Degree of ramification divisor and number of fixed points under a group action.

Question

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

$2g_C-2=|G|(2g_{C'}-2)+\deg R,$

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?

Answer 1

The ramification divisor $R$ is supported at those points $x\in C$ such that $f^{-1}(f(x))$ consists of strictly less than $|G|$ points. But $f^{-1}(f(x))\subset C$ is the orbit of $x$ (as $C\to C/G$ is a geometric quotient). As such, it has cardinality $|G|/|G_x|$. Thus, $$|f^{-1}(f(x))|<|G|\iff G_x\neq 1\iff \textrm{the action is not free at }x.$$ So the degree of $R$ should count these $x\in C$ where the action is not free (including fixed points). Of course, a fixed point $x\in C$ has stabilizer $G_x=G$, which means that $f$ is totally ramified at $x$.