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I'm following the proof of Prop. 4.4., Chapter 7.4. Liu-Algebraic Geometry and Arithmetic Curves:

$\textbf{Proposition 4.4.}$ Let $X$ be a smooth, geometrically connected, projective curve over a field $k$ of genus $g$. Let $\mathcal{L}$ be an invertible sheaf on X. (a) If $\text{deg}\mathcal{L}\geq2g$, then $\mathcal{L}$ is generated by its global sections.

The proof goes as follows: We may suppose that $k$ is algebraically closed, and $\mathcal{O}_X(D)\simeq \mathcal{L}$ for a divisor $D$ on $X$. We then can show that \begin{equation}l(D-E)=l(D)-\text{deg}E \end{equation} for all effective divisors $E$ such that $\text{deg}E\leq 1$. Liu then mentions that $\mathbf{\text{in particular, }l(D)\neq 0\text{, so by linear equivalence } D\geq 0}$ . The claim should then follow by Lemma 4.2., so I follow its proof applied to my problem to deduce the claim:

For all closed points $x\in X$, we should have $\mathcal{O}_X(D-x)_x=\mathfrak{m}_x\mathcal{O}_X(D)_x$. Further, since all closed $x\in X$ are effective divisor of degree 1, we have $l(D-x)<l(D)$. Consequently, we find an $s\in H^0(X,\mathcal{O}(D))\setminus H^0(X,\mathcal{O}(D-x))$. Thus, $s_x\notin \mathcal{O}_X(D-x)_x$ so $s_x$ is a generator of $\mathcal{O}(D)_x$.

$\textbf{My question}$: Where do I need the information $D\geq 0$ that Liu was deducing in the proof of the proposition?

$\textbf{Lemma 4.2. }$ Let $X$ be a projective curve over a field $k$ and let $D$ be an effective divisor on $X$ with support in the regular locus of $X$. Then $\mathcal{O}_X(D)$ is generated by its global sections iff for any $x\in \text{supp}(D)$, we have $l(D-x)<l(D)$.

Thank you very much for all help!!!

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You need to know that $D\geq 0$ in order to apply Lemma 4.2. The lemma requires that $D$ be effective, ie $D\geq 0$.

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    $\begingroup$ Thank you for your answer! I know that the Lemma requires $D\geq 0$. Yet, as far as I understand, this is because we don't have $l(D-x)<l(D)$ for all $x\notin\text{supp}(D)$ in the Lemma. To prove the claim for these $x$ one then requires $D\geq 0$. However, in my special case in the proposition, we have also $l(D-x)<l(D)$ for all $x\notin\text{supp}(D)$ and, as outlined above, I think one then doesn't use $D\geq 0$. $\endgroup$
    – user220164
    Mar 2, 2015 at 0:24
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    $\begingroup$ It's entirely possible that you don't need to use $D\geq 0$ because of the special circumstances of your problem. But it should not be particularly offensive to you that we applied a fact that was true (remember, $l(D)\neq 0$) to use a lemma that was perhaps a little overkill. It's a nicer, shorter proof than writing out all of the details of the proof of 4.2 again. $\endgroup$
    – KReiser
    Mar 2, 2015 at 1:55
  • $\begingroup$ Why is it possible to assume that the field $k$ is algebraically closed? This seems to be a major difference compared to other books (e.g., Hartshorne, Vakil's notes, Görtz and Wedhorn). Thank you! $\endgroup$
    – J224
    Dec 31, 2020 at 15:53

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