In Liu's answer to this MO question there is a characterization of smooth affine varieties which are étale over affine space. I was wondering if one can give a similar characterization for projective varieties.

Let $X$ be a smooth projective variety of dimension $n$. When is $X$ étale over $\mathbb A^n$? How can one construct an étale map $X\to \mathbb A^n$?

I should maybe point out that I'm primarily interested in the case when the base field is $\mathbb C$.

Thank you for any help!

  • $\begingroup$ Alternatively, you could ask which projective varieties are etale over projective space. The answer would be equally disappointing. $\endgroup$ – Nefertiti Feb 4 '16 at 14:02
  • $\begingroup$ Thanks for the comment. Could you explain it a little bit more? Now proper subvarieties of projective space are not just points. So what is the contradiction? $\endgroup$ – Brenin Feb 6 '16 at 11:57
  • $\begingroup$ Dear Brenin, there is no basic contradiction (that I know of), but nevertheless it is true that projective space has no etale covers by a projective variety. For $\mathbf P^1$ this is an easy application of Riemann--Hurwitz. One can then extend this to higher dimensions by an inductive argument. See this MO question and (great) answers for details: mathoverflow.net/questions/62282/mathbbpn-is-simply-connected $\endgroup$ – Nefertiti Feb 8 '16 at 13:48
  • $\begingroup$ Thanks! Those answers are really instructive. $\endgroup$ – Brenin Feb 8 '16 at 20:46

The image of a projective variety under any morphism is proper, and the only subvarieties of $\mathbb{A}^n$ that are proper are those of dimension $0$. So a projective variety of dimension $n$ is etale over $\mathbb{A}^n$ iff $n=0$.


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