# 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 $$Y$$?