The Stacks project

Remark 55.12.2. In the situation of Lemma 55.12.1 we can also say exactly how the genus $g_ i$ of $C_ i$ and the genus $g'_ i$ of $C'_ i$ are related. The formula is

\[ g'_ i = \frac{w_ i}{w'_ i}(g_ i - 1) + 1 + \frac{(C_ i \cdot C_ n)^2 - w_ n(C_ i \cdot C_ n)}{2w'_ iw_ n} \]

where $w_ i = [\kappa _ i : k]$, $w_ n = [\kappa _ n : k]$, and $w'_ i = [\kappa '_ i : k]$. To prove this we consider the short exact sequence

\[ 0 \to \mathcal{O}_{X'}(-C'_ i) \to \mathcal{O}_{X'} \to \mathcal{O}_{C'_ i} \to 0 \]

and its pullback to $X$ which reads

\[ 0 \to \mathcal{O}_ X(-C'_ i - e_ iC_ n) \to \mathcal{O}_ X \to \mathcal{O}_{C_ i + e_ i C_ n} \to 0 \]

with $e_ i$ as in the proof of Lemma 55.12.1. Since $Rf_*f^*\mathcal{L} = \mathcal{L}$ for any invertible module $\mathcal{L}$ on $X'$ (details omitted), we conclude that

\[ Rf_*\mathcal{O}_{C_ i + e_ i C_ n} = \mathcal{O}_{C'_ i} \]

as complexes of coherent sheaves on $X'_ k$. Hence both sides have the same Euler characteristic and this agrees with the Euler characteristic of $\mathcal{O}_{C_ i + e_ i C_ n}$ on $X_ k$. Using the exact sequence

\[ 0 \to \mathcal{O}_{C_ i + e_ i C_ n} \to \mathcal{O}_{C_ i} \oplus \mathcal{O}_{e_ iC_ n} \to \mathcal{O}_{C_ i \cap e_ iC_ n} \to 0 \]

and further filtering $\mathcal{O}_{e_ iC_ n}$ (details omitted) we find

\[ \chi (\mathcal{O}_{C'_ i}) = \chi (\mathcal{O}_{C_ i}) - {e_ i + 1 \choose 2}(C_ n \cdot C_ n) - e_ i(C_ i \cdot C_ n) \]

Since $e_ i = -(C_ i \cdot C_ n)/(C_ n \cdot C_ n)$ and $(C_ n \cdot C_ n) = -w_ n$ this leads to the formula stated at the start of this remark. If we ever need this we will formulate this as a lemma and provide a detailed proof.

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