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This question is the two-dimensional analogue of http://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 simple 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 $\mathbf{P}^2_{\mathbf{Q}}$?

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This question is the two-dimensional analogue of http://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 $\mathbf{P}^1_{\mathbf{Z}}$. X=\mathbf{P}^1_{\mathbf{Z}}$. Let$D$be a simple normal crossings divisor on$\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$\mathbf{P}^2_{\mathbf{Q}}$? 2 Removed some stuff. Should be clearer now. Basically, there 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$K$. (This is bad notation, but it means that we invert the primes of$D$.) Let me be more precise. Let$D$be a simple normal crossings divisor on$X=\mathbf{P}^1_{\mathbf{Z}}$\mathbf{P}^1_{\mathbf{Z}}$ and let $U= X-D$ U$be its complement. Main question: Can we "describe" the etale coverings of$U$? (Here describe should be taken lightly.) Basically, can one say something like "these are the extensions complement of$\mathbf{Q}$unramified outside D..." Questions that naturally arise from the discussion in my previous question (see above link) are the followingits support. Q2. What is known about the etale fundamental group in this case, apart from being ? Is it "finitely generatedgenerated"? Has anybody studied the maximal pro-p-quotients of these groups? (I have to admit that I don't really know what that means.) Q3. The analytic analogue would be to consider the same question for$\mathbf{P}^2_{\mathbf{C}}$. Here we have a Grauert-Remmert theorem, but do we also have information about the fundamental group? \mathbf{P}^1_{\mathbf{C}} \times \mathbf{P}^1_{\mathbf{C}}$.

Q5. Does the notion of rational singularity still make sense in this context? If yes and if $Y$ is the normalization of $X$ in the function field of $V$, can we show that $Y$ has only rational singularities? This would result from an arithmetic Abyankar's lemma by Angelo's answer. (Here I'm talking about the question http://mathoverflow.net/questions/23091/is-there-an-obvious-way-for-showing-singularities-are-quotient )

I think these questions are really interesting. I strongly believe that the results in the analytic case (not just for Riemann surfaces) "should" generalize to the arithmetic setting.

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