Lemma 37.45.5 (086R)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.45: Applications of Zariski's Main Theorem, II
Lemma 37.45.5 (cite)

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Lemma 37.45.5. Let $f : X \to S$ be a flat, locally of finite presentation, separated, and locally quasi-finite morphism of schemes. Then there exist open subschemes

$S = U_0 \supset U_1 \supset U_2 \supset \ldots$

such that a morphism $\mathop{\mathrm{Spec}}(k) \to S$ where $k$ is a field factors through $U_ d$ if and only if $X \times _ S \mathop{\mathrm{Spec}}(k)$ has degree $\geq d$ over $k$ .

Proof. The statement simply means that the collection of points where the degree of the fibre is $\geq d$ is open. Thus we can work locally on $S$ and assume $S$ is affine. In this case, for every $W \subset X$ quasi-compact open, the set of points $U_ d(W)$ where the fibres of $W \to S$ have degree $\geq d$ is open by Lemma 37.45.4. Since $U_ d = \bigcup _ W U_ d(W)$ the result follows. $\square$

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