The eleventh cohomology group of $\overline {\mathcal {M}}_{g,n}$
Forum of Mathematics, Sigma, Tome 11 (2023)
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We prove that the rational cohomology group $H^{11}(\overline {\mathcal {M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\overline {\mathcal {M}}_{g,n})$ is pure Hodge–Tate for all even $k \leq 12$ and deduce that $\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$ is surprisingly well approximated by a polynomial in q. In addition, we use $H^{11}(\overline {\mathcal {M}}_{1,11})$ 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 nontautological algebraic cycle classes in Chow cohomology.
@article{10_1017_fms_2023_59,
author = {Samir Canning and Hannah Larson and Sam Payne},
title = {The eleventh cohomology group of $\overline {\mathcal {M}}_{g,n}$},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.59},
language = {en},
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AU - Hannah Larson
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Samir Canning; Hannah Larson; Sam Payne. The eleventh cohomology group of $\overline {\mathcal {M}}_{g,n}$. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.59
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