Nearly Gorenstein rational surface singularities

TL;DR

This paper characterizes nearly Gorenstein rational surface singularities by analyzing the canonical trace ideal and minimal anti-nef cycles, providing a classification in specific cases such as almost reduced fundamental cycles and quotient singularities.

Contribution

It introduces a criterion for nearly Gorenstein rational surface singularities and classifies them in cases with almost reduced fundamental cycles and quotient singularities.

Findings

The canonical trace ideal is integrally closed and represented by the minimal anti-nef cycle.

For non-Gorenstein singularities, the minimal anti-nef cycle exceeds the fundamental cycle.

Complete classification of nearly Gorenstein rational singularities in specific cases.

Abstract

In this paper, we show that for any rational surface singularity $A$, the canonical trace ideal $\mathrm{Tr}_A(K_A)$ is integrally closed ideal which is represented by the minimal anti-nef cycle $F$ on the minimal resolution of singularities so that $K_X+F$ is anti-nef. Then $F \ge \mathbb Z$ if $A$ is not Gorenstein, where $\mathbb Z$ is the fundamental cycle. As a result, we give a criterion for rational surface singularity $A$ to be nearly Gorenstein. Moreover, we classify all nearly Gorenstein rational singularities in terms of resolution of singularities in the following cases: (a) the fundamental cycle $\mathbb Z$ is almost reduced; (b) quotient singularity.