On varieties of commuting nilpotent matrices

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