A new approach to the Hilbert function of a 0-dimensional projective scheme

To any 0-dimensional, reduced, degree s, projective scheme X we associate a set SX of sequences of s natural numbers; these sequences turn out to be a suitable permutations of a nondecreasing sequence (d1,…,ds), which is in 1–1 correspondence with the first difference ΓX of the Hilbert function of X. The structure of SX allows us to read geometric properties of X. In its turn, the set image of all sequences allowed for at least one scheme X with ΓX=Γ can be partitioned into equivalence classes, giving information on all the schemes X with ΓX=Γ. We start to afford the problem of producing all the sequences equivalent to a given one.