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For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

Let $f:X\to Y$ be a nonconstant holomorphic map between compact Riemann surfaces that are connected spaces, without boundary. Write $g_X,g_Y$ for their genera and $d\ge 1$ for the degree of a holomorphic map. With analytic ramification index $e_p$ at each $p\in X$, the formula isHere $\chi =2-2g$ is the Euler characteristic. Only finitely many ramification points contribute. The holomorphic map is automatically surjective; the valency theorem gives ${\sum }_{p\in f^{-1}(y)}{e}_p=d$ even at a branch value of a holomorphic map. Constant maps are excluded.^[ref]

The useful interpretation is that total ramification of a holomorphic map measures how much the source Euler characteristic falls short of $d$ copies of the target. The ramification of a power map of the Riemann sphere spends that entire correction at two points; the double cover of the sphere with four branch values spends it at four points and produces a genus-one surface.