I am currently studying various Moduli problems and in order to check whether some families have non-trivial automorphisms, I have the strong intuition that the following should hold:
Let $k$ be an algebraically closed field, $S$ any scheme over $k$ and $\pi_X: X \to S, \pi_Y: Y \to S$ families of a given class of varieties (curves of given genus, K3 surfaces, etc.), so proper, surjective, finitely presented morphisms s.t. all fibres over geometric points of $S$ are varieties over $k$. Let $f,g \in Hom_S(X,Y)$ be morphisms of these families s.t. for any geometric point $s \in S$ it holds that $f_s =g_s$ as elements of $Hom_k(X_s,Y_s)$, then it follows $f=g$.
If $S$ satisfies some nice properties, I could argue that the set of points of $S$ where $f_s=g_s$ is a closed subset and the set of closed points in $S$ is dense, similar as in this question: Is a morphism of reduced schemes over an algebraically closed field determined by its values on closed points?
However, if $S$ gets crazy enough, for example not being finitely presented over $k$, this argument fails. Can this be fixed by different methods or are there indeed families of varieties where all geometric fibres do not admit any non-trivial autmorphisms but the family does?
