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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