Definition 66.3.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$. Let $f : U \to X$ be a morphism over $S$. We say the *fibres of $f$ are universally bounded*^{1} if there exists an integer $n$ such that for all fields $k$ and all morphisms $\mathop{\mathrm{Spec}}(k) \to X$ the fibre product $\mathop{\mathrm{Spec}}(k) \times _ X U$ is a finite scheme over $k$ whose degree over $k$ is $\leq n$.

[1] This is probably nonstandard notation.

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