[Lemma 2, Jongmin]

Lemma 53.21.2. In Situation 53.6.2 assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $Z \subset X$ be a $0$-dimensional closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. If $\deg (\mathcal{L}) > 2g - 2 + \deg (Z)$, then $H^1(X, \mathcal{I}\mathcal{L}) = 0$ and one of the following possibilities occurs

1. $H^0(X, \mathcal{I}\mathcal{L}) \not= 0$, or

2. $g = 0$ and $\deg (\mathcal{L}) = \deg (Z) - 1$.

In case (2) if $Z = \emptyset$, then $X \cong \mathbf{P}^1_ k$ and $\mathcal{L}$ corresponds to $\mathcal{O}_{\mathbf{P}^1}(-1)$.

Proof. The vanishing of $H^1(X, \mathcal{I}\mathcal{L})$ follows from Lemma 53.21.1. If $H^0(X, \mathcal{I}\mathcal{L}) = 0$, then $\chi (\mathcal{I}\mathcal{L}) = 0$. From the short exact sequence $0 \to \mathcal{I}\mathcal{L} \to \mathcal{L} \to \mathcal{O}_ Z \to 0$ we conclude $\deg (\mathcal{L}) = g - 1 + \deg (Z)$. Thus $g - 1 + \deg (Z) > 2g - 2 + \deg (Z)$ which implies $g = 0$ hence (2) holds. If $Z = \emptyset$ in case (2), then $\mathcal{L}^{-1}$ is an invertible sheaf of degree $1$. This implies there is an isomorphism $X \to \mathbf{P}^1_ k$ and $\mathcal{L}^{-1}$ is the pullback of $\mathcal{O}_{\mathbf{P}^1}(1)$ by Lemma 53.10.2. $\square$

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