Lemma 53.23.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \subset X$ be a rational bridge (Example 53.23.1). Then $\deg (\omega _ X|_ C) = 0$.
Proof. Let $X' \subset X$ be as in the example. Then we have a short exact sequence
\[ 0 \to \omega _ C \to \omega _ X|_ C \to \mathcal{O}_{C \cap X'} \to 0 \]
See Lemmas 53.4.6, 53.19.16, and 53.19.17. With $k''/k'/k$ as in the example we see that $\deg (\omega _ C) = -2[k' : k]$ as $C$ has genus $0$ (Lemma 53.5.2) and $\deg (C \cap X') = [k'' : k] = 2[k' : k]$. Hence $\deg (\omega _ X|_ C) = 0$. $\square$
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