Bounds on the second Hilbert coefficient and the depth of the associated graded ring
Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d \geq 1$ with $\mathrm{depth} R \geq d-1,$ and let $I$ be an $\mathfrak m$-primary ideal. In this paper, we study bounds on the second Hilbert coefficient of $I$, denoted by $e_{2}(I)$. Under the assumption that the associated graded ring $G(I)$ has depth at least $d-1,$ we first establish a lower bound for $e_{2}(I).$ We then extend several known results from the Cohen-Macaulay case to this general setting and obtain upper bounds for $e_{2}(I)$ in terms of the sectional genus denoted by $\mathrm{g}_{s}(I)$ and the Hilbert coefficients of $I$ and those of a minimal reduction $Q$ of $I$. We further analyze the extremal case when $e_{2}(I)$ attains this bound and relate it to the depth of $G(I)$. In addition, for Buchsbaum local rings, we establish a sharp upper bound for $e_{2}(\mathfrak m)$ using the technique of $S_{2}$-fication. Finally, in the Cohen-Macaulay case, we give sufficient conditions to ensure good properties on the depth of $G(I)$ and of $G(I^n)$ under the assumption that $e_{2}(I)=0$.