Let H ab be the equivariant Hilbert scheme parametrizing the 0-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus T ab :We compute the irreducible components of H ab : they are in one-one correspondence with a set of Hilbert functions. As a by-product of the proof, we give new proofs of results by Ellingsrud and Strømme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme H l parametrizing the 0-dimensional subschemes of the affine plane of length l [4], and a description of Bialynicki-Birula cells on H l by means of explicit flat families [5]. In particular, we precise conditions of applications of this last description.

Alexander and Hirschowitz [1] determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points which approach a divisor. On the other hand, Nagata [10], in connection with its counter example to the fourteenth problem of Hilbert determined the Hilbert function H(d) of the union of k 2 points of the same multiplicity m in the plane up to degree d = km. We introduce a new method to determine limits of linear systems. This generalizes the result by Alexander and Hirschowitz. Our main application of this method is the conclusion of the work initiated by Nagata: we compute H(d) for all d. As a second application, we determine the generic successive collision of four fat points of the same multiplicity in the plane.

Let H ab (H) be the equivariant Hilbert scheme parametrizing the zero dimensional subschemes of the affine plane k 2 , fixed under the one dimensional torus T ab = {(t −b , t a ), t ∈ k * } and whose Hilbert function is H. This Hilbert scheme admits a natural stratification in Schubert cells which extends the notion of Schubert cells on Grassmannians. However, the incidence relations between the cells become more complicated than in the case of Grassmannians. In this paper, we give a necessary condition for the closure of a cell to meet another cell. In the particular case of Grassmannians, it coincides with the well known necessary and sufficient incidence condition. There is no known example showing that the condition wouldn't be sufficient.