Multiplicities and log canonical threshold

Authors:
Tommaso de Fernex, Lawrence Ein and Mircea Mustaţǎ

Journal:
J. Algebraic Geom. **13** (2004), 603-615

DOI:
https://doi.org/10.1090/S1056-3911-04-00346-7

Published electronically:
February 25, 2004

MathSciNet review:
2047683

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Abstract |
References |
Additional Information

Abstract: Given an $n$-dimensional local ring $R$ of a smooth variety, and a zero-dimensional ideal $I\subset R$, we prove the following inequality involving the Samuel multiplicity and the log canonical threshold: $e(I)\geq n^n/\operatorname {lc}(I)^n$. Moreover, equality holds if and only if the integral closure of $I$ is a power of the maximal ideal in $R$. When $n=2$, we give a similar inequality for an arbitrary ideal $I$.

[Co]corti A. Corti, Singularities of linear systems and $3$-fold birational geometry, in *Explicit birational geometry of $3$-folds*, 259–312, Cambridge Univ. Press, Cambridge, 2000.
[CPR]CPR A. Corti, A.V. Pukhlikov and M. Reid, Fano $3$-fold hypersurfaces, in *Explicit birational geometry of $3$-folds*, 175–258, Cambridge Univ. Press, Cambridge, 2000.
[DEM]DEM T. de Fernex, L. Ein and M. Mustaţǎ, Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett. 10 (2003), 219–236.
[DK]DK J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) **34** (2001), 525–556.
[Ei]eisenbud D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math. **150**, Springer, New York, 1995.
[GG]GG T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geom. **8** (1999), 695–736.
[Ho]howald J. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. **353** (2001), 2665–2671.
[IM]IM V.A. Iskovskikh and Yu. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Math. USSR Sbornik **15** (1971), 141–166.
[Ko]kollar J. Kollár, Singularities of pairs, in *Algebraic Geometry—Santa Cruz 1995*, 221–287, Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1997.
[Mu]mustata M. Mustaţǎ, Singularities of pairs via jet schemes, J. Amer. Math. Soc. **15** (2002), 599–615.
[Pu1]Pu1 A.V. Pukhlikov, Birationally rigid Fano hypersurfaces, preprint 2002, math.AG/ 0201302.
[Pu2]Pu2 A.V. Pukhlikov, Essentials of the method of maximal singularities, in *Explicit birational geometry of $3$-folds*, 73–100, Cambridge Univ. Press, Cambridge, 2000.
[Pu3]Pu3 A.V. Pukhlikov, Birational automorphisms of a four-dimensional quintic, Invent. Math. **87** (1987), 303–329.
[Re]Rees D. Rees, ${\mathfrak a}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. **57** (1961), 8–17.

[Co]corti A. Corti, Singularities of linear systems and $3$-fold birational geometry, in *Explicit birational geometry of $3$-folds*, 259–312, Cambridge Univ. Press, Cambridge, 2000.
[CPR]CPR A. Corti, A.V. Pukhlikov and M. Reid, Fano $3$-fold hypersurfaces, in *Explicit birational geometry of $3$-folds*, 175–258, Cambridge Univ. Press, Cambridge, 2000.
[DEM]DEM T. de Fernex, L. Ein and M. Mustaţǎ, Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett. 10 (2003), 219–236.
[DK]DK J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) **34** (2001), 525–556.
[Ei]eisenbud D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math. **150**, Springer, New York, 1995.
[GG]GG T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geom. **8** (1999), 695–736.
[Ho]howald J. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. **353** (2001), 2665–2671.
[IM]IM V.A. Iskovskikh and Yu. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Math. USSR Sbornik **15** (1971), 141–166.
[Ko]kollar J. Kollár, Singularities of pairs, in *Algebraic Geometry—Santa Cruz 1995*, 221–287, Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1997.
[Mu]mustata M. Mustaţǎ, Singularities of pairs via jet schemes, J. Amer. Math. Soc. **15** (2002), 599–615.
[Pu1]Pu1 A.V. Pukhlikov, Birationally rigid Fano hypersurfaces, preprint 2002, math.AG/ 0201302.
[Pu2]Pu2 A.V. Pukhlikov, Essentials of the method of maximal singularities, in *Explicit birational geometry of $3$-folds*, 73–100, Cambridge Univ. Press, Cambridge, 2000.
[Pu3]Pu3 A.V. Pukhlikov, Birational automorphisms of a four-dimensional quintic, Invent. Math. **87** (1987), 303–329.
[Re]Rees D. Rees, ${\mathfrak a}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. **57** (1961), 8–17.

Additional Information

**Tommaso de Fernex**

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

MR Author ID:
635850

Email:
defernex@math.uic.edu

**Lawrence Ein**

Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 Morgan St., M/C. 249, Chicago, Illinois 60607-7045

MR Author ID:
62255

Email:
ein@math.uic.edu

**Mircea Mustaţǎ**

Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138

Email:
mirceamustata@yahoo.com

Received by editor(s):
May 23, 2002

Published electronically:
February 25, 2004

Additional Notes:
Research of the first author was partially supported by MURST of Italian Government, National Research Project (Cofin 2000) “Geometry of Algebraic Varieties”. Research of the second author was partially supported by NSF Grant DMS 99-70295. The third author served as a Clay Mathematics Institute Long-Term Prize Fellow while this research was done.