I am looking for examples of fibrations $f:X\to Y$ where the fibers are all isomorphic, but $f$ is not Zariski locally trivial. In particular, I am interested in understanding how much such examples are "rare". (I believe they are not that rare)
First of all, by fibration I mean a proper flat surjective morphism of (complex) varieties. But I am not sure this is the correct definition of fibration in Algebraic Geometry; in that case, any correction is much appreciated.
By $f:X\to Y$ being Zariski locally trivial, I mean that there is a variety $F$ such that every point in the base $Y$ has a Zariski open neighborhood $U$ such that $f^{-1}(U)\to U$ is isomorphic to the projection $F\times U\to U$. Here $F$ is called the fiber of $f$ (in particular, Zariski locally trivial fibrations do have isomorphic fibers).
One example I came up with is that of an étale cover of curves: the fibers are discrete of the same size, hence isomorphic, but it is not Zariski locally trivial in general.
Another example might be the Hirzebruch surface $\mathbb F_n\to \mathbb P^1$, with $n\neq 0$.
As for projective bundles $\mathbb P(E)\to Y$, I do not know whether they are Zariski locally trivial or not.
Probably there are many important examples that I am missing here. I would very much appreciate if you could help me to fill in this picture!
Thank you.
Best Answer
In complex analytic geometry the theorem of Fischer and Grauert states: A smooth family of compact complex manifolds is locally trivial if and only if all fibres are analytically isomorphic.
Hence in complex analytic geometry there are no examples - with smooth fibre - of the type you are searching for.
Could you please detail your statement about étale cover of curves?