  This note is devoted to the study of the links between the Hilbert function
of a subscheme X of the projective space, and its geometric properties. We will
assume that X is arithmetically Cohen-Macaulay, which allows us to characterize
its Hilbert function by a d integers, where d is the degree of X. We study in
this context the geometric description of special linear systems of dimension
maximal with respect to their degree on projective Gorenstein curves.
