Nearly Gorenstein rational surface singularities

Abstract

In this paper, we show that for any rational surface singularity $A$, the canonical trace ideal ${\mathrm{Tr}}_A(K_A)$ is an 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 $F\ge Z_f$ if $A$ is not Gorenstein, where $Z_f$ is the fundamental cycle. As a result, we give a criterion for the 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 $Z_f$ is almost reduced; (b) quotient singularities.

20 pages. To appear in Journal of Pure and Applied Algebra