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 $C=\mathbb{P}^1$?