Cohomological invariants of $\mathscr{M}_{3,n}$ via level structures
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916946705907712 |
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| author | Di Lorenzo, Andrea |
| author_facet | Di Lorenzo, Andrea |
| contents | We show that mod $2$ cohomological invariants of the moduli stack $\mathscr{M}_{3,n}$ of smooth pointed curves of genus three contain a free module with generators in degree $0$, $2$, $3$, $4$ and $6$, formed by the invariants of the symplectic group $\mathrm{Sp}_6(2)$. We achieve this by showing that the torsor of full level two structures $\mathscr{M}_{3,n}(2) \to \mathscr{M}_{3,n}$ is versal. Along the way, we prove that the invariants of the stack of del Pezzo surfaces of degree two contain the invariants of the Weyl group $W(\mathsf{E}_7)$ and that the mod $2$ cohomology of $\mathscr{M}_{3,n}$ is non-zero in degree three. Our main result holds also for the stack $\mathscr{A}_3$ of principally polarized abelian threefolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomological invariants of $\mathscr{M}_{3,n}$ via level structures Di Lorenzo, Andrea Algebraic Geometry 14F20, 14H1, 14D23 We show that mod $2$ cohomological invariants of the moduli stack $\mathscr{M}_{3,n}$ of smooth pointed curves of genus three contain a free module with generators in degree $0$, $2$, $3$, $4$ and $6$, formed by the invariants of the symplectic group $\mathrm{Sp}_6(2)$. We achieve this by showing that the torsor of full level two structures $\mathscr{M}_{3,n}(2) \to \mathscr{M}_{3,n}$ is versal. Along the way, we prove that the invariants of the stack of del Pezzo surfaces of degree two contain the invariants of the Weyl group $W(\mathsf{E}_7)$ and that the mod $2$ cohomology of $\mathscr{M}_{3,n}$ is non-zero in degree three. Our main result holds also for the stack $\mathscr{A}_3$ of principally polarized abelian threefolds. |
| title | Cohomological invariants of $\mathscr{M}_{3,n}$ via level structures |
| topic | Algebraic Geometry 14F20, 14H1, 14D23 |
| url | https://arxiv.org/abs/2509.09661 |