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arXiv · 2512.21461

Nearly Gorenstein rational surface singularities

Kyosuke Maeda·Tomohiro Okuma·Kei-ichi Watanabe·Ken-ichi Yoshida
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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.

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Kyosuke Maeda, Tomohiro Okuma, Kei-ichi Watanabe, Ken-ichi Yoshida. 2025-12-25. Nearly Gorenstein rational surface singularities. https://arxiv.org/abs/2512.21461

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