TITLE: Flatness over non-reduced schemes
QUESTION [6 upvotes]: Is any family of hypersurfaces of fixed degree d (in a projective space of dimension n) over a non-reduced base flat? This is true over reduced bases (assume everything Noetherian), since the Hilbert polynomial is constant. Is there any criterion of flatness of the form: If the Hibert polynomial is constant and (extra conditions that do not assume the base is reduced) then the family is flat? 
Is there any result that guarantees that push forwards and higher push-forwards (via a proper mophism of Noetherian schemes) of a flat over the base sheaf are locally free if the base is non-reduced? This is about the Cohomology and Base Change statements that appear in Mumford (Abelian Varieties) and Hartshorne. In both places the base is assumed to be reduced. However, it seems that this is used often even when the base is non-reduced (such as when constructing the Hilbert scheme - although I must be misunderstanding this proof).  

Thanks a lot!

REPLY [3 votes]: In response to #2:  you mention that cohomology and base change in Mumford and Hartshorne requires reducedness of the target.  I don't have either with me right now, but I agree with VA:  it's possible that you saw versions in these books that had this hypothesis (the proof is easier in this case --- I've seen it called Grauert's Theorem), but that there is a later version (in the same sources) that works for a nonreduced base.  Here is one possible statement (whose hypotheses can be relaxed as usual, e.g. Noetherian hypotheses often can be traded in for finite presentation ones).  Suppose $\pi: X \rightarrow Y$ is a proper morphism to a locally Noetherian scheme, $\mathcal{F}$ is coherent and flat over $Y$, and for each point $y \in Y$, the natural map $\phi^p_y$ from the fiber of the $p$th pushforward to the $p$th cohomology of the fiber is surjective.  Then (i) for any $\rho: Z \rightarrow Y$, restriction to $Z$ commutes with the $p$th pushforward.  In particular, $\phi^p_y$ is an isomorphism.  (ii) Furthermore, $\phi^{p-1}_y$ is surjective for all $y$ if and only if the $R^p \pi \mathcal{F}$ is locally free.  This in turn implies that $h^p$ is constant.  
Better discussion and proof is given here (see for example 25.8-9 in the October 21, 2011 version).