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 local submersion after extending to $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$.

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?

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?