let $X$ be a smooth projective irreducible curve of genus $g$ over the complex numbers. Assume that $X$ comes with an action of $\mu_d$.
Is the quotient $Y:=X/\mu_d$ always smooth?
Let $\pi: X \to Y$ be the quotient map. Is it possible to calculate the genus of $Y$ by considering the map induced on jacobians $J(X) \to J(Y)$ and lifting the action of $\mu_d$ to an action on $J(X)$?
Thanks for your help