Pairs of commuting nilpotent matrices, and Hilbert function

Abstract

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 + λ B , λ ∈ K has Jordan partition the maximum partition P ( H ) whose diagonal lengths are the Hilbert function H of K [ A , B ] .