The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912176534454272 |
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| author | Canning, Samir Larson, Hannah Payne, Sam |
| author_facet | Canning, Samir Larson, Hannah Payne, Sam |
| contents | We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# \bar{\mathcal{M}}_{g,n}(\mathbb{F}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_03113 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$ Canning, Samir Larson, Hannah Payne, Sam Algebraic Geometry 14C15, 14C17, 14C25 We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# \bar{\mathcal{M}}_{g,n}(\mathbb{F}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology. |
| title | The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$ |
| topic | Algebraic Geometry 14C15, 14C17, 14C25 |
| url | https://arxiv.org/abs/2209.03113 |