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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 finitely or infinitely many of them. 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?

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?

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

Given a natural number $n$ can we always find a smooth proper variety $V$ over $\mathbb{F}_q$ such that the number of isomorphism classes of smooth proper varieties that can be put in a common proper flat family with $V$ is finite and equal to $n$?

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

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 finitely or infinitely many of them. 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?

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

Given a natural number $n$ can we always find a smooth proper variety $V$ over $\mathbb{F}_q$ such that the number of isomorphism classes of smooth proper varieties that can be put in a common proper flat family with $V$ is finite and equal to $n$?

Can we at least get $2$ in this way?

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

Given a natural number $n$ can we always find a smooth proper variety $V$ over $\mathbb{F}_q$ such that the number of isomorphism classes of smooth proper varieties that can be put in a common proper flat family with $V$ is finite and equal to $n$?

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