Let $f:X\to S$ be a smooth proper morphism of schemes with geometrically connected fibres. Assume $S$ is a smooth irreducible variety over $\mathbb{C}$. Assume that there is a sequence of closed points $(s\_i)\_{i=1}^\infty$ in $S$ such that the fibres $X\_{s\_i}$ are pairwise non-isomorphic over $\mathbb{C}$.


Is there a curve $C\subset S$ such that the restricted family $f|\_{X\_C}:X\_C\to C$ is non-isotrivial? (That is, are there infinitely many points $c\_i$ such that the fibres $X\_{c\_i}$ are pairwise non-isomorphic?)


I think the answer is yes, and that one can construct $C$ by taking a "general" curve. But how does one make this precise?



On non-isotriviality of families
Let $k$ be an algebraically closed field that is uncountable. Let $S$ be finite type $k$-scheme. Let $f:X\to S$ be a proper, flat morphism of schemes.


**Proposition.** For every closed subset $T\subset S$, if there exists an infinite subset of $T(k)$ such that the corresponding fibers of $f$ are pairwise non-isomorphic, then there exists an irreducible curve $C\subset T$ such that $X\_C\to C$ is non-isotrivial. In particular, for $T$ equals $S,$ if $f$ has pairwise non-isomorphic fibers over an infinite subset of $S(k)$, then there exists an irreducible curve $C\subset S$ such that $X\_C\to C$ is non-isotrivial.


**Proof.** This is proved by Noetherian induction on the closed subset $T$ of $S.$ If $T$ is reducible, then the infinite subset of $T$ has infinite intersection with one of the finitely many irreducible components of $T.$ Thus, to prove the proposition, it suffices to assume that $T$ is irreducible.


Denote $X\times\_S T$ by $X\_T.$ On $T\times T,$ form the two pullback families $X\_1=X\_T\times T$ and $X\_2 = T\times X\_T.$ **Edit. Thanks to Laurent Moret-Bailly who pointed out the projectivity issue.** First assume that $X$ is **projective**. Then by the existence of the Hilbert scheme, there is a locally finite type morphism of schemes $\rho:I\to T\times T$, whose domain $I =\text{Isom}\_{T\times T}(X\_1,X\_2)$ is a countable increasing union of quasi-compact opens $I\_j$, and there exists a universal $I$-isomorphism of the pullback families, $$ \phi: I\times\_{T\times T} X\_1 \to I\times\_{T\times T} X\_2.$$ In case $f:X\to S$ is only proper, flat (and locally finitely presented), then Artin's paper Algebraization of Formal Moduli, I (reference to follow) proves that there is such $I$ that is an algebraic space. Every algebraic space has an fppf representable morphism $I'\to I$ such that $I'$ is a scheme, which means that $\phi'$ over $I'$ will not be universal, but for every isomorphism $\phi\_J$ after base change by a morphism $J\to T\times T$, there will exist an fppf morphism of schemes $J'\to J$ and a morphism $J'\to I'$ such that the pullback of $\phi\_J$ to $J'$ equals the pullback of $\phi'$ to $J'$. In both cases, the image of $I\to S\times S$, resp. of $I'\to S\times S$, is a countable increasing union of constructible subsets $Z\_j=\rho(I\_j).$ 


Since $k$ is uncountable, if the union contains all $k$-points of a nonempty Zariski open (or even just a nonempty analytic open when $k=\mathbb{C}$), then one of the $Z\_j$ contains a dense Zariski open. In this case, the union of all opens contained in the image of $\rho$ is a dense open of the form $U\times U$ for $U\subset T$ a dense Zariski open. Thus, the infinite subset of $T$ must be in the complement $T'=T\setminus U.$ By Noetherian induction, the claim holds in this case.


Thus, assume that the image of $\rho$ contains no nonempty open, i.e., every closure $\overline{Z}\_j$ is a nowhere dense closed subset of $T\times T.$ Since $k$ is uncountable, there exists a point $p\in T(k)$ such that $\{p\}\times T$ has nowhere dense intersection with every $\overline{Z}\_j.$ For a very general complete intersection curve $C\subset T$ that contains $p,$ also $\{p\}\times C$ has nowhere dense intersection with every $\overline{Z}\_j.$ Thus, $C\times C$ has nowhere dense intersection with every $\overline{Z}\_j.$ Therefore, the restriction of $X$ over $C$ is non-isotrivial. Thus, the result is proved by Noetherian induction on $T.$ **QED**


**Example.** There are examples where you need to remove a dense open $U$ as in the proof. For instance, let $S$ be the Hilbert scheme of length $4$, zero-dimensional closed subschemes of $\mathbb{P}^2$ that are reduced. Let $X\to S$ be the blowing up of $S\times \mathbb{P}^2$ along the universal closed subscheme of $S\times \mathbb{P}^2.$ This is isotrivial over a dense Zariski open subset of $S$, but it is non-isotrivial over the locus parameterizing linearly degenerate closed subschemes.


