Let $A \colon V \to W$ be a surjective linear map (defined over $\mathbb{Z}$), inducing a projection $\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$. Let $X \subseteq \mathbb{P}(V)$ and $Y \subseteq \mathbb{P}(W)$ be some absolutely irreducible projective varieties (defined over $\mathbb{Z}$) that we know well.

Suppose that when considered over large fields, the restriction of the map $\alpha$ is close to a surjection. For example, suppose that $\alpha$ restricted to $X$ is a submersion almost everywhere when considered as a differentiable function over $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$.

My question is whether or not such a behaviour implies that we can deduce that $\alpha$ is close to being a surjection also over finite fields.

How to produce a lower bound on the number of $\mathbb{F}_p$ points of $\alpha(X)$?

Is there a standard procedure of how to pass from information over $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$ to answering the above question? Could someone please point out some relevant theorems? Which other properties of $\alpha$ restricted to $X$ should one prove in order to obtain the desired lower bound?

*Remark 1:* The concrete example of $X$ that I have in mind is a product of Grassmannians.

*Remark 2:* I am aware of the fibre dimension theorem, saying that over $\bar{\mathbb{F}}_p$, the generic fibers are of the same dimension. Together with Lang-Weil estimates, this would bring us close to giving an answer, as long as one could say something about the *set over which generic fibres are*, the fact that *the fibres are absolutely irreducible* and have *known dimension* (that is, the same dimension as they have over $\mathbb{C}$). Are any of these known?