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 $A=S/(f)$ where $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in (x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{ b+1}$, and $a, \, b, \, m$ are integers. We show that the normal tangent cone $\overline{ G}(\mathfrak{ m})$ is Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of the Gorensteinness of $\overline{ G}(\mathfrak{ m})$.
Categories: math.AC