Multiplicities and log canonical threshold

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 $I$.