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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 $U$?

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  • $\begingroup$ To be pedantic: the empty subset is open everywhere. If you mean “on all fibers of $π$ over $U$”, then wouldn’t it be vacuously true for empty $U$? $\endgroup$ Apr 13, 2021 at 11:09

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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.

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