Pairs of commuting nilpotent matrices, and Hilbert function

Let K be an infinite field. There has been recent study of the family R(n, K) of pairs of commuting nilpotent n x n matrices, relating this family to the fibre H-vertical bar n vertical bar of the punctual Hilbert scheme A(vertical bar n vertical bar) = Hilb(n) (A(2)) over the point np of the symmetric product Sym(n) (A(2)), where p is a point of the affine plane A(2) [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-vertical bar n vertical bar by the Hilbert function as [J. Briancon, Description de Hilb(n) C[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 J(P). We then explore the relation between H(n, K) and its stratification by the Hilbert function of K[A, B]. Suppose that dim(K) K[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 + lambda B, lambda is an element of K has Jordan partition the maximum partition P(H) whose diagonal lengths are the Hilbert function H of K [A, B]. (C) 2008 Elsevier Inc. All rights reserved.