ag.algebraic geometry – Closed immersion hitting all $mathbb{Q}$-points

Let $i:Xto Y$ be a closed immersion of smooth projective varieties over $mathbb{Q}$.

Assume that $Y(mathbb{Q})$ is infinite and $X(mathbb{Q})to Y(mathbb{Q})$ is surjective. Also assume that $X$ is not ruled.

Then can we relate the geometric invariants (e.g. plurigenera, Hodge numbers) of $X$ to those of $Y$?