This question is about properties of Isom-schemes that are well-known over algebraically closed fields.


Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically connected curve over $K$ and let $P$ a $K$-rational point of $C$. Let $X$ be a smooth projective geometrically connected surface over $K$, and $X\to C$ a curve of genus at least two over $K$. Define $F:= X\_P$. Then $F$ is a curve of genus at least two over $K$.


Consider the Isom-scheme:


$\underline{\mathrm{Isom}}\_{C}(X, C\times\_K F) \to C $.


This map is finite unramified. It is etale if $X\to C$ is "isotrivial", i.e., all geometric fibres of $X\to C$ are isomorphic. This is proven in Deligne-Mumford. (Are there any other references?)


Assume $X\to C$ is isotrivial, **and non-trivial.**


Is the Isom-scheme $\underline{\mathrm{Isom}}\_{C}(X, C\times\_K F) $ geometrically connected over the ground field $K$? What if $K$ is a number field?



Are Isom-schemes geometrically connected
If $X\to C$ is trivial, i.e. $X=C\times\_K F$, then $\underline{\mathrm{Isom}}\_C(X,C\times\_K F)$ is just $C\times\_K \underline{\mathrm{Aut}}\_K(F)$ which is geometrically connected if and only if $\underline{\mathrm{Aut}}\_K(F)$ is trivial.


