Lemma 50.12.2. Let $f : X \to Y$ be a morphism of smooth proper curves over a field $k$ which satisfies the equivalent conditions of Lemma 50.12.1. If $k = H^0(Y, \mathcal{O}_ Y) = H^0(X, \mathcal{O}_ X)$ and $X$ and $Y$ have genus $g_ X$ and $g_ Y$, then

$2g_ X - 2 = (2g_ Y - 2) \deg (f) + \deg (R)$

where $R \subset X$ is the effective Cartier divisor cut out by the different of $f$.

Comment #4013 by Dario Weißmann on

typo: $g_y$ should be $g_Y$

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