On the irreducibility of commuting varieties of nilpotent matrices

Abstract

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 .

Nilpotent commuting varieties of reductive Lie algebras

“…This corollary generalises earlier results by Briançon [6] for p = 0, Iarrobino [14] for p > n, and Basili [1] for p ≥ n/2. In conjunction with an observation in [22, (1.2)] it also shows that H r is isomorphic (as a reduced scheme) to the closure in Gr r (A) of the (r − 1)-dimensional family of principal ideals J = A x − (t 1 y + t 2 y 2 + · · · + t r−1 y r−1 ) | (t 1 , t 2 , .…”

The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix

“…It was Guralnick [7] who showed that this is no longer the case for the variety of triples of commuting matrices (see also Guralnick and Sethuraman [8], Holbrook and Omladič [11], Omladič [14], Han [10],Šivic [16]). Recently, it was proved that the variety of commuting pairs of nilpotent matrices was irreducible (Baranovsky [1], Basili [2]). Our motivation to study the problem is to contribute to better understanding of the structure of this variety and which might also help in understanding the (ir)reducibility of the variety of triples of commuting matrices.…”

Les degrés des variétés des matrices nilpotentes

“…Vue queJ (3,2) (E) s'applique birationnellement sur J (3,2) (E), on obtient la classe de ceci comme l'image directe de la classe de la première. Or, la classe deJ (3,2) (E) est connue au moins après restriction à l'ouvert Z \ W ; donc on la connaît même dans Z, parce qu'on a dim W < dimJ (3,2) …”

“…Or, la classe deJ (3,2) (E) est connue au moins après restriction à l'ouvert Z \ W ; donc on la connaît même dans Z, parce qu'on a dim W < dimJ (3,2) …”

“…Il serait intéressant de pousser l'étude énumératif fait ici pour les cas des variétés de couples de matrices considérées dans [2] ou [12]. Le rapporteur nous a gentiment suggéré qu'il serait désirable d'étendre le résultat pour d'autres algèbres de Lie classiques, objet d'un futur travail.…”