A variety of commuting triples Original Research Article

The problem of approximating triples of commuting n×n matrices by generic matrices is equivalent to the problem of whether the variety C(3,n) of commuting triples is irreducible. The answer to the problem is known to be positive for dimensions no greater than 5 and negative for dimensions no smaller than 30. Using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimension 6 as well. All the proofs are given over an arbitrary algebraically closed field of characteristic zero.