In M-n(k), k an algebraically closed field, we call a matrix I-regular if e
ach eigenspace is at most l-dimensional. We prove that the variety of commu
ting pairs in the centralizer of a 2-regular matrix is the direct product o
f various affine spaces and various determinantal varieties L-l,L-m obtaine
d from matrices over truncated polynomial rings, We prove that these variet
ies L-l,L-m, are irreducible and apply this to the case of the: k-algebra g
enerated by three commuting matrices: we show that if one of the three matr
ices is 2-regular, then the algebra has dimension at most n. We also show t
hat such an algebra is always contained in a commutative subalgebra of M-n(
k) of dimension exactly n. (C) 1999 Academic Press.