The Stacks project

Lemma 33.45.12. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $Z \subset X$ be a closed subscheme of dimension $\leq 1$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

\[ (\mathcal{L} \cdot Z) = \deg (\mathcal{L}|_ Z) \]

where $\deg (\mathcal{L}|_ Z)$ is as in Definition 33.44.1. If $\mathcal{L}$ is ample, then $\deg _\mathcal {L}(Z) = \deg (\mathcal{L}|_ Z)$.

Proof. This follows from the fact that the function $n \mapsto \chi (Z, \mathcal{L}|_ Z^{\otimes n})$ has degree $1$ and hence the leading coefficient is the difference of consecutive values. $\square$

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