Quasi-socle ideals in local rings with Gorenstein tangent cones

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$inaNoetherianlocalring$(A, \mathfrak{m})$ with the Gorenstein tangent cone $$\mathrm{G}$(\mathfrak{m}) = \bigoplus_{n \geq 0}{\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$areexplored,where$q \geq 1$ is an integer and $Q$ is a parameter ideal of $A$ generated by monomials of a system $x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_d$ of elements in $A$ such that $(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_d)$ is a reduction of $\mathfrak{m}$. The questions of when $I$ is integral over $Q$ and of when the graded rings $\mathrm{G}(I)={⨁}_{n\ge 0}{I}^n\mathrm{/}{I}^{n+1}$ and $\mathrm{F}(I)={⨁}_{n\ge 0}{I}^n\mathrm{/}\mathfrak{m}{I}^n$ are Cohen-Macaulay are answered. Criteria for $\mathrm{G}(I)$ and $$\mathcal{R}$ (I) = \bigoplus_{n \geq 0}I^n$ to be Gorenstein rings are given.