Local cyclic quotient of a Riemann surface
= Local cyclic quotient of a Riemann surface
Let a finite group $H$ of conformal automorphisms of the Riemann sphere fix $p$. The derivative representation at $p$ embeds $H$ into $\mathbb C^\times$, so $H$ is cyclic. Averaging a local coordinate linearizes the action to $z\mapsto\zeta z$, and the quotient has coordinate $u=z^{|H|}$.