Parametrizations of ideals in K[x,y] and K[x,y,z]

We parametrize the affine space of Artinian affine ideals of K[x,y] which have a given initial ideal with respect to the degree reverse-lexicographic term order. The fact that the term order is degree-compatible allows us to extend the parametrization to the projective case, namely zero-dimensional subschemes of P2 with some extra assumption and to determine the Betti strata of these Gröbner cells. This allows us to prove a formula due to A. Iarrobino for the codimension of the Betti strata of codimension two punctual schemes in P2.