Lemma 33.44.17. Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Set $g = \dim _ k H^1(X, \mathcal{O}_ X)$. For every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ with $\deg (\mathcal{L}) \geq 2g - 1$ we have $H^1(X, \mathcal{L}) = 0$.
Proof. Let $\mathcal{N}$ be the invertible module we found in Lemma 33.44.16 part (2). The degree of $\mathcal{N}$ is $\chi (X, \mathcal{N}) - \chi (X, \mathcal{O}_ X) = 0 - (1 - g) = g - 1$. Hence the degree of $\mathcal{L} \otimes \mathcal{N}^{\otimes - 1}$ is $\deg (\mathcal{L}) - (g - 1) \geq g$. Hence $\chi (X, \mathcal{L} \otimes \mathcal{N}^{\otimes -1}) \geq g + 1 - g = 1$. Thus there is a nonzero global section $s$ whose zero scheme is an effective Cartier divisor $D$ of degree $\deg (\mathcal{L}) - (g - 1)$. This gives a short exact sequence
\[ 0 \to \mathcal{N} \xrightarrow {s} \mathcal{L} \to i_*(\mathcal{L}|_ D) \to 0 \]
where $i : D \to X$ is the inclusion morphism. We conclude that $H^0(X, \mathcal{L})$ maps isomorphically to $H^0(D, \mathcal{L}|_ D)$ which has dimension $\deg (\mathcal{L}) - (g - 1)$. The result follows from the definition of degree. $\square$
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