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Let $\mathbb{F}_q$ be a finite field.

Let $V$ be a smooth proper variety over $\mathbb{F}_q$. Consider the isomorphism classes of smooth proper varieties that can be put in a common proper flat family with $V$. There may be finitely or infinitely many. For example if $V$ is one-dimensional there's finitely many.

Given a natural number $n$ can we always find $V$ such that the number is finite and equal to $n$?

Weaker questions: can we get $2$ in this way? Can we get a subset of positive Dirichlet density?

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  • $\begingroup$ Do you allow V to be, for instance, the union of two points? $\endgroup$
    – Asvin
    Commented Aug 6, 2021 at 15:58
  • $\begingroup$ @Asvin we require varieties to be geometrically connected $\endgroup$
    – uio
    Commented Aug 7, 2021 at 4:41
  • $\begingroup$ I find the question a bit difficult to parse. Could you try rephrasing it by breaking up the long sentence? $\endgroup$
    – Daniel Loughran
    Commented Aug 7, 2021 at 15:30
  • $\begingroup$ @DanielLoughran I edited $\endgroup$
    – uio
    Commented Aug 8, 2021 at 5:12

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