Jet Schemes of the Commuting Matrix Pairs Scheme
نویسنده
چکیده
We show that for all k ≥ 1, there exists an integer N(k) such that for all n ≥ N(k) the k-th order jet scheme over the commuting n × n matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for all k ≥ 1, the k-th order jet scheme over the commuting 2× 2 matrices is irreducible: we show that the same holds for n = 3.
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Some Schemes Related to the Commuting Variety
The commuting variety is the pairs of n × n matrices (X, Y) such that XY = YX. We introduce the diagonal commutator scheme, { (X, Y) : XY − YX is diagonal } , which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme { (X, Y) : X...
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