On the irreducibility of varieties of commuting matrices

We will show that the variety of all the pairs (A,B) of n x n matrices over an algebraically closed field K such that [A,B] = 0, rank B less than or e qual to h is irreducible for all n and h = 0,...,n. The same result holds f or symmetric matrices (but when h not equal 0 we will assume char K not equ al 2) and, if char K not equal 2, for antisymmetric matrices (if char K = 0 when h not equal 0, 1). (C) 2000 Elsevier Science B.V. All rights reserved . MSC. 15A30; 14L30.