Jet schemes of the commuting matrix pairs scheme
Authors:
B. A. Sethuraman and Klemen Sivic
Journal:
Proc. Amer. Math. Soc. 137 (2009), 39533967
MSC (2000):
Primary 14M99
Published electronically:
July 30, 2009
MathSciNet review:
2538555
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that for all there exists an integer such that for all the th order jet scheme over the commuting matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for all the th order jet scheme over the commuting matrices is irreducible; we show that the same holds for .
 1.
F.
R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated
by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649
(21 #6372c)
 2.
Murray
Gerstenhaber, On dominance and varieties of commuting
matrices, Ann. of Math. (2) 73 (1961), 324–348.
MR
0132079 (24 #A1926)
 3.
Russell
A. Goward Jr. and Karen
E. Smith, The jet scheme of a monomial scheme, Comm. Algebra
34 (2006), no. 5, 1591–1598. MR 2229478
(2007e:13037), http://dx.doi.org/10.1080/00927870500454927
 4.
Robert
M. Guralnick, A note on commuting pairs of matrices, Linear
and Multilinear Algebra 31 (1992), no. 14,
71–75. MR
1199042 (94c:15021), http://dx.doi.org/10.1080/03081089208818123
 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, J. Pure Appl. Algebra 195 (2005), no. 1,
75–95. MR
2100311 (2005h:13020), http://dx.doi.org/10.1016/j.jpaa.2004.06.001
 7.
Tomaž
Košir and B.
A. Sethuraman, A Groebner basis for the 2×2 determinantal
ideal \mod𝑡², J. Algebra 292 (2005),
no. 1, 138–153. MR 2166800
(2006f:13011), http://dx.doi.org/10.1016/j.jalgebra.2004.12.005
 8.
Qing
Liu, Algebraic geometry and arithmetic curves, Oxford Graduate
Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002.
Translated from the French by Reinie Erné; Oxford Science
Publications. MR
1917232 (2003g:14001)
 9.
Mircea
Mustaţă, Jet schemes of locally complete intersection
canonical singularities, Invent. Math. 145 (2001),
no. 3, 397–424. With an appendix by David Eisenbud and Edward
Frenkel. MR
1856396 (2002f:14005), http://dx.doi.org/10.1007/s002220100152
 10.
Mircea
Mustaţǎ, Singularities of pairs via jet
schemes, J. Amer. Math. Soc.
15 (2002), no. 3,
599–615 (electronic). MR 1896234
(2003b:14005), http://dx.doi.org/10.1090/S0894034702003910
 11.
T.
S. Motzkin and Olga
Taussky, Pairs of matrices with property
𝐿. II, Trans. Amer. Math. Soc. 80 (1955), 387–401.
MR
0086781 (19,242c), http://dx.doi.org/10.1090/S00029947195500867815
 12.
John
F. Nash Jr., Arc structure of singularities, Duke Math. J.
81 (1995), no. 1, 31–38 (1996). A celebration
of John F. Nash, Jr. MR 1381967
(98f:14011), http://dx.doi.org/10.1215/S0012709495081034
 13.
Michael
G. Neubauer and David
J. Saltman, Twogenerated commutative subalgebras of
𝑀_{𝑛}(𝐹), J. Algebra 164
(1994), no. 2, 545–562. MR 1271255
(95g:16030), http://dx.doi.org/10.1006/jabr.1994.1077
 14.
Michael
G. Neubauer and B.
A. Sethuraman, Commuting pairs in the centralizers of 2regular
matrices, J. Algebra 214 (1999), no. 1,
174–181. MR 1684884
(2000h:15024), http://dx.doi.org/10.1006/jabr.1998.7703
 15.
Cornelia Yuen, Jet Schemes and Truncated Wedge Schemes, Ph.D. thesis, University of Michigan (2006).
 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), 324348. MR 0132079 (24:A1926)
 3.
 Russell Goward and Karen Smith, The jet scheme of a monomial scheme. Comm. Algebra, 34 (2006), 15911598. MR 2229478 (2007e:13037)
 4.
 R. Guralnick, A note on commuting pairs of matrices, Linear and Multilinear Algebra, 31 (1992), 7175. 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), 7595. MR 2100311 (2005h:13020)
 7.
 Tomaž Košir and B.A. Sethuraman, A Groebner basis for the determinantal ideal , J. Algebra, 292 (2005), 138153. 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), 397424. MR 1856396 (2002f:14005)
 10.
 Mircea Mustaţa. Singularities of pairs via jet schemes. J. Amer. Math. Soc., 15(3) (2002), 599615. MR 1896234 (2003b:14005)
 11.
 T. Motzkin and O. TausskyTodd, Pairs of matrices with property . II, Trans. Amer. Math. Soc., 80 (1955), 387401. 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), 3138 (1996). MR 1381967 (98f:14011)
 13.
 Michael G. Neubauer and David J. Saltman, Twogenerated commutative subalgebras of , J. Algebra, 164 (1994), 545562. MR 1271255 (95g:16030)
 14.
 M.J. Neubauer and B.A. Sethuraman, Commuting pairs in the centralizers of regular matrices, Journal of Algebra, 214 (1999), 174181. 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.unilj.si
DOI:
http://dx.doi.org/10.1090/S0002993909100291
PII:
S 00029939(09)100291
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 DMS0700904.
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.
