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Can one take quotient by finite group in the category of schemes? Will the singularities be visible? For example, it looks like $\mathbb{C}/\{z \sim -z\}$ is isomorphic to $\mathbb{C}$. What about complex analytic spaces - can we take quotients by finite groups and are the singularities visible?

From the comments, I know that quotient singularities are visible on schemes/analytic spaces of complex dimension $\geq 2$. Are there examples of quotient singularities on curves?

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  • $\begingroup$ I do not complete understand what you means by sayng "are the singularities visibles?". Are you asking whether the quotient space (scheme or analytic variety) can be singular? Of course it can be. The quotient of $\mathbb{C}^2$ by the involution $(x, y) \cong (-x, -y)$ is isomorphic to a quadric cone (look at the ring of invariants). The key words are "Quotient singularities". There is a huge literature about. $\endgroup$
    – Francesco Polizzi
    Commented Sep 24, 2013 at 10:02
  • $\begingroup$ @francesco-polizzi : Actually I was only looking at curves. Is it the case that quotient singularities won't occur on schemes (or analytic spaces) corresponding to curves? $\endgroup$
    – Anon
    Commented Sep 24, 2013 at 11:36
  • $\begingroup$ If you take the quotient of a smooth curve by a finite group, you will get a smooth curve. $\endgroup$
    – S. Carnahan ♦
    Commented Sep 24, 2013 at 12:10
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    $\begingroup$ The quotient of a smooth curve for a finite group of automorphism is smooth. The point is that quotient singularities are normal, so they can only occur in codimension $\geq 2$. $\endgroup$
    – Francesco Polizzi
    Commented Sep 24, 2013 at 12:11

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