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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2505.08211 |
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| _version_ | 1866916735014141952 |
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| author | Gorsky, Eugene Kim, Soyeon Scroggin, Tonie Simental, José |
| author_facet | Gorsky, Eugene Kim, Soyeon Scroggin, Tonie Simental, José |
| contents | For a positive braid $β\in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(β)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(β)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater than the length of $β$. For fixed $r$, the sets $\mathcal{U}_{r, w}$ form an open cover of $X(β)$. We conjecture that $\mathcal{U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb{C}[X(β)]$ and that $\mathcal{U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal{U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splicing braid varieties Gorsky, Eugene Kim, Soyeon Scroggin, Tonie Simental, José Algebraic Geometry Combinatorics 13F60 For a positive braid $β\in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(β)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(β)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater than the length of $β$. For fixed $r$, the sets $\mathcal{U}_{r, w}$ form an open cover of $X(β)$. We conjecture that $\mathcal{U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb{C}[X(β)]$ and that $\mathcal{U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal{U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures. |
| title | Splicing braid varieties |
| topic | Algebraic Geometry Combinatorics 13F60 |
| url | https://arxiv.org/abs/2505.08211 |