The Stacks project

Weak version of [Lemma 4, Jongmin]

Lemma 53.21.6. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and of dimension $1$. 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 $H^1(X, \mathcal{I}\mathcal{L}) \not= 0$, then there exists a reduced connected closed subscheme $Y \subset X$ of dimension $1$ such that

\[ \deg (\mathcal{L}|_ Y) \leq -2\chi (Y, \mathcal{O}_ Y) + \deg (Z \cap Y) \]

where $Z \cap Y$ is the scheme theoretic intersection.

Proof. If $H^1(X, \mathcal{I}\mathcal{L})$ is nonzero, then there is a nonzero map $\varphi : \mathcal{I}\mathcal{L} \to \omega _ X$, see Lemma 53.4.2. Let $Y \subset X$ be the union of the irreducible components $C$ of $X$ such that $\varphi $ is nonzero in the generic point of $C$. Then $Y$ is a reduced closed subscheme. Let $\mathcal{J} \subset \mathcal{O}_ X$ be the ideal sheaf of $Y$. Since $\mathcal{J}\mathcal{I}\mathcal{L}$ has no embedded associated points (as a submodule of $\mathcal{L}$) and as $\varphi $ is zero in the generic points of the support of $\mathcal{J}$ (by choice of $Y$ and as $X$ is reduced), we find that $\varphi $ factors as

\[ \mathcal{I}\mathcal{L} \to \mathcal{I}\mathcal{L}/\mathcal{J}\mathcal{I}\mathcal{L} \to \omega _ X \]

We can view $\mathcal{I}\mathcal{L}/\mathcal{J}\mathcal{I}\mathcal{L}$ as the pushforward of a coherent sheaf on $Y$ which by abuse of notation we indicate with the same symbol. Since $\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Y, \omega _ X)$ by Lemma 53.4.5 we find a map

\[ \mathcal{I}\mathcal{L}/ \mathcal{J}\mathcal{I}\mathcal{L} \to \omega _ Y \]

of $\mathcal{O}_ Y$-modules which is injective in the generic points of $Y$. Let $\mathcal{I}' \subset \mathcal{O}_ Y$ be the ideal sheaf of $Z \cap Y$. There is a map $\mathcal{I}\mathcal{L}/\mathcal{J}\mathcal{I}\mathcal{L} \to \mathcal{I}'\mathcal{L}|_ Y$ whose kernel is supported in closed points. Since $\omega _ Y$ is a Cohen-Macaulay module, the map above factors through an injective map $\mathcal{I}'\mathcal{L}|_ Y \to \omega _ Y$. We see that we get an exact sequence

\[ 0 \to \mathcal{I}'\mathcal{L}|_ Y \to \omega _ Y \to \mathcal{Q} \to 0 \]

of coherent sheaves on $Y$ where $\mathcal{Q}$ is supported in dimension $0$ (this uses that $\omega _ Y$ is an invertible module in the generic points of $Y$). We conclude that

\[ 0 \leq \dim \Gamma (Y, \mathcal{Q}) = \chi (\mathcal{Q}) = \chi (\omega _ Y) - \chi (\mathcal{I}'\mathcal{L}) = -2\chi (\mathcal{O}_ Y) - \deg (\mathcal{L}|_ Y) + \deg (Z \cap Y) \]

by Lemma 53.5.1 and Varieties, Lemma 33.33.3. If $Y$ is connected, then this proves the lemma. If not, then we repeat the last part of the argument for one of the connected components of $Y$. $\square$

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