Lemma 53.8.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. If $X$ is Gorenstein, then
\[ \deg (\omega _ X) = 2g - 2 \]
where $g$ is the genus of $X$ and $\omega _ X$ is as in Lemma 53.4.1.
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