Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth.


My question is the following: can we always find a Zariski open subset $U\subset\mathbb{P}^1$ such that the group $Aut(X\_U)$, of automorphisms of $X\_U = \pi^{-1}(U)\subset X$, acts transitively on the fibers of $\pi$ over $U$?



Moving general fibers of a fibration
Definitely not; it may be that none of the fibres are isomorphic to each other.  
 The typical situation is that a given fibre is isomorphic to only finitely many others.


