On some Gorenstein loci in Hilb_6(P^4_k)

Let k be an algebraically closed field and let $ Hilb^{aG}_d (P^{d−2}_k ) $ be the open locus inside the Hilbert scheme $ Hilb_d (P^{d−2}_k ) $ corresponding to arithmetically Gorenstein subschemes.We prove the irreducibility and characterize the singularities of $ Hilb^{aG}_6 (P^4_k).$ 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 $ Hilb^{aG}_6 (P^4_k) $ obtained via some geometric construction. Finally we prove the obstructedness of some families of points in $ Hilb^{aG}_d (P^{d−2}_k ) $ for each $ d \geq 6.$