thresholds of hypersurfaces
Authors:
Manuel Blickle, Mircea Mustata and Karen E. Smith
Journal:
Trans. Amer. Math. Soc. 361 (2009), 65496565
MSC (2000):
Primary 13A35; Secondary 14B05
Published electronically:
July 16, 2009
MathSciNet review:
2538604
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Abstract: We use the module theoretic description of generalized test ideals to show that in any finite regular ring the thresholds of hypersurfaces are discrete and rational. Furthermore we show that any limit of pure thresholds of principal ideals in bounded dimension is again an pure threshold; hence in particular the limit is rational.
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 [Bli]
 M. Blickle, The intersection homology module in finite characteristic, Ph.D. thesis, University of Michigan, 2001, math. AG/0110244.
 [BMS]
 M. Blickle, M. Mustaţă and K. E. Smith, Discreteness and rationality of thresholds, Michigan Math. J. 57 (2008), 4361.
 [dFM]
 T. de Fernex and M. Mustaţă, Limits of log canonical thresholds, Ann. Sci. École Norm. Sup., to appear.
 [Gol]
 R. Goldblatt, Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics 188, SpringerVerlag, New York, 1998. MR 1643950 (2000a:03113)
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 N. Hara, with an appendix by P. Monsky, pure thresholds and jumping coefficients in dimension two, Math. Res. Lett. 13 (2006), 747760. MR 2280772 (2007m:14032)
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 N. Hara and K.i. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 31433174. MR 1974679 (2004i:13003)
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 J. Kollár, Singularities of pairs, in Algebraic geometry, Santa Cruz 1995, 221286, volume 62 of Proc. Symp. Pure Math., Amer. Math. Soc., 1997. MR 1492525 (99m:14033)
 [Laz]
 R. Lazarsfeld, Positivity in algebraic geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 49, SpringerVerlag, Berlin, 2004. MR 2095472 (2005k:14001b)
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 G. Lyubeznik, modules: applications to local cohomology and modules in characteristic , J. Reine Angew. Math. 491 (1997), 65130. MR 1476089 (99c:13005)
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 M. Mustaţă, S. Takagi and K.i. Watanabe, thresholds and BernsteinSato polynomials, European Congress of Mathematics, 341364, Eur. Math. Soc., Zürich, 2005. MR 2185754 (2007b:13010)
 [Ta1]
 S. Takagi, Formulas for multiplier ideals on singular varieties, Amer. J. Math. 128 (2006), 13451362. MR 2275023 (2007i:14006)
 [Ta2]
 S. Takagi, Adjoint ideals along closed subvarieties of higher codimension, J. Reine Angew. Math., to appear.
 [TW]
 S. Takagi and K.i. Watanabe, On pure thresholds, J. Algebra 282 (2004), 278297. MR 2097584 (2006a:13010)
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Additional Information
Manuel Blickle
Affiliation:
Fachbereich Mathematik, Universität DuisburgEssen, Standort Essen, 45117 Essen, Germany
Email:
manuel.blickle@uniessen.de
Mircea Mustata
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
mmustata@umich.edu
Karen E. Smith
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
kesmith@umich.edu
DOI:
http://dx.doi.org/10.1090/S0002994709047199
PII:
S 00029947(09)047199
Keywords:
$F$thresholds,
test ideals,
$F$modules,
nonstandard extension
Received by editor(s):
July 23, 2007
Received by editor(s) in revised form:
January 2, 2008
Published electronically:
July 16, 2009
Additional Notes:
Partial support was provided by grant SFB/TR 45 of the DFG (first author), NSF grants DMS0758454, DMS 0111298 and a Packard Fellowship (second author), and NSF grant DMS0500823(third author)
Article copyright:
© Copyright 2009
American Mathematical Society
