Pairs of commuting nilpotent matrices, and Hilbert function

Let K be an infinite field. There has been recent study of the family H(n,K) of pairs of commuting nilpotent n×n matrices, relating this family to the fibre H[n] of the punctual Hilbert scheme A[n]=Hilbn(A2) over the point np of the symmetric product Symn(A2), where p is a point of the affine plane A2 [V. Baranovsky, The variety of pairs of commuting nilpotent matrices is irreducible, Transform. Groups 6 (1) (2001) 3–8; R. Basili, On the irreducibility of commuting varieties of nilpotent matrices, J. Algebra 268 (1) (2003) 56–80; A. Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (3) (2003) 653–683]. In this study a pair of commuting nilpotent matrices (A,B) is related to an Artinian algebra K[A,B]. There has also been substantial study of the stratification of the local punctual Hilbert scheme H[n] by the Hilbert function as [J. Briançon, Description de HilbnC[x,y], Invent. Math. 41 (1) (1977) 45–89], and others. However these studies have been hitherto separate.We first determine the stable partitions: i.e. those for which P itself is the partition Q(P) of a generic nilpotent element of the centralizer of the Jordan nilpotent matrix JP. We then explore the relation between H(n,K) and its stratification by the Hilbert function of K[A,B]. Suppose that dimKK[A,B]=n, and that K is algebraically closed of characteristic 0 or large enough p. We show that a generic element of the pencil A+λB,λ∈K has Jordan partition the maximum partition P(H) whose diagonal lengths are the Hilbert function H of K[A,B].