On varieties of commuting nilpotent matrices

Let N(d,n) be the variety of all d-tuples of commuting nilpotent n×n matrices. It is well-known that N(d,n) is irreducible if d=2, if n⩽3 or if d=3 and n=4. On the other hand N(3,n) is known to be reducible for n⩾13. We study in this paper the reducibility of N(d,n) for various values of d and n. In particular, we prove that N(d,n) is reducible for all d,n⩾4. In the case d=3, we show that it is irreducible for n⩽6.