TITLE: Flat morphisms whose fibers are affine spaces
QUESTION [8 upvotes]: Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an  ètale affine bundle?
If not, is there some reasonable natural ipotesis to add to make this true?
My definition of affine bundle is a map $f:X \to Y$ such that there is an open cover ${U_{\alpha}}$ of $Y$ with $f^{-1} U_{\alpha} \cong U_{\alpha} \times \mathbb{A}^n$, and $f$ restricted to $f^{-1}(U_{\alpha})$ corresponds to the first projection. 
The question is somewhat related to:
Affine bundles over varieties

REPLY [4 votes]: The question is discussed in detail in the paper "Families of group actions, generic isotriviality, and linearization" (Transform. Groups 19 (2014), no. 3, 779–792. MR 3233525) by Peter Russell and myself. There is e.g. an example, due to Nori, of a flat morphism $X \to Y$ with fibers $A^1$ and base $Y$ equal the cuspidal curve which is not locally trivial in the etale topology.