Toric Varieties in Hilbert Schemes

For an arbitrary field K, let I be an ideal in the ring K[[x,y]] expressible as a polynomial in either the pair of ideals (x, y^4) and (x,y) or the pair (x,y^3) and (x^2, y). Let G be the group of automorphisms of K[[x,y]] sending the ideal (x,y^2) to itself. Then the normalization of the closure of the G orbit of I is a toric variety. We use the interplay of the properties of Hilbert schemes and toric varieties to study these spaces.