The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909845185101824 |
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| author | Canning, Samir |
| author_facet | Canning, Samir |
| contents | We prove that the tautological rings $\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})$ and $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq 2$ and $2g+n\geq 24$, extending results of Petersen and Tommasi in genus $2$. The proof uses the intersection of tautological classes with non-tautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when $g=0,1$ or $g\geq 2$ and $2g+n<24$. The conjecture is known for $g=0,1$ by work of Keel and Petersen, and we prove several new cases of this conjecture for $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ when $g\geq 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_10516 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein Canning, Samir Algebraic Geometry 14C15, 14C17 We prove that the tautological rings $\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})$ and $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq 2$ and $2g+n\geq 24$, extending results of Petersen and Tommasi in genus $2$. The proof uses the intersection of tautological classes with non-tautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when $g=0,1$ or $g\geq 2$ and $2g+n<24$. The conjecture is known for $g=0,1$ by work of Keel and Petersen, and we prove several new cases of this conjecture for $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ when $g\geq 2$. |
| title | The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein |
| topic | Algebraic Geometry 14C15, 14C17 |
| url | https://arxiv.org/abs/2406.10516 |