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

TL;DR

This paper characterizes when the associated graded ring of an integrally closed ideal in a two-dimensional Gorenstein singularity is Gorenstein, linking it to Cohen-Macaulayness and a specific numerical condition involving the ideal's cycle.

Contribution

It provides a precise criterion for the Gorenstein property of the tangent cone of integrally closed ideals in two-dimensional normal Gorenstein singularities.

Findings

Gorenstein property of the tangent cone is equivalent to Cohen-Macaulayness and a numerical condition.

The criterion involves the normal reduction number and intersection numbers with the canonical divisor.

The result characterizes Gorenstein tangent cones in terms of geometric and algebraic invariants.

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 $X$.