## 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.## References

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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