Commuting pairs in the centralizers of 2-regular matrices

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.