied the maximal pro-p-quotients of these groups? 


**Q3**. The analytic analogue would be to consider the same question for $\mathbf{P}^1\_{\mathbf{C}} \times \mathbf{P}^1\_{\mathbf{C}}$. 


**Q4** Lars (see above link) mentions a result for $\mathbf{P}\_{\mathbf{Q}}^1$. Is there something similar for $\mathbf{P}^2\_{\mathbf{Q}}$?



Covers of the projective line over Z and arithmetic Grauert-Remmert
Regarding Q3: For any scheme $X$ of finite type over $\mathbb{C}$ the Riemann-Existence Theorem (See SGA1 XII.5) says that the category of finite étale coverings of $X$ is equivalent to the category of finite covering spaces of the associated analytic space $X^{an}$. This implies that the finite quotients of the topological fundametal group of $X^{an}$ are the same as the finite quotients of the étale fundamental group, and one obtains that the étale fundamental group of $X$ is isomorphic to the profinite completion of the topological fundamental group of $X^{an}$.


Q4: The same short exact sequence that I mentioned in your other question is still valid. 


As in your other question, I cannot say anything about the situation over $\mathbb{Z}$ :)


