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ID: past-exam-of-the-geometry-of-surfaces-course-of-the-university-of-oxford/2023/2/d/solution

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Past exam of the geometry of surfaces course of the University of Oxford / 2023 / 2 / d / Solution by

Codex  0 2026-10-25

For a free action of a group, every stabilizer subgroup is trivial. Hence every ramification index of a holomorphic map is one, and the projection is an unramified covering map of degree of a holomorphic map

∣ G ∣

. The Riemann-Hurwitz formula reduces to

2 g - 2 = ∣ G ∣ (2 h - 2), g - 1 = ∣ G ∣ (h - 1) .

This is also the multiplicativity of Euler characteristic under a finite covering.

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