Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917591644110848 |
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| author | Chen, William Y. Salter, Nick |
| author_facet | Chen, William Y. Salter, Nick |
| contents | We show that the configuration space of four unordered points in $\mathbb{C}$ with barycenter 0 is isomorphic to the space of triples $(E,Q,ω)$, where $E$ is an elliptic curve, $Q\in E^\circ$ a nonzero point, and $ω$ a nonzero holomorphic differential on $E$. At the level of fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group $B_4$: namely, the sequence $1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1$, where $F_2$ is a free group of rank 2, related to Ferrari's solution of the quartic, and the sequence $1\rightarrow \mathbb{Z} \rightarrow B_4\rightarrow\operatorname{Aut}^+(F_2)\rightarrow 1$ of Dyer-Formanek-Grossman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands Chen, William Y. Salter, Nick Algebraic Geometry Geometric Topology We show that the configuration space of four unordered points in $\mathbb{C}$ with barycenter 0 is isomorphic to the space of triples $(E,Q,ω)$, where $E$ is an elliptic curve, $Q\in E^\circ$ a nonzero point, and $ω$ a nonzero holomorphic differential on $E$. At the level of fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group $B_4$: namely, the sequence $1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1$, where $F_2$ is a free group of rank 2, related to Ferrari's solution of the quartic, and the sequence $1\rightarrow \mathbb{Z} \rightarrow B_4\rightarrow\operatorname{Aut}^+(F_2)\rightarrow 1$ of Dyer-Formanek-Grossman. |
| title | Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands |
| topic | Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2402.11081 |