Degree of variety over $\mathbb{Q}$ versus over $\mathbb{F}_p$

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.

• What do you mean by 'degree'? – Marc Paul Oct 20 '18 at 22:31
• 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. – Marc Paul Oct 20 '18 at 22:36
• 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. – darko Oct 21 '18 at 7:11