The multiplicity and the number of generators of an integrally closed ideal

Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay $R$ of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on $p_g$ ideals by Okuma-Watanabe-Yoshida, multiplicity of the fiber cone, and the $h$-vector of the associated graded ring.