Lemma 53.12.2 (0C1D)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 53: Algebraic Curves
Section 53.12: Riemann-Hurwitz
Lemma 53.12.2 (cite)

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Lemma 53.12.2. Let $f : X \to Y$ be a morphism of smooth proper curves over a field $k$ which satisfies the equivalent conditions of Lemma 53.12.1. 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 $f$.

Proof. See discussion above; we used Discriminants, Lemma 49.12.6, Lemma 53.8.4, and Varieties, Lemmas 33.44.7 and 33.44.11. $\square$

Comments (2)

Comment #4013 by Dario Weißmann on February 20, 2019 at 08:40

typo: $g_y$ should be $g_Y$

Comment #4123 by Johan on April 06, 2019 at 11:19

Thanks and fixed here.

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