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Let $X$ be a surface (algebraic projective smooth complex) and suppose $\sigma$ is an automorphism of finite order $d$. Let $Y=X/\sigma$.

I wonder under which simple conditions on $\sigma$ is $Y$ a smooth surface.

For example it seems reasonable that this is the case when $\sigma$ has no fixed points, is that true?

What about more general conditions (i.e. allowing a fixed locus for $\sigma$)? Also, what about the canonical bundle of $Y$ in relation to that of $X$?

Basically I would like to get a general basic picture of this situation for the particular case of complex surfaces.

Heitor Fontana
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(Reference for these results: SGA 1, Expose 5.2. I don't know of a source outside of SGA for just complex varieties, but that source probably exists.)

It is indeed the case that $Y$ is a smooth surface if $\sigma$ has no fixed points. The key to proving this is to note that in this case, $f:X\rightarrow Y$ will be etale, so $Y$ will be smooth by SGA 1, Expose 1.9.1.

As this map is etale, we just have $K_X=f^*K_Y$.

David Yang
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  • thanks for your answer David. Could you add some comments for the case when the fixed locus has no isolated points, but only curves (the quotient should be smooth here too, right)? – Heitor Fontana Nov 08 '13 at 15:18
  • I don't think that's true; just take some variety where the fixed locus has some isolated point, and then multiply it by $\mathbb{A}^1$ with the trivial action and you get a counterexample for that. – David Yang Nov 08 '13 at 16:33