Lemma 53.6.7. In Situation 53.6.2. Let $\mathcal{L}$ be a very ample invertible $\mathcal{O}_ X$-module with $\deg (\mathcal{L}) \geq 2$. Then $\omega _ X \otimes _{\mathcal{O}_ X} \mathcal{L}$ is globally generated.

Proof. Assume $k$ is algebraically closed. Let $x \in X$ be a closed point. Let $C_ i \subset X$ be the irreducible components and for each $i$ let $x_ i \in C_ i$ be the generic point. By Varieties, Lemma 33.22.2 we can choose a section $s \in H^0(X, \mathcal{L})$ such that $s$ vanishes at $x$ but not at $x_ i$ for all $i$. The corresponding module map $s : \mathcal{O}_ X \to \mathcal{L}$ is injective with cokernel $\mathcal{Q}$ supported in finitely many points and with $H^0(X, \mathcal{Q}) \geq 2$. Consider the corresponding exact sequence

$0 \to \omega _ X \to \omega _ X \otimes \mathcal{L} \to \omega _ X \otimes \mathcal{Q} \to 0$

By Lemma 53.6.5 we see that the module generated by global sections surjects onto $\omega _ X \otimes \mathcal{Q}$. Since $x$ was arbitrary this proves the lemma. Some details omitted.

We will reduce the case where $k$ is not algebraically closed, to the algebraically closed field case. We suggest the reader skip the rest of the proof. Choose an algebraic closure $\overline{k}$ of $k$ and consider the base change $X_{\overline{k}}$. Let us check that $X_{\overline{k}} \to \mathop{\mathrm{Spec}}(\overline{k})$ is an example of Situation 53.6.2. By flat base change (Cohomology of Schemes, Lemma 30.5.2) we see that $H^0(X_{\overline{k}}, \mathcal{O}) = \overline{k}$. The scheme $X_{\overline{k}}$ is proper over $\overline{k}$ (Morphisms, Lemma 29.41.5) and equidimensional of dimension $1$ (Morphisms, Lemma 29.28.3). The pullback of $\omega _ X$ to $X_{\overline{k}}$ is the dualizing module of $X_{\overline{k}}$ by Lemma 53.4.4. The pullback of $\mathcal{L}$ to $X_{\overline{k}}$ is very ample (Morphisms, Lemma 29.38.8). The degree of the pullback of $\mathcal{L}$ to $X_{\overline{k}}$ is equal to the degree of $\mathcal{L}$ on $X$ (Varieties, Lemma 33.44.2). Finally, we see that $\omega _ X \otimes \mathcal{L}$ is globally generated if and only if its base change is so (Varieties, Lemma 33.22.1). In this way we see that the result follows from the result in the case of an algebraically closed ground field. $\square$

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