Lemma 53.22.2. 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 tail (Example 53.22.1). 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 $k'$ as in the example we see that $\deg (\omega _ C) = -2[k' : k]$ as $C \cong \mathbf{P}^1_{k'}$ by Proposition 53.10.4 and $\deg (C \cap X') = [k' : k]$. Hence $\deg (\omega _ X|_ C) = -[k' : k]$ which is negative. $\square$
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