Past exam of the geometry course of the University of Oxford / 2011 / 2 / a / Solution

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, let $n$ be its degree and $e_p$ its local degree at $p$. The Riemann-Hurwitz formula isEquivalently, $2g_X-2=n(2g_Y-2)+{\sum }_p(e_p-1)$. Only finitely many terms are nonzero; their sum is the total ramification. In local coordinates the map has the form $w=z^{e_p}$ after suitable coordinate changes.