Lemma 54.16.7 (0C2J)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 54: Resolution of Surfaces
Section 54.16: Contracting exceptional curves
Lemma 54.16.7 (cite)

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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 $H^1(X, \mathcal{L}) \subset H^1(X, \mathcal{L}(E))$ .

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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