Theorem
Let \(f:X\to Y\) be a cover of degree \(d\) and let \(R\) be the ramification divisor \[R = \sum_{p\in Y} (\nu_p(t) - 1)[p]\] where \(t\) is the local coordinate at \(p\in Y\). Then \[2g(Y) - 2 = d(2g(X) - 2) + \deg R.\]
Theorem
Let \(f:X\to Y\) be a cover of degree \(d\) and let \(R\) be the ramification divisor \[R = \sum_{p\in Y} (\nu_p(t) - 1)[p]\] where \(t\) is the local coordinate at \(p\in Y\). Then \[2g(Y) - 2 = d(2g(X) - 2) + \deg R.\]