Cohen-Macaulay local rings of embedding dimension $e+d-k$

Abstract

In this paper, we prove the following. Let $(R, \mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with multiplicity $e$ and embedding dimension $v=e+d-k$, where $k \geq 3$ and $e-k>1$. If $\lambda(\mathfrak{m}^3/J\mathfrak{m}^2)=1$ and $\mathfrak{m}^3\subseteq J\mathfrak{m}$, where $J$ is a minimal reduction of $\mathfrak{m}$, then $3 \leq s \leq \tau +k-1$, where $s$ is the degree of the $h$-polynomial of $R$ and $\tau$ is the Cohen-Macaulay type of $R$.