Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Multiplicities and log canonical threshold


Authors: Tommaso de Fernex, Lawrence Ein and Mircea Mustata
Translated by:
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 $I$.


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Additional Information

Tommaso de Fernex
Affiliation: Department of Mathematics, University of Michigan, 525 East University Avenue, Ann Arbor, Michigan 48109-1109
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
Email: ein@math.uic.edu

Mircea Mustata
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Email: mirceamustata@yahoo.com

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

American Mathematical Society