Unramified covers of the affine line

by Anselm   Last Updated May 15, 2018 20:20 PM

Let $\kappa$ be an algebraically closed field of characteristic 0. Let $X$ be a smooth connected curve defined over $\kappa$, such that there is an unramified cover

$$\pi:X\to \mathbb{A}^1_\kappa$$ Is it $\pi$ necessarily an isomorphism?

Of course the assertion is easy if $\kappa=\mathbb{C}$ (by the simply connectedness of $\mathbb{C}$), but is false if $ch(\kappa)=p$: just take the map

$$V(y^p-y-1)\to \mathbb{A}^1_\kappa,\;\; (x,y)\mapsto x$$

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