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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.

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  • $\begingroup$ Brenin, I apologize if this is a naive question. What do you mean by "Zariski locally trivial"? I, in particular, don't know what the "trivial"ness would mean? Something like a birational morphism? Thanks! $\endgroup$ – Alex Youcis Feb 22 '14 at 10:27
  • $\begingroup$ Oh, you mean a trivial fibration? $\endgroup$ – Alex Youcis Feb 22 '14 at 10:30
  • $\begingroup$ Sorry Alex. Not naive at all. I planned to write it down but I forgot. Now I added (what I believe is) the correct definition of "Zariski locally trivial". Essentially, locally a product. $\endgroup$ – Brenin Feb 22 '14 at 10:36
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    $\begingroup$ Projective bundles are by definition Zariski locally trivial, since vector bundles are. In particular, Hirzebruch surfaces are indeed Zariski locally trivial, since they are projectivisations of rank 2 bundles on $\mathbf P^1$. $\endgroup$ – user64687 Feb 22 '14 at 11:09
  • $\begingroup$ By the way, I don't have a good answer to your question, but a search for "Severi--Brauer variety" might be a useful starting point for the case when the fibres are projective spaces. Alternatively, if you post this question on MathOverflow, I'm sure you'll get a thorough answer from Jason Starr. $\endgroup$ – user64687 Feb 23 '14 at 11:54
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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?

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  • $\begingroup$ Thank you for your answer! By smooth family you mean a smooth morphism (in the sense of algebraic geometry)? Also, I think of complex manifolds as smooth objects (when viewed in the algebraic category), so I am confused when you say there are no examples with smooth fiber. By the way, for an étale cover of curves $f:X\to Y$, I just wanted to say that it is not isomorphic to one of the form $F\times U\to U$ for open sets $U\subset Y$ covering $Y$. Nevertheless, the fibers are isomorphic because they all consists of $\deg f$ reduced points. $\endgroup$ – Brenin Jan 8 '15 at 8:58
  • $\begingroup$ Yes, smooth family means smooth morphism in the sense of algebraic geometry. Fischer-Grauert says: The family is locally trivial iff all fibres are isomorphic. Hence there are no smooth families which are not locally trivial and with all fibres isomorphic. Which part of my statement is confusing? $\endgroup$ – Jo Wehler Jan 8 '15 at 9:27
  • $\begingroup$ Nothing is confusing anymore! But could you please indicate a reference for this theorem? Thanks! $\endgroup$ – Brenin Jan 8 '15 at 21:15
  • $\begingroup$ Fischer, W.; Grauert, H.: Lokal triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachr. Akad. Wiss. Göttingen, II. Math. Phys. Kl. (1965), 89-94. It is in German. In case you do not have access to the Journal I could try to download a copy. $\endgroup$ – Jo Wehler Jan 8 '15 at 21:20
  • $\begingroup$ Right now I do not have access (my university does I guess), but the main obstacle is the language... I will look around for some other reference maybe. But thank you very much! $\endgroup$ – Brenin Jan 9 '15 at 11:04

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