Gorenstein Normal tangent cones of integrally closed ideals in two-dimensional normal singularities

Abstract

Let $(A,\mathfrak m)$ be a two-dimensional excellent normal Gorenstein local domain containing an algebraically closed filed. Let $I =H^0(X,\mathcal{O}_X(-Z)) \subset A$ be an $\mathfrak m$-primary integrally closed ideal represented by an anti-nef cycle $Z$ on some resolution $X\to \mathrm{Spec} A$. In this paper, we prove that $\overline{G}(I)$ is Gorenstein if and only if it is Cohen-Macaulay and $(r-1)Z^2+K_XZ=0$, where $r=\bar{\athrm{r}}(I)$ denotes the normal reduction number of $I$ and $K_X$ denotes the canonical divisor on $X$.