Nearly Gorenstein rational surface singularities
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911338661412864 |
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| author | Maeda, Kyosuke Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi |
| author_facet | Maeda, Kyosuke Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearly Gorenstein rational surface singularities Maeda, Kyosuke Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi Algebraic Geometry Primary: 14J17, Secondary: 14B05, 13B22, 13H10 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. |
| title | Nearly Gorenstein rational surface singularities |
| topic | Algebraic Geometry Primary: 14J17, Secondary: 14B05, 13B22, 13H10 |
| url | https://arxiv.org/abs/2512.21461 |