On the irreducibility of commuting varieties of nilpotent matrices

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 charK=0 or charK 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 charK=0 or charK n/2.