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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Conjugacy classes of commuting nilpotents
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by William J. Haboush and Donghoon Hyeon PDF
Trans. Amer. Math. Soc. 372 (2019), 4293-4311 Request permission

Abstract:

We consider the space $\mathcal {M}_{q,n}$ of regular $q$-tuples of commuting nilpotent endomorphisms of $k^n$ modulo simultaneous conjugation. We show that $\mathcal {M}_{q,n}$ admits a natural homogeneous space structure, and that it is an affine space bundle over ${\mathbb {P}}^{q-1}$. A closer look at the homogeneous structure reveals that, over ${\mathbb {C}}$ and with respect to the complex*1pt topology, $\mathcal {M}_{q,n}$ is a smooth vector bundle over ${\mathbb {P}}^{q-1}$. We prove that, in this case, $\mathcal {M}_{q,n}$ is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that $\mathcal {M}_{q,n}$ possesses a universal property and represents a functor of ideals, and we use it to identify $\mathcal {M}_{q,n}$ with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that $\mathcal {M}_{q,n} \to {\mathbb {P}}^{q-1}$ is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.
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Additional Information
  • William J. Haboush
  • Affiliation: Department of Mathematics, The University of Illinois at Urbana Champaign, 1409 West Green Street, 273 Altgeld Hall, Urbana, Illinois 61801
  • MR Author ID: 79055
  • Email: haboush@math.uiuc.edu
  • Donghoon Hyeon
  • Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
  • MR Author ID: 673409
  • Email: dhyeon@snu.ac.kr
  • Received by editor(s): February 14, 2017
  • Received by editor(s) in revised form: August 16, 2018
  • Published electronically: April 4, 2019
  • Additional Notes: The second author was partially supported by NRF grants No. 2017R1A5A1015626 and No. 2017R1E1A1A03071042, funded by the government of Korea, and Samsung Science & Technology Foundation grant SSTF-BA1601-05.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4293-4311
  • MSC (2010): Primary 14L30; Secondary 14C05, 15A27, 15A72
  • DOI: https://doi.org/10.1090/tran/7782
  • MathSciNet review: 4009390