Length, multiplicity, and multiplier ideals

Author:
Tommaso de Fernex

Journal:
Trans. Amer. Math. Soc. **358** (2006), 3717-3731

MSC (2000):
Primary 14B05; Secondary 13H05, 14B07, 13H15

DOI:
https://doi.org/10.1090/S0002-9947-06-03862-1

Published electronically:
March 24, 2006

MathSciNet review:
2218996

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

**[Bli]**M. Blickle, Multiplier ideal and module on a toric variety,*Math. Z.***248**(2004), no. 1, 113-121. MR**2092724****[dFEM1]**T. de Fernex, L. Ein and M. Mustata, Multiplicities and log canonical threshold,*J. Algebraic Geom.***13**(2004), 603-615. MR**2047683 (2005b:14008)****[dFEM2]**T. de Fernex, L. Ein and M. Mustata, Bounds on log canonical thresholds with application to birational rigidity,*Math. Res. Lett.***10**(2003), 219-236. MR**1981899 (2004e:14060)****[Eis]**D. Eisenbud,*Commutative Algebra with a View toward Algebraic Geometry*, Grad. Texts in Math.**150**, Springer, New York, 1995. MR**1322960 (97a:13001)****[HY]**N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals,*Trans. Amer. Math. Soc.***355**(2003), 3143-3174. MR**1974679 (2004i:13003)****[HT]**N. Hara and S. Takagi, On a generalization of test ideals, Nagoya Math. J.**175**(2004), 59-74. MR**2085311****[HH]**M. Hochster and C. Huneke, Tight closure, invariant theory and the Briançon-Skoda theorem,*J. Amer. Math. Soc.***3**(1990), 21-116. MR**1017784 (91g:13010)****[How]**J. Howald, Multiplier ideals of monomial ideals,*Trans. Amer. Math. Soc.***353**(2001), 2665-2671. MR**1828466 (2002b:14061)****[Laz]**R. Lazarsfeld,*Positivity in Algebraic Geometry*, book in print.**[TW]**S. Takagi and K.-i. Watanabe, On F-pure thresholds, J. Algebra**282**(2004), no. 1, 278-297. MR**2097584****[Tei]**B. Teissier, Monomes, volumes et multiplicités, in*Introduction à la Théorie des Singularités, II*, pp. 127-141, Travaux en Cours, 37, Hermann, Paris, 1988. MR**1074593 (92e:14003)**

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Additional Information

**Tommaso de Fernex**

Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109

Address at time of publication:
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540

Email:
defernex@umich.edu, defernex@math.ias.edu

DOI:
https://doi.org/10.1090/S0002-9947-06-03862-1

Keywords:
Multiplier ideal,
Samuel multiplicity,
monomial ideal

Received by editor(s):
June 21, 2004

Received by editor(s) in revised form:
September 20, 2004

Published electronically:
March 24, 2006

Additional Notes:
The author’s research was partially supported by the University of Michigan Rackham Research Grant and Summer Fellowship, and by the MIUR of the Italian Government in the framework of the National Research Project “Geometry on Algebraic Varieties" (Cofin 2002)

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.