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 P(2) with some extra assumption and to determine the Betti strata of these Grobner cells. This allows us to prove a formula due to A. larrobino for the codimension of the Betti strata of codimension two punctual schemes in P(2). (C) 2011 Elsevier Inc. All rights reserved.