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

geometricallyprojective spaces. Maybe what you say works if the fibres are actually projective spaces. – Damian Rössler Feb 3 '13 at 14:34