 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2011/2/a/solution

f : X \to Y

between compact connected Riemann surfaces, let

n

be its degree and

e_{p}

its local degree at

p

χ (X) = n χ (Y) - p \in X \sum (e_{p} - 1) .

Equivalently,

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

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