  Let k be an algebraically closed field and let HaG(d) be the open locus
inside H(d) (the Hilbert scheme of 0-dimensional length d subschemes of the
projective (d-2)-space over k) corresponding to arithmetically Gorenstein
subschemes. We prove the irreducibility and characterize the singularities of
HaG(6). In order to achieve these results we also classify all Artinian,
Gorenstein, not necessarily graded, k-algebras up to degree 6. Moreover we
describe the loci in HaG(6) obtained via some geometric construction. Finally
we prove the obstructedness of some families of points in HaG(d) for each d>5.
