Holomorphic forms and non-tautological cycles on moduli spaces of curves
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908306248826880 |
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| author | Arena, Veronica Canning, Samir Clader, Emily Haburcak, Richard Li, Amy Q. Mok, Siao Chi Tamborini, Carolina |
| author_facet | Arena, Veronica Canning, Samir Clader, Emily Haburcak, Richard Li, Amy Q. Mok, Siao Chi Tamborini, Carolina |
| contents | We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results show that there exist non-tautological algebraic cohomology classes on $\mathcal{M}_g$ for $g=12$ and all $g \geq 16$. These results generalize the work of Graber--Pandharipande and van Zelm, who proved that the classes of particular loci of bielliptic curves are non-tautological and thereby exhibited the only previously-known non-tautological class on any $\mathcal{M}_g$: the bielliptic cycle on $\mathcal{M}_{12}$. We extend their work by using the existence of holomorphic forms on certain moduli spaces $\overline{\mathcal{M}}_{g,n}$ to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new double-cover loci are non-tautological. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_03874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic forms and non-tautological cycles on moduli spaces of curves Arena, Veronica Canning, Samir Clader, Emily Haburcak, Richard Li, Amy Q. Mok, Siao Chi Tamborini, Carolina Algebraic Geometry 14C15, 14C17 We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results show that there exist non-tautological algebraic cohomology classes on $\mathcal{M}_g$ for $g=12$ and all $g \geq 16$. These results generalize the work of Graber--Pandharipande and van Zelm, who proved that the classes of particular loci of bielliptic curves are non-tautological and thereby exhibited the only previously-known non-tautological class on any $\mathcal{M}_g$: the bielliptic cycle on $\mathcal{M}_{12}$. We extend their work by using the existence of holomorphic forms on certain moduli spaces $\overline{\mathcal{M}}_{g,n}$ to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new double-cover loci are non-tautological. |
| title | Holomorphic forms and non-tautological cycles on moduli spaces of curves |
| topic | Algebraic Geometry 14C15, 14C17 |
| url | https://arxiv.org/abs/2402.03874 |