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Let $V$ be a projective variety (possibly reducible) in $\mathbb{P}^n$ defined over $\mathbb{Z}$.

What is the relation between the degree of $V$ seen as a variety over $\overline{\mathbb{Q}}$ and the degree of $V$ seen as a variety over ${\overline{\mathbb{F}}_p}$ for a prime $p$?

It seems that the degree could fall at some primes, probably finitely many. If true, I would be grateful for a reference of this.

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  • $\begingroup$ What do you mean by 'degree'? $\endgroup$ – Marc Paul Oct 20 '18 at 22:31
  • $\begingroup$ Also, what is your definition of variety over a ring (I've seen some different definitions)? In particular, do you want to assume that $V$ is flat over $\mathbb Z$? If not, you will probably find that things like $\mathbb P^n_{\mathbb F_p} \subset \mathbb P^n_{\mathbb Z}$ become a counterexample. $\endgroup$ – Marc Paul Oct 20 '18 at 22:36
  • $\begingroup$ I mean the standard degree defined using the Hilbert series or equivalently by counting points in the intersection of $V$ with general subspaces. I am certainly assuming that $V_{\mathbb{Q}}$ is non empty. $\endgroup$ – darko Oct 21 '18 at 7:11

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