Definition 33.44.1 (0AYR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.44: Degrees on curves
Definition 33.44.1 (cite)

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Definition 33.44.1. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$ , and let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. The degree of $\mathcal{L}$ is defined by

$\deg (\mathcal{L}) = \chi (X, \mathcal{L}) - \chi (X, \mathcal{O}_ X)$

More generally, if $\mathcal{E}$ is a locally free sheaf of rank $n$ we define the degree of $\mathcal{E}$ by

$\deg (\mathcal{E}) = \chi (X, \mathcal{E}) - n\chi (X, \mathcal{O}_ X)$

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