Given an n × n nilpotent matrix over an algebraically closed field K, we prove some properties of the set of all the n × n nilpotent matrices over K which commute with it. Then we give a proof of the irreducibility of the variety of all the pairs (A, B) of n × n nilpotent matrices over K such that [A, B] = 0 if either char K = 0 or char K ≥ n 2. We get as a consequence a proof of the irreducibility of the local Hilbert scheme of n points of a smooth algebraic surface over K if either char K = 0 or char K ≥ n 2 .

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] = 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 [n] by the Hilbert function as [J. Briançon, 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 + λB, λ ∈ K has Jordan partition the maximum partition P (H ) whose diagonal lengths are the Hilbert function H of K [A, B].

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We establish a connection between commuting varieties Cg (G) (potentially of higher genus), which are associated with a group scheme G consisting of upper triangular matrices, and the representation homology HR * (Σg , G) of a Riemann surface Σg with coefficients in the group G. As an outcome, we provide a numerical criterion for determining whether commuting varieties C(Un) form a complete intersection for groups consisting of upper triangular unipotent matrices.