On the irreducibility of varieties of commuting matrices

We will show that the variety of all the pairs (A,B) of n×n matrices over an algebraically closed field K such that [A,B]=0, rank B≤h is irreducible for all n and h=0,…,n. The same result holds for symmetric matrices (but when h≠0 we will assume char K≠2) and, if char K≠2, for antisymmetric matrices (if char K=0 when h≠0,1).