Let $k$ be a an algebraically closed field of char. $0$. Is there a morphism of varieties over $k$ which is:
1) etale
2) such that fibers at closed points all have the same size
3) yet not finite?

  • 10
    $\begingroup$ No. This EGA IV, Prop.18.2.8 and Corollary 18.2.9. $\endgroup$ – abx Mar 14 '14 at 12:54
  • 7
    $\begingroup$ And a very stupid example shows a separatedness hypothesis is necessary : take the base to be $\mathbb{A}^1$ and the total space be the disjoint union of $\mathbb{A}^1$ with double origine and $\mathbb{A}^1$ with the origin removed. $\endgroup$ – Olivier Benoist Mar 14 '14 at 13:03
  • 4
    $\begingroup$ A beautiful generalization is given as Lemma 1.19 in Ch. II of the paper of Deligne and Rapoport on generalized elliptic curves (and proved via valuative criterion for properness and Zariski's Main Theorem): if $f:X \rightarrow Y$ is a separated quasi-finite flat map of finite presentation and the fiber-rank function is locally constant on $Y$ then $f$ is finite. (They assume $X$ and $Y$ are noetherian, but the proof can be adapted to work without that condition.) $\endgroup$ – user76758 Mar 14 '14 at 14:18

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