Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 24G / a

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- Solution a

Solution

For a nonconstant holomorphic map

f : X \to Y

of degree

d

between compact Riemann surfaces, Riemann-Hurwitz formula is

2 g_{X} - 2 = d (2 g_{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.

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