Conjugacy classes of commuting nilpotents

William J. Haboush, Donghoon Hyeon

Research output: Contribution to journalArticlepeer-review


We consider the space Mq,n of regular q-tuples of commuting nilpotent endomorphisms of kn modulo simultaneous conjugation. We show that Mq,n admits a natural homogeneous space structure, and that it is an affine space bundle over Pq−1. A closer look at the homogeneous structure reveals that, over C and with respect to the complex topology, Mq,n is a smooth vector bundle over Pq−1. We prove that, in this case, Mq,n is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that Mq,n possesses a universal property and represents a functor of ideals, and we use it to identify Mq,n with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that Mq,n → Pq−1 is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.

Original languageEnglish (US)
Pages (from-to)4293-4311
Number of pages19
JournalTransactions of the American Mathematical Society
Issue number6
StatePublished - Sep 15 2019

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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