On Gorensteinness of associated graded rings of filtrations

Let $(A,\mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F}=\{F_n{\}}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the Hilbert coefficients of $\mathcal{F}$ in some cases. As a consequence we recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of $A=S\mathrm{/}(f)$ where $S=K[\text{\hspace{0.17em}⁣}[x_0,x_1,\dots ,x_m]\text{\hspace{0.17em}⁣}]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\dots ,x_m)$, where $g$ is a polynomial with $g\in (x_1,\dots ,x_m{)}^b\setminus (x_1,\dots ,x_m{)}^{b+1}$, and $a,b,m$ are integers. We show that the normal tangent cone $\bar{G}(\mathfrak{m})$ is Cohen-Macaulay if $A$ is normal and $a\le b$. Moreover, we give a criterion of the Gorensteinness of $\bar{G}(\mathfrak{m})$.