TY - JOUR

T1 - Conjugacy classes of commuting nilpotents

AU - Haboush, William J.

AU - Hyeon, Donghoon

N1 - Funding Information:
2017R1E1A1A03071042, funded by the government of Korea, and Samsung Science & Technology Foundation grant SSTF-BA1601-05.
Funding Information:
Received by the editors February 14, 2017, and, in revised form, August 16, 2018. 2010 Mathematics Subject Classification. Primary 14L30; Secondary 14C05, 15A27, 15A72. The second author was partially supported by NRF grants No. 2017R1A5A1015626 and No.
Publisher Copyright:
© 2019 American Mathematical Society.

PY - 2019/9/15

Y1 - 2019/9/15

N2 - 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.

AB - 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.

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U2 - 10.1090/tran7782

DO - 10.1090/tran7782

M3 - Article

AN - SCOPUS:85075182531

SN - 0002-9947

VL - 372

SP - 4293

EP - 4311

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -