A note on discreteness of jumping numbers
Author:
Karl Schwede
Journal:
Proc. Amer. Math. Soc. 139 (2011), 38953901
MSC (2000):
Primary 13A35, 14F18, 14B05
Published electronically:
June 28, 2011
MathSciNet review:
2823035
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Suppose that is a ring essentially of finite type over a perfect field of characteristic and that is an ideal. We prove that the set of jumping numbers of has no limit points under the assumption that is normal and Gorenstein  we make no assumption as to whether the Gorenstein index is divisible by . Furthermore, we also show that the jumping numbers of are discrete under the more general assumption that is Cartier.
 [Bli09]
M. Blickle: Test ideals via algebras of liner maps, arXiv:0912.2255. To appear in the Journal of Algebraic Geometry.
 [BMS08]
Manuel
Blickle, Mircea
Mustaţǎ, and Karen
E. Smith, Discreteness and rationality of
𝐹thresholds, Michigan Math. J. 57 (2008),
43–61. Special volume in honor of Melvin Hochster. MR 2492440
(2010c:13003), 10.1307/mmj/1220879396
 [BMS09]
Manuel
Blickle, Mircea
Mustaţă, and Karen
E. Smith, 𝐹thresholds of
hypersurfaces, Trans. Amer. Math. Soc.
361 (2009), no. 12, 6549–6565. MR 2538604
(2011a:13006), 10.1090/S0002994709047199
 [BSTZ10]
Manuel
Blickle, Karl
Schwede, Shunsuke
Takagi, and Wenliang
Zhang, Discreteness and rationality of 𝐹jumping numbers on
singular varieties, Math. Ann. 347 (2010),
no. 4, 917–949. MR 2658149
(2011k:13008), 10.1007/s0020800904612
 [DH09]
Tommaso
de Fernex and Christopher
D. Hacon, Singularities on normal varieties, Compos. Math.
145 (2009), no. 2, 393–414. MR 2501423
(2010c:14013), 10.1112/S0010437X09003996
 [ELSV04]
Lawrence
Ein, Robert
Lazarsfeld, Karen
E. Smith, and Dror
Varolin, Jumping coefficients of multiplier ideals, Duke Math.
J. 123 (2004), no. 3, 469–506. MR 2068967
(2005k:14004), 10.1215/S0012709404123334
 [Har06]
Nobuo
Hara, Fpure thresholds and Fjumping exponents in dimension
two, Math. Res. Lett. 13 (2006), no. 56,
747–760. With an appendix by Paul Monsky. MR 2280772
(2007m:14032), 10.4310/MRL.2006.v13.n5.a6
 [HT04]
Nobuo
Hara and Shunsuke
Takagi, On a generalization of test ideals, Nagoya Math. J.
175 (2004), 59–74. MR 2085311
(2005g:13009)
 [HY03]
Nobuo
Hara and KenIchi
Yoshida, A generalization of tight closure and
multiplier ideals, Trans. Amer. Math. Soc.
355 (2003), no. 8,
3143–3174 (electronic). MR 1974679
(2004i:13003), 10.1090/S0002994703032859
 [Hoc07]
M. Hochster: Foundations of tight closure theory, lecture notes from a course taught at the University of Michigan, fall 2007.
 [HH90]
Melvin
Hochster and Craig
Huneke, Tight closure, invariant theory, and
the BriançonSkoda theorem, J. Amer.
Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784
(91g:13010), 10.1090/S08940347199010177846
 [KLZ09]
Mordechai
Katzman, Gennady
Lyubeznik, and Wenliang
Zhang, On the discreteness and rationality of 𝐹jumping
coefficients, J. Algebra 322 (2009), no. 9,
3238–3247. MR 2567418
(2011c:13005), 10.1016/j.jalgebra.2008.11.032
 [MTW05]
Mircea
Mustaţǎ, Shunsuke
Takagi, and Keiichi
Watanabe, Fthresholds and BernsteinSato polynomials,
European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005,
pp. 341–364. MR 2185754
(2007b:13010)
 [MY09]
Mircea
Mustaţă and KenIchi
Yoshida, Test ideals vs. multiplier ideals, Nagoya Math. J.
193 (2009), 111–128. MR 2502910
(2010a:13010)
 [Sch09a]
Karl
Schwede, 𝐹adjunction, Algebra Number Theory
3 (2009), no. 8, 907–950. MR 2587408
(2011b:14006), 10.2140/ant.2009.3.907
 [Sch09b]
K. Schwede: Test ideals in nonQGorenstein rings, arXiv:0906.4313, to appear in Trans. Amer. Math. Soc.
 [Sch10]
Karl
Schwede, Centers of 𝐹purity, Math. Z.
265 (2010), no. 3, 687–714. MR 2644316
(2011e:13011), 10.1007/s0020900905365
 [Tak04]
Shunsuke
Takagi, An interpretation of multiplier ideals
via tight closure, J. Algebraic Geom.
13 (2004), no. 2,
393–415. MR 2047704
(2005c:13002), 10.1090/S1056391103003667
 [Tak06]
Shunsuke
Takagi, Formulas for multiplier ideals on singular varieties,
Amer. J. Math. 128 (2006), no. 6, 1345–1362. MR 2275023
(2007i:14006)
 [Urb10]
S. Urbinati: Discrepancies of non Gorenstein varieties, arXiv:1001.2930. To appear in the Michigan Mathematical Journal.
 [Bli09]
 M. Blickle: Test ideals via algebras of liner maps, arXiv:0912.2255. To appear in the Journal of Algebraic Geometry.
 [BMS08]
 M. Blickle, M. Mustaţă, and K. Smith: Discreteness and rationality of Fthresholds, Michigan Math. J. 57 (2008), 4361. MR 2492440 (2010c:13003)
 [BMS09]
 M. Blickle, M. Mustaţă, and K. E. Smith: thresholds of hypersurfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 65496565. MR 2538604 (2011a:13006)
 [BSTZ10]
 M. Blickle, K. Schwede, S. Takagi, and W. Zhang: Discreteness and rationality of jumping numbers on singular varieties, Math. Ann. 347 (2010), no. 4, 917949. MR 2658149
 [DH09]
 T. De Fernex and C. Hacon: Singularities on normal varieties, Compos. Math. 145 (2009), no. 2, 393414. MR 2501423 (2010c:14013)
 [ELSV04]
 L. Ein, R. Lazarsfeld, K. E. Smith, and D. Varolin: Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), no. 3, 469506. MR 2068967 (2005k:14004)
 [Har06]
 N. Hara: Fpure thresholds and Fjumping exponents in dimension two, Math. Res. Lett. 13 (2006), no. 56, 747760. With an appendix by Paul Monsky. MR 2280772 (2007m:14032)
 [HT04]
 N. Hara and S. Takagi: On a generalization of test ideals, Nagoya Math. J. 175 (2004), 5974. MR 2085311 (2005g:13009)
 [HY03]
 N. Hara and K.I. Yoshida: A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), no. 8, 31433174 (electronic). MR 1974679 (2004i:13003)
 [Hoc07]
 M. Hochster: Foundations of tight closure theory, lecture notes from a course taught at the University of Michigan, fall 2007.
 [HH90]
 M. Hochster and C. Huneke: Tight closure, invariant theory, and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31116. MR 1017784 (91g:13010)
 [KLZ09]
 M. Katzman, G. Lyubeznik, and W. Zhang: On the discreteness and rationality of jumping coefficients, J. Algebra 322 (2009), no. 9, 32383247. MR 2567418 (2011c:13005)
 [MTW05]
 M. Mustaţa, S. Takagi, and K. Watanabe: Fthresholds and BernsteinSato polynomials, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 341364. MR 2185754 (2007b:13010)
 [MY09]
 M. Mustaţă and K.I. Yoshida: Test ideals vs. multiplier ideals, Nagoya Math. J. 193 (2009), 111128. MR 2502910 (2010a:13010)
 [Sch09a]
 K. Schwede: adjunction, Algebra Number Theory 3 (2009), no. 8, 907950. MR 2587408 (2011b:14006)
 [Sch09b]
 K. Schwede: Test ideals in nonQGorenstein rings, arXiv:0906.4313, to appear in Trans. Amer. Math. Soc.
 [Sch10]
 K. Schwede: Centers of purity, Math. Z. 265 (2010), no. 3, 687714. MR 2644316 (2011e:13011)
 [Tak04]
 S. Takagi: An interpretation of multiplier ideals via tight closure, J. Algebraic Geom. 13 (2004), no. 2, 393415. MR 2047704 (2005c:13002)
 [Tak06]
 S. Takagi: Formulas for multiplier ideals on singular varieties, Amer. J. Math. 128 (2006), no. 6, 13451362. MR 2275023 (2007i:14006)
 [Urb10]
 S. Urbinati: Discrepancies of non Gorenstein varieties, arXiv:1001.2930. To appear in the Michigan Mathematical Journal.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
13A35,
14F18,
14B05
Retrieve articles in all journals
with MSC (2000):
13A35,
14F18,
14B05
Additional Information
Karl Schwede
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email:
schwede@math.psu.edu
DOI:
http://dx.doi.org/10.1090/S000299392011108256
Keywords:
Test ideal,
jumping number,
$\mathbb{Q}$Gorenstein,
multiplier ideal
Received by editor(s):
April 8, 2010
Received by editor(s) in revised form:
October 4, 2010
Published electronically:
June 28, 2011
Additional Notes:
The author was partially supported by a National Science Foundation postdoctoral fellowship and by NSF grant DMS1064485/0969145.
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
