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I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic fibers over closed points of $Y$ are reduced. I can assume that the morphism $f$ is flat and finite.

As a commutative algebra question this becomes the following: Let $\phi: A \to B$ be a flat, finite morphism between finite type $k$-algebras over a perfect field. Is there a criteria on $\phi$ which will ensure that for every maximal ideal $m$ of $A$ the algebra $B / \phi(m)$ is reduced?

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Do you need a weaker condition than etale? –  Ramsey Mar 4 '11 at 21:31
    
Yes, étale is a bit too strong. –  name Mar 6 '11 at 14:10
    
Actually, I'm beginning to wonder whether in this setting étale is equivalent to having all such fibers reduced. –  name Mar 6 '11 at 14:14
    
If your field is characteristic 0, then most of the fibers will be reduced. If you're then willing to do finite flat base changes ramified over a point in Y with nonreduced fiber, and to normalize the total space at those fibers, you can make those fibers reduced. We do this in front.math.ucdavis.edu/0602.5626 –  Allen Knutson Mar 6 '11 at 17:31
    
Hi Allen. Actually, I am working in characteristic p, but what you suggest sounds interesting anyway, although I had trouble understanding it completely. Would you be willing to give a reference to a precise statement in the link, or write one here? –  name Mar 7 '11 at 19:09

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