Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913008510304256 |
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| author | Battistella, Luca Di Lorenzo, Andrea |
| author_facet | Battistella, Luca Di Lorenzo, Andrea |
| contents | For $n\leq 6$, we compute the integral Chow ring of every modular compactification of $\mathcal{M}_{1,n}$ parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne--Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We further deduce that all these modular compactifications satisfy the Chow-Künneth generation property, that the cycle class map is an isomorphism, and for $n=4$ we study whether the Getzler's relation hold integrally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$ Battistella, Luca Di Lorenzo, Andrea Algebraic Geometry 14C15, 14H10 For $n\leq 6$, we compute the integral Chow ring of every modular compactification of $\mathcal{M}_{1,n}$ parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne--Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We further deduce that all these modular compactifications satisfy the Chow-Künneth generation property, that the cycle class map is an isomorphism, and for $n=4$ we study whether the Getzler's relation hold integrally. |
| title | Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$ |
| topic | Algebraic Geometry 14C15, 14H10 |
| url | https://arxiv.org/abs/2505.04587 |