Abstract We investigate geometric properties of algebraic sets defined by matrix pairs with special commutators. Specifically, we prove that the set consisting of the n × n matrix pairs whose commutators are diagonal matrices is a complete intersection with two components, one of which is the so called commuting variety that consists of commuting matrix pairs, and that the algebraic set as a scheme is reduced when the base field is of characteristic zero. We also prove that the algebraic set comprising the n × n matrix pairs whose commutators have vanishing diagonal entries is an irreducible complete intersection, and, as a scheme, is reduced when the base field is of characteristic zero.

中文翻译：

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关于具有对角线换向器的矩阵对

摘要 我们研究了由具有特殊交换子的矩阵对定义的代数集的几何性质。具体来说，我们证明由交换子为对角矩阵的 n × n 矩阵对组成的集合是两个分量的完全交集，其中一个是由交换矩阵对组成的所谓交换簇，代数集为当基场的特征为零时，方案被简化。我们还证明了由 n × n 矩阵对组成的代数集是一个不可约的完全交集，并且作为一种方案，当基场的特征为零时被约简。