 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2009/a/2/i/solution

For a nonconstant holomorphic map of degree

d

2 - 2 g_{X} = d (2 - 2 g_{Y}) - p \in X \sum (e_{p} - 1) .

Here

e_{p}

is the local degree, characterized by local coordinates in which the map is

z \mapsto z^{e_{p}}

. The sum is finite and its terms are nonnegative; a ramification point has

e_{p} > 1

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