Lemma 54.16.7. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $n$ be the integer such that $\mathcal{L}|_ E$ has degree $n$ viewed as an invertible module on $\mathbf{P}^1$. Then

If $H^1(X, \mathcal{L}) = 0$ and $n \geq 0$, then $H^1(X, \mathcal{L}(iE)) = 0$ for $0 \leq i \leq n + 1$.

If $n \leq 0$, then $H^1(X, \mathcal{L}) \subset H^1(X, \mathcal{L}(E))$.

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