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

Let $k$ be an algebraically closed field and let $\Hilb_{d}^{aG}(\p{d-2})$ be the open locus inside the Hilbert scheme $\Hilb_{d}(\p{d-2})$ corresponding to arithmetically Gorenstein subschemes. We prove the irreducibility and characterize the singularities of $\Hilb_{6}^{aG}(\p{4})$. 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 $\Hilb_{6}^{aG}(\p{4})$ obtained via some geometric construction. Finally we prove the obstructedness of some families of points in $\Hilb_{d}^{aG}(\p{d-2})$ for each $d\ge6$.