The Stacks project

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

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

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

Proof. Observe that $\mathcal{L}|_ E = \mathcal{O}(n)$ by Divisors, Lemma 31.28.5. Use induction, the long exact cohomology sequence associated to the short exact sequence

\[ 0 \to \mathcal{L} \to \mathcal{L}(E) \to \mathcal{L}(E)|_ E \to 0, \]

and use the fact that $H^1(\mathbf{P}^1, \mathcal{O}(d)) = 0$ for $d \geq -1$ and $H^0(\mathbf{P}^1, \mathcal{O}(d)) = 0$ for $d \leq -1$. Some details omitted. $\square$

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