Pairs of commuting nilpotent matrices, and Hilbert function

Let K be an infinite field and denote by H(n,K) the family of pairs (A,B) of commuting nilpotent n by n matrices with entries in K. There has been substantial recent study of the connection between H(n,K) and the fibre H[n] of the punctual Hilbert scheme of the plane, over an n-fold point of the symmetric product, by V. Baranovsky, R. Basili, and A. Premet. We study the stratification of H(n,K) by the Hilbert function of the Artinian ring K[A,B]. We show that when dim_K K[A,B] = n, then the generic element of the pencil A+\lambda B, \lambda \in K, has Jordan partition the maximum partition P(H) whose diagonal lengths are the Hilbert function of K[A,B]. We denote by Q(P) the maximum Jordan partition of a nilpotent A commuting with a nilpotent B of Jordan partition P. We show that the stable partitions - those such that Q(P)=P - are those whose parts differ by at least two. In characteristic zero, the latter is a special case of a result of D. Panyushev. Our result on pencils shows that Q(P) has decreasing parts. In related work, T. Kosir and P. Oblak have shown further that Q(P) is itself stable.