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Let $X$ be a smooth projective variety with $\omega_X$ ample. Let $C$ be a smooth curve.

Can there be a relatively ample line bundle $\mathcal{L}$ on the trivial family $X\times C\to C$ such that, for every ample line bundle $L$ on $X$, the set of closed points $c$ of $C$ with $(X, \mathcal{L}_c) \cong (X,L)$ is finite?

In other words, can a relatively ample line bundle on $X\times C\to C$ truly vary in the trivial family? Can this happen even with $C=\mathbb{P}^1$?

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    $\begingroup$ Let $X$ equals $C$ equal a curve of genus $g>1$, and let $\mathcal{L}$ be the invertible sheaf associated to the diagonal in $C\times C$. The Picard group of $\text{Pic}(X\times \mathbb{P}^1)$ equals $\text{Pic}(X)\oplus \text{Pic}(\mathbb{P}^1)$, so this cannot happen with $C$ equal to $\mathbb{P}^1$. $\endgroup$ May 12, 2022 at 15:45
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    $\begingroup$ This can happen if $C$ has genus $>1$ if we take $X = C$ and $\mathcal L = \mathcal O(\Delta)$, and in fact for any $C$ other than $\mathbb P^1$ if $X$ has a nontrivial map to the Jacobian of $C$. $\endgroup$
    – Will Sawin
    May 12, 2022 at 18:29
  • $\begingroup$ More generally, you can always start with a given ample line bundle on $X$ and twist it with a varying family of flat line bundles. $\endgroup$ May 14, 2022 at 10:23

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