On varieties of commuting triples III

The problem of irreducibility of the variety C(3,n) of triples of commuting n×n matrices is equivalent to that whether each triple of commuting n×n matrices can be approximated arbitrary well by triples of commuting generic matrices (i.e. matrices having n distinct eigenvalues). It has been proved that the variety C(3,n) is irreducible for n⩽8 and reducible for n⩾30. Using simultaneous commutative approximation of pairs of matrices in the centralizer of the third matrix we prove that the varieties C(3, 9) and C(3, 10) are also irreducible.