Let $X\subset \mathbb{A}^N_k$ be an irreducible smooth variety over an algebraically closed field $k$. Suppose we have an etale map $\pi:X\to \mathbb{A}^1_k$. Are there any bounds on the degree of $\pi$? Here etale means flat with smooth, finite fibers. I'm interested mainly in the characteristic zero case.

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    $\begingroup$ If $X \to \mathbb{A}^1_k$ is a finite étale cover and $X$ is connected then it is an isomorphism. No? For $k = \mathbb{C}$ this corresponds to the fact that $\mathbb{C}$ is simply connected. $\endgroup$ – Zhen Lin Aug 12 '15 at 19:17
  • $\begingroup$ @ZhenLin: Thanks! you are right. I think I confused myself by thinking that degree 2 plane curves have genus zero and must give an etale degree 2 map to the affine line. $\endgroup$ – adrido Aug 13 '15 at 0:49

Here your $X$ is just a covering space of an open subset of $\mathbb{A}^1_k$ (this is because etale maps are open). If $k$ is characteristic 0, then this is basically reduces to the case of $k = \mathbb{C}$. In this case the Riemann-existence theorem says that everything can be done topologically.

In particular you may be interested in the theory of covering spaces, which among many things says that isomorphism classes of degree $n$ covering spaces of a "nice" topological space $X$ are in bijection with equivalence classes of permutation representations $\pi_1(X)\rightarrow S_n$, where the equivalence relation is "modulo composition by inner automorphisms of $S_n$". Here connected covers correspond to transitive permutation representations (ie, where the image of $\pi_1(X)$ is a transitive subgroup of $S_n$).

You're basically asking for a bound on the degrees of finite covers of subsets of $\mathbb{C}$, for which there is none.

Suppose for example you want covers of $\mathbb{C}\setminus\{0\}$. The fundamental group of the punctured plane is $\mathbb{Z}$, and there are transitive representations $\mathbb{Z}\rightarrow S_n$ for every $n$. In fact the universal cover here is connected and has infinite degree.

If you're only interested in surjective etale maps $X\rightarrow\mathbb{A}^1_k$, then since $\mathbb{C}$ is simply connected, there are no nontrivial covers of $\mathbb{A}^1_k$.

Even if you allow $k$ to be a number field you can use the etale fundamental group to compute that every cover of a subset of $\mathbb{A}^1$ realizable over $\mathbb{C}$ is realizable over $\mathbb{Q}$ (and hence over any number field).

Though, note that if $k$ is not algebraically closed then there exist plenty of covers of $\mathbb{A}^1_k$ coming from taking separable extensions of $k$.

If $k$ is characteristic $p$ then things get quite interesting. For example see https://mathoverflow.net/questions/868/etale-covers-of-the-affine-line


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