Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Jet schemes of the commuting matrix pairs scheme


Authors: B. A. Sethuraman and Klemen Sivic
Journal: Proc. Amer. Math. Soc. 137 (2009), 3953-3967
MSC (2000): Primary 14M99
DOI: https://doi.org/10.1090/S0002-9939-09-10029-1
Published electronically: July 30, 2009
MathSciNet review: 2538555
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that for all $ k\ge 1$ there exists an integer $ N(k)$ such that for all $ n\ge N(k)$ the $ k$-th order jet scheme over the commuting $ n\times n$ matrix pairs scheme is reducible.

At the other end of the spectrum, it is known that for all $ k\ge 1$ the $ k$-th order jet scheme over the commuting $ 2\times 2$ matrices is irreducible; we show that the same holds for $ n=3$.


References [Enhancements On Off] (What's this?)

  • 1. F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea Publishing Company, New York, 1959. MR 0107649 (21:6372c)
  • 2. M. Gerstenhaber, On dominance and varieties of commuting matrices, Annals of Mathematics (2), 73 (1961), 324-348. MR 0132079 (24:A1926)
  • 3. Russell Goward and Karen Smith, The jet scheme of a monomial scheme. Comm. Algebra, 34 (2006), 1591-1598. MR 2229478 (2007e:13037)
  • 4. R. Guralnick, A note on commuting pairs of matrices, Linear and Multilinear Algebra, 31 (1992), 71-75. MR 1199042 (94c:15021)
  • 5. Boyan Jonov, Shellability of a complex associated to the first order jet scheme of a determinantal variety, in preparation.
  • 6. Tomaž Košir and B.A. Sethuraman, Determinantal varieties over truncated polynomial rings, Journal of Pure and Applied Algebra, 195 (2005), 75-95. MR 2100311 (2005h:13020)
  • 7. Tomaž Košir and B.A. Sethuraman, A Groebner basis for the $ 2\times2$ determinantal ideal $ \mod t^2$, J. Algebra, 292 (2005), 138-153. MR 2166800 (2006f:13011)
  • 8. Qing Liu, Algebraic Geometry and Arithmetic Curves, Oxford Grad. Texts Math., 6, Oxford University Press, 2002. MR 1917232 (2003g:14001)
  • 9. Mircea Mustaţa. Jet schemes of locally complete intersection canonical singularities. Invent. Math., 145 (3) (2001), 397-424. MR 1856396 (2002f:14005)
  • 10. Mircea Mustaţa. Singularities of pairs via jet schemes. J. Amer. Math. Soc., 15(3) (2002), 599-615. MR 1896234 (2003b:14005)
  • 11. T. Motzkin and O. Taussky-Todd, Pairs of matrices with property $ L$. II, Trans. Amer. Math. Soc., 80 (1955), 387-401. MR 0086781 (19:242c)
  • 12. John F. Nash, Jr., Arc structure of singularities. A celebration of John F. Nash, Jr., Duke Math. J., 81 (1995) (1), 31-38 (1996). MR 1381967 (98f:14011)
  • 13. Michael G. Neubauer and David J. Saltman, Two-generated commutative subalgebras of $ M_n(F)$, J. Algebra, 164 (1994), 545-562. MR 1271255 (95g:16030)
  • 14. M.J. Neubauer and B.A. Sethuraman, Commuting pairs in the centralizers of $ 2$-regular matrices, Journal of Algebra, 214 (1999), 174-181. MR 1684884 (2000h:15024)
  • 15. Cornelia Yuen, Jet Schemes and Truncated Wedge Schemes, Ph.D. thesis, University of Michigan (2006).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14M99

Retrieve articles in all journals with MSC (2000): 14M99


Additional Information

B. A. Sethuraman
Affiliation: Department of Mathematics, California State University, Northridge, Northridge, California 91330
Email: al.sethuraman@csun.edu

Klemen Sivic
Affiliation: Institute of Mathematics, Physics, and Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Email: klemen.sivic@fmf.uni-lj.si

DOI: https://doi.org/10.1090/S0002-9939-09-10029-1
Received by editor(s): November 4, 2008
Received by editor(s) in revised form: February 19, 2009
Published electronically: July 30, 2009
Additional Notes: The first author was supported by the National Science Foundation grant DMS-0700904.
The second author was supported by the Slovenian Research Agency.
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society