Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 24G (Riemann Surfaces)

For a nonconstant holomorphic map $f:X\to Y$ of degree $d$ between compact Riemann surfaces, Riemann-Hurwitz formula is $2g_X-2=d(2g_Y-2)+{\sum }_p(e_p-1)$. Triangulate $Y$ with branch values as vertices and lift the triangulation: faces and edges lift $d$ times, while the vertex deficit is exactly the ramification sum. Comparing Euler characteristics proves the formula.

Compactify $y^d=g(z)$ by adding the points over infinity. Since $d\mid n$, there are $d$ such points and they are unramified. Each of the $n$ simple finite roots is totally ramified with index $d$. Riemann-Hurwitz for the degree-$d$ map to ${\mathbb{P}}^1$ gives