Nearly Gorenstein rational surface singularities

In this paper, we show that for any rational surface singularity $A$, the canonical trace ideal ${\mathrm{T}\mathrm{r}}_A(K_A)$ 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.