  Given an nxn nilpotent matrix over an algebraically closed field K, we prove
some properties of the set of all the nxn 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 nxn nilpotent matrices over K if either char K = 0 or
char K isn't less than 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 with the previous condition on char K.
