On commuting varieties of upper triangular matrices

It is known that the variety of the pairs of n×n commuting upper triangular matrices is not 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 m could be found by determining the dimension of the variety of the pairs of commuting strictly upper triangular matrices. Then, we define an embedding of any commuting variety into a grassmannian of subspaces of codimension 2.