Integrally closed ideals with $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)$

Abstract: Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d.$ We introduce and study the notion of the generalized type of $A$ with respect to an $\mathfrak{m}$-primary ideal $I$ denoted by $\operatorname{type}<em>I(A).$ Let $e_i(I)$ denote $i$th Hilbert coefficients of $A$ w.r.t. $I$. Assuming $I$ is integrally closed and $e</em>{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I) \neq 0$, we establish a sharp lower bound for $\operatorname{type}<em>I(A)$ in terms of the multiplicity and certain lengths associated to $I.$ We further show that when this lower bound is attained, the associated graded ring $G(I)$, is Cohen Macaulay. In the case of Buchsbaum local rings of dimension $d$ and depth at least $d-1$, we obtain an optimal lower bound for $e</em>{2}(\mathfrak{m})$ using the technique of $S_{2}$-fication. Additionally, for an integrally closed $\mathfrak{m}$-primary ideal $I,$ we also study the second extremal case $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)+1$ and its consequences on $G(I).$ We also investigate bounds on $e_3(I)$ and for $d=3,$ we study the consequences when these bounds are attained for.