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Let $p \in \mathbb{Z}$ be a prime. Suppose we have a quasi-finite map $f \colon X \to \mathbb{A}^n$ of schemes over $\mathbb{Z}$, and let $f_p \colon X_{\mathbb{F}_p} \to \mathbb{A}_{\mathbb{F}_p}^n$ be the basechange of $f$ along $\operatorname{Spec} \mathbb{F}_p \to \operatorname{Spec} \mathbb{Z}$. Let $H \subset \mathbb{A}_{\mathbb{F}_p}^n$ be a hypersurface defined over $\mathbb{F}_p$, and let $U = \mathbb{A}_{\mathbb{F}_p}^n \setminus H$ be the complement. Suppose that neither $U(\mathbb{F}_p)$ nor $H(\mathbb{F}_p)$ is empty.

Question: Suppose that there is some number $N$ such that the fiber cardinality $\# f_p^{-1}(x) = N$ for every $x \in U$. Can anything be said about the fiber cardinality for $x \in H$? Is it true, for example, that $f_p^{-1}(x) \geq N$ for every $x \in H$?

What I know: I'm aware of the standard results on upper-semicontinuity of fiber cardinality / dimension over algebraically closed fields, so I was wondering what can be said over finite fields.

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    $\begingroup$ Don't the standard results hold as long you replace cardinality by degree? And is it really important that $X$ is defined over $\mathbb Z$? $\endgroup$
    – Asvin
    Jul 24, 2019 at 1:11
  • $\begingroup$ @Asvin Thanks for the response! I guess it is not important that $X$ is defined over $\mathbb{Z}$. About whether the results hold for degree, I guess I'm confused because the topology on $\mathbb{A}^n(\mathbb{F}_p)$ is discrete. If I compute the degree at the generic fiber (say I get a number $M$), how can I compare that to the degrees of fibers over points in $\mathbb{A}^n(\mathbb{F}_p)$? Are all of these degrees equal to $M$? $\endgroup$ Jul 24, 2019 at 1:17
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    $\begingroup$ I think the degrees should all be at least $M$. Instead of working with the $\mathbb F_p$ points, you can work with the zariski topology (aka,points over any finite field of char p and non closed points). Here is an example where cardinality goes down: Take $X = \mathbb F_p[x,y]/(x^2-y^2)$ and the projection to $x$ map. Then, the fiber over $0$ has cardinality $1$ with multiplicity $2$ while other points have cardinality $2$ (for $p\neq 2$). $\endgroup$
    – Asvin
    Jul 24, 2019 at 1:27
  • $\begingroup$ Haven't thought about the problem very much, but if you assume that your map is smooth proper (over the locus of interest to you) then you should get the result by smooth proper base change and the Grothendieck-Lefschetz trace formula (unless I am misunderstanding you). $\endgroup$ Jul 24, 2019 at 13:50

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As $f$ is quasi finite it is locally of finite type (EGA II 6.2.2 and 6.2.3) and this ensures (EGA IV 13.1.3, Chevalley's theorem) that $x \mapsto {\textrm{dim}}_x(f^{-1}\left(f(x)\right))$ is upper semi-continuous.

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