## Abstract

We consider the space M_{q,n} of regular q-tuples of commuting nilpotent endomorphisms of k^{n} modulo simultaneous conjugation. We show that M_{q,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, M_{q,n} is a smooth vector bundle over Pq−^{1}. We prove that, in this case, M_{q,n} is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that M_{q,n} possesses a universal property and represents a functor of ideals, and we use it to identify M_{q,n} with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that M_{q,n} → Pq−^{1} is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.

Original language | English (US) |
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Pages (from-to) | 4293-4311 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Issue number | 6 |

DOIs | |

State | Published - Sep 15 2019 |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics