Moduli spaces of curves with polynomial point counts
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908978757238784 |
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| author | Canning, Samir Larson, Hannah Payne, Sam Willwacher, Thomas |
| author_facet | Canning, Samir Larson, Hannah Payne, Sam Willwacher, Thomas |
| contents | We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19913 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moduli spaces of curves with polynomial point counts Canning, Samir Larson, Hannah Payne, Sam Willwacher, Thomas Algebraic Geometry Number Theory Primary 14H10, Secondary 14G15, 14F20, 14F25 We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n. |
| title | Moduli spaces of curves with polynomial point counts |
| topic | Algebraic Geometry Number Theory Primary 14H10, Secondary 14G15, 14F20, 14F25 |
| url | https://arxiv.org/abs/2410.19913 |