It is known that the variety of the pairs of n x n commuting upper triangular matrices is not a complete intersection for infinitely many values of n; we show that there exists in 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.
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