Put another way: is every polynomial in $2 n^2$ variables that vanishes on a pair of commuting matrices, in the ideal generated by the obvious $n^2$ quadratic relations? Alas, we do not know (and I cannot answer the question either). I will introduce several other related schemes that seem easier to study, like the space of pairs of matrices whose commutator is diagonal, which I will prove is a reduced complete intersection, one of whose components is the commuting variety. Conjecturally, it has only one other component, and I will explain where that one comes from. Along the way we will also see a rather curious invariant of permutations, and much simple linear algebra.