  It is known that the variety of pairs of n x n commuting upper triangular
matrices isn't a complete intersection for infinitely many values of n; we show
that there exists m such that this happens if and only if n > m. We also show
that m < 18 and that it could be found by determining the dimension of the
variety of pairs of commuting strictly upper triangular matrices. Then we
define a natural map from the variety of pairs of commuting n x n matrices onto
a subvariety defined by linear equations of the grassmannian of subspaces of
codimension 2 of a vector space of dimension n x n.

Algebraic Geometry;Operator Algebras