Lemma 53.21.1. In Situation 53.6.2 assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $Z \subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. If $H^1(X, \mathcal{I}\mathcal{L})$ is nonzero, then

\[ \deg (\mathcal{L}) \leq 2g - 2 + \deg (Z) \]

with strict inequality unless $\mathcal{I}\mathcal{L} \cong \omega _ X$.

**Proof.**
Any curve, e.g. $X$, is Cohen-Macaulay. If $H^1(X, \mathcal{I}\mathcal{L})$ is nonzero, then there is a nonzero map $\mathcal{I}\mathcal{L} \to \omega _ X$, see Lemma 53.4.2. Since $\mathcal{I}\mathcal{L}$ is torsion free, this map is injective. Since a field is Gorenstein and $X$ is reduced, we find that the Gorenstein locus $U \subset X$ of $X$ is nonempty, see Duality for Schemes, Lemma 48.24.4. This lemma also tells us that $\omega _ X|_ U$ is invertible. In this way we see we have a short exact sequence

\[ 0 \to \mathcal{I}\mathcal{L} \to \omega _ X \to \mathcal{Q} \to 0 \]

where the support of $\mathcal{Q}$ is zero dimensional. Hence we have

\begin{align*} 0 & \leq \dim \Gamma (X, \mathcal{Q})\\ & = \chi (\mathcal{Q}) \\ & = \chi (\omega _ X) - \chi (\mathcal{I}\mathcal{L}) \\ & = \chi (\omega _ X) - \deg (\mathcal{L}) - \chi (\mathcal{I}) \\ & = 2g - 2 - \deg (\mathcal{L}) + \deg (Z) \end{align*}

by Lemmas 53.5.1 and 53.5.2, by (53.8.1.1), and by Varieties, Lemmas 33.33.3 and 33.44.5. We have also used that $\deg (Z) = \dim _ k \Gamma (Z, \mathcal{O}_ Z) = \chi (\mathcal{O}_ Z)$ and the short exact sequence $0 \to \mathcal{I} \to \mathcal{O}_ X \to \mathcal{O}_ Z \to 0$. The lemma follows.
$\square$

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