1. Is any family of hypersurfaces of fixed degree d 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 for families over non-reduced schemes?

2. 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?

Thanks a lot!

1. 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?

2. 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!