Let $X$ be a smooth projective curve over the field of complex numbers. Assume it comes with an action of $\mu_3$. Could someone explain to me why is the quotient $X/\mu_3$ a smooth curve of genus zero?
No, the result is not true for genus $1$.
Consider an elliptic curve $E$ (which has genus one) and a non-zero point $a\in E$ of order three .