Branch Points (reviewed)

If $G$ is a finite group of automorphisms acting on a Riemann surface $X$, and $\phi\colon X\to X/G$ is the natural projection to the quotient $X/G$ (the compact Riemann surface whose points are the $G$-orbits of $X$), then all but finitely many fibers of the map $\phi$ have cardinality $d:=|G|$.

The points $P$ of $X/G$ whose fibers $\phi^{-1}(P)$ have cardinality strictly less than $d$ are called branch points.