The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 086R. Beware of the difference between the letter 'O' and the digit '0'.