Lemma 33.44.15 (0B5Y)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.44: Degrees on curves
Lemma 33.44.15 (cite)

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Lemma 33.44.15. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $C_ i \subset X$, $i = 1, \ldots , t$ be the irreducible components of dimension $1$. The following are equivalent:

$\mathcal{L}$ is ample, and
$\deg (\mathcal{L}|_{C_ i}) > 0$ for $i = 1, \ldots , t$.

Proof. Let $x_1, \ldots , x_ r \in X$ be the isolated closed points. Think of $x_ i = \mathop{\mathrm{Spec}}(\kappa (x_ i))$ as a scheme. Consider the morphism of schemes

\[ f : C_1 \amalg \ldots \amalg C_ t \amalg x_1 \amalg \ldots \amalg x_ r \longrightarrow X \]

This is a finite surjective morphism of schemes proper over $k$ (details omitted). Thus $\mathcal{L}$ is ample if and only if $f^*\mathcal{L}$ is ample (Cohomology of Schemes, Lemma 30.17.2). Thus we conclude by Lemma 33.44.14. $\square$

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