Past exam of the geometry course of the University of Oxford / 2009 / A / 2 / i / Solution

For a nonconstant holomorphic map of degree $d$ between compact connected Riemann surfaces, the Riemann-Hurwitz formula isHere $e_p$ is the local degree, characterized by local coordinates in which the map is $z\mapsto z^{e_p}$. The sum is finite and its terms are nonnegative; a ramification point has $e_p>1$.