Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient
$f: C\rightarrow C/G=:C'$.
Then, by Riemann–Hurwitz
where $R$ is ramification divisor.
I guess that the number $\deg R$ is somehow connected to the number of fixed points of $G$ action, but I can't make this precise. Am I right?