Let $C$ be a nonsingular curve over an algebraically closed field of characteristic zero and $G$ a finite group acting on $C$. Then the quotient scheme $C/G$ exists and is again nonsingular, projective and of dimension one. Does this quotient have the same genus as $C$ or do we have another genus...And if so, what is the genus?
The genus of the quotient will be given by the Riemann--Hurwitz formula: if the quotient map $C \rightarrow C/G$ has degree $d$, then R--H says
$$2g(C)-2 = d \cdot \left( 2 g(C/G) - 2 \right) + \sum_p (n(p)-1) $$
where the sum is over the ramification points $p$ of the quotient map and $n(p)$ is the so-called ramification index at $p$. So in fact if $g(C)>1$ and $d>1$, the genus of the quotient must be smaller.