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Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$.

Assume that $f$ is a covering space topologically of degree $d<\infty$.

Can we give $X$ the structure of a smooth affine curve such that $f:X\to \mathbf{P}^1_k-B$ becomes a finite etale morphism?

This is possible if $k=\mathbf{C}$.

  • 0
    What is the topology on $P^1_k \setminus B$ for your covering space ?2012-03-14
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    I consider $U:=\mathbf{P}^1_k\backslash B$ with the open subscheme structure. Although this isn't clear from the way I wrote it, I assume that $B$ is closed.2012-03-15
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    If you us the Zariski topology, then the covering is just trivial : $X$ is a disjoint union of $d$ copies of the base space (exercise :)).2012-03-15
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    Doesn't $\mathbf{P}^1-\{0,1,\infty\}$ have many non-trivial finite etale covers?2012-03-16
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    But the étale covers in algebraic geometry are not topological covers.2012-03-16
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    What a silly mistake of mine!! I was confused because the analytification of an etale cover is a topological cover.2012-03-16

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