When a scheme theoretical fiber is reduced? I'd like to ask some basic things about algebraic geometry. 
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). 
Thank you very much!
 A: 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.
