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Let $C$ be a nonsingular complex projective curve and a finite group $G$ acts on $C$, then if $C/G$ is a irreducible nonsingular curve?

I know from Shafarevich's Basic Algebraic geometry 1, if $G$ acts on $C$ freely then $C/G$ is nonsigular. Here I wonder if its' true in general and if $C/G$ is irreducible.

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  • Assuming $C$ is irreducible, the quotient will be irreducible since the image of an irreducible variety is always irreducible. Not sure about smoothness. The first naive examples with non-trivial stabilizer gives a smooth curve ($C = \Bbb A^1, G = {\pm 1}$). – Nicolas Hemelsoet Apr 30 '19 at 00:42
  • https://math.stackexchange.com/questions/160170/riemann-surface-and-discontinuous-group-action?rq=1 Also check Miranda's book. – Moishe Kohan Apr 30 '19 at 01:25

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