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Transactions of the American Mathematical Society

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



Conjugacy classes of commuting nilpotents

Authors: William J. Haboush and Donghoon Hyeon
Journal: Trans. Amer. Math. Soc. 372 (2019), 4293-4311
MSC (2010): Primary 14L30; Secondary 14C05, 15A27, 15A72
Published electronically: April 4, 2019
MathSciNet review: 4009390
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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. Iarrobinoโ€™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

Donghoon Hyeon
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
MR Author ID: 673409

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.
Article copyright: © Copyright 2019 American Mathematical Society