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Let $i:X\to Y$ 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 and that $Y$ is a complete intersection.

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

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    $\begingroup$ Don't you want to assume that $Y(Q)$ is Zariski dense? Otherwise just assuming it's infinite doesn't say much. $\endgroup$
    – YCor
    Commented May 27, 2021 at 12:29
  • $\begingroup$ @YCor I feel that assuming density would be too much $\endgroup$
    – guido
    Commented May 27, 2021 at 12:55
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    $\begingroup$ Say, take $Y_0$ with a single rational point, $f_0:X_0\to Y_0$ a closed immersion with $X_0(Q)$ nonempty. Then for $X=X_0\times P^1$, $Y=Y_0\times P^1$, $f=f_0\times\mathrm{id}$ your assumptions are fulfilled. Hence infiniteness of $Y(Q)$ is not really more (after changing the variety) than assuming it's nonempty. $\endgroup$
    – YCor
    Commented May 27, 2021 at 13:01
  • $\begingroup$ @YCor The assumption that $Y(\mathbb Q)$ is Zariski dense doesn't seem to lead to an interesting question either -- since $X$ is supposed to be a closed subvariety this would lead to $X=Y$. $\endgroup$
    – Dan Petersen
    Commented May 28, 2021 at 8:07
  • $\begingroup$ @DanPetersen you're right: maybe rather $X(Q)$ Zariski-dense in $X$? Anyway I'm puzzled how these arithmetic assumptions pertain to the question (e.g., what are phenomena that could occur in a general setting $X\subset Y$ but seem to rule out the presence of too many rational points)? $\endgroup$
    – YCor
    Commented May 28, 2021 at 8:11

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