# When a scheme theoretical fiber is reduced?

Suppose I have a map $$\phi:V\to W$$, between affine varieties over $$k=\mathbb{C}$$. For any point $$y \in W$$, the scheme theoretical fiber is defined to be $$Spec(k[V]\otimes_{k[W]} k )$$. The question is when is this reduced?
I'm seeking an answer like: For the generic point in the image of $$\phi$$, this is true. I also like to ask a proper reference for this (hope not in EGA, but sections in some textbook. such as Hartshorne).
If you are over a field of char. $$0$$, then yes, it is true that the fibre over a generic point of the image is reduced. To see this, first replace $$W$$ with the closure of the image of $$\phi$$, so as to assume that $$\phi$$ is dominant. Then the morphism $$\phi$$ corresponds to an injection of finite-type domains over $$k$$, say $$A \hookrightarrow B$$. (The injectivity follows from the dominance of $$\phi$$.) Now let's look at the generic fibre of $$\phi$$: this is Spec of the tensor product $$K(A) \otimes_A B$$ (here $$K(A)$$ denotes the fraction field of $$A$$), which is a localization of $$B$$, and so a domain (and hence reduced). Now if we are in char. zero, the reduced $$K(A)$$-algebra $$K(A)\otimes_A B$$ is necessarily geometrically reduced (i.e. stays reduced after extending to an algebraic closure of $$K(A)$$), and the property of fibres being geometrically reduced is open on the base, i.e. on Spec $$A$$; thus the fibres over an open subset of the image of $$\phi$$ will be reduced. (See e.g. Thm. 2.2 of these notes. You can find this in many places (though mabye not Hartshorne); these notes are just what turned up near the top of a quick google search.)
In char. $$p$$, this need not be true. E.g. consider the Frobenius map $$\mathbb A^1 \to \mathbb A^1$$, given by $$x \mapsto x^p$$. Then the fibre over every (closed) point is non-reduced. (The fibre over the generic point is reduced, but not geometrically reduced.) Nevertheless, if the generic fibre (in the scheme-theoretic sense) is geometrically reduced, then the fibre over an open set of closed points will be reduced too.