Separating sequences of 0-dimensional schemes

In two previous papers, we defined, for every projective, 0-dimensional, reduced scheme X, a set of numerical sequences, which turns out to be a refinement of the Hilbert function of X. Here, we extend that definition to the case of a scheme X not necessarily reduced; the aim is reached by replacing a point by its corresponding “separating ideal” in its coordinate ring. The numerical sequences are obtained by taking the degrees of the elements appearing in suitable sequences of separating ideals. These latter sequences are themselves a good tool in the search for subschemes of X not in general position.