Lemma 33.44.10. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced and connected. Let $\kappa = H^0(X, \mathcal{O}_ X)$. Then $\kappa /k$ is a finite extension of fields and $w = [\kappa : k]$ divides

$\deg (\mathcal{E})$ for all locally free $\mathcal{O}_ X$-modules $\mathcal{E}$,

$[\kappa (x) : k]$ for all closed points $x \in X$, and

$\deg (D)$ for all closed subschemes $D \subset X$ of dimension zero.

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