On some Gorenstein loci in

Let k be an algebraically closed field and let be the open locus inside the Hilbert scheme corresponding to arithmetically Gorenstein subschemes. We prove the irreducibility and characterize the singularities of . 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 obtained via some geometric construction. Finally we prove the obstructedness of some families of points in for each d⩾6.