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Let $k$ be a field, not necessarily algebraically closed, not necessarily of characteristic 0 (actually, the example I have in mind is $k=F_2$). Let $V,W$ be varieties over $k$, and $W\to V$ a morphism defined over $k$, such that every fiber is isomorphic to an affine space of the same dimension.

What conditions imply that $W$ is actually $V\times A^n$?

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you may enjoy reading mathoverflow.net/questions/54502/affine-bundles-over-varieties –  Dan Petersen Mar 17 '11 at 21:21
Here's another possibly relevant MO post: mathoverflow.net/questions/58009/non-locally-trivial-an-bundles –  Kevin H. Lin Mar 21 '11 at 22:14

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