Suppose I have $R$ polynomial equations $F_1, ..., F_R$ and say they all have integer coefficients. Let us denote $V_{\mathbb{C}}$ to be the affine variety defined by these polynomials over $\mathbb{C}$.
Let $\mathbb{F}_p$ be the finite field of $p$ elements. Since the polynomials can be interpreted as polynomials over $\mathbb{F}_p$ (by reducing the coefficients) let us denote $V_{\bar{\mathbb{F}_p}}$ to be the affine variety defined by these equations over the algbraic closure of $\mathbb{F}_p$. I was wondering, are the quantities such as dimension, the number of irreducible components and degree of $V_{\mathbb{C}}$ and $V_{\bar{\mathbb{F}_p}}$ related somehow? I can see that for small $p$ they could be different, but maybe they will be the same once $p$ is sufficiently large or something? I would greatly appreciate any comments. Thank you very much.
ps I have learned that these quantities can be bounded uniformly (in $p$) and I am interested in how one can prove this. Thank you!
Good question! The answers are a lot nicer (or atleast I know how to prove them) if you assume that we are actually working with a projective variety. In this case, everything that you want is true. Suppose that the dimension of $\mathbb{V}_{\mathbb{C}}=2$, then there are infinitely many primes where the $V_{\overline{\mathbb{F}_p}}=2$. Reversely, if there are infinitely many primes such that the dimension of $V_{\overline{\mathbb{F}_p}}=2$, then $V_{\mathbb{C}}$ also has dimension $2$. Similar results will be true for the number of connected components, the number of irreducible components and the degree.
In order to prove things like this we have to work with schemes, so I will assume from now on that you are familiar with schemes. What we have is a projective scheme $V \to \mathbb{Z}$ and we are interested in the way the numerical invariants of the fibers vary.
The first thing we can do (which is still somewhat elementary), is to consider the polynomials $f_1, \cdots, f_R$ which cut out our projective scheme. Suppose that $f_1$ has degree $d$ in $\mathbb{Z}[X_1, \cdots, X_n]$, then it is definitely possible that its degree is lower in $\overline{\mathbb{F}_p}[X_1, \cdots, X_n]$ (for example some coefficient could be divisible by $p$). So the first thing we want to do is to just throw out all the primes where this happens. I will leave it as an exercise to prove that there are only finitely many such primes. So now we have shown that the degree of $V_{\mathbb{C}}$ agrees with the degree of $V_{\overline{\mathbb{F}_p}}$ for infinitely many $p$ (and vice versa).
Now I am going to start throwing around some theorems from algebraic geometry, which I need because I don't know how to do things in an elementary way, as above. By generic flatness there is an open subset $U \subset \operatorname{Spec} \mathbb{Z}$ such that $X_U$ is flat over $U$. Since fiber dimension is constant in flat families, this means that all the fibers of $X_U$ over $U$ will have the same dimension. Translation: We can throw away finitely many primes where the dimension is bad (it will be too high, can you see why?), and then for the rest of the primes the dimension of $V_{\overline{\mathbb{F}_p}}$ is the same as the dimension of $V_{\mathbb{C}}$.
For the number of connected components, the situtation is a little more complicated. Once we restrict to our $U$ such that $X_U/U$ is flat then the number of connected components will be an upper semi-continuous function on $\operatorname{Spec} \mathbb{Z}$, which means that it can jump up at some primes. Still there will be infinitely many primes $p$ where the number of connected components $V_{\overline{\mathbb{F}_p}}$ and $V_{\mathbb{C}}$ agree (and vice versa)
The last thing that I will say is about smoothness: The variety $V_{\mathbb{C}}$ is smooth if and only if there are infinitely many primes $p$ such that $V_{\overline{\mathbb{F}_p}}$ is smooth if and only if there is one $p$ such that $V_{\overline{\mathbb{F}_p}}$ is smooth. This is something you can think about yourself using Jacobians.
Remark: When I say 'infinitely many primes' I really mean a Zariski open set of primes.