2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66300Given 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.LaTex, 27 pages, to appear in J. AlgebraAlgebraic GeometryCommutative Algebratext