Moduli Theory of the $r$-Braid Arrangement
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912444491759616 |
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| author | Blankers, Vance Clader, Emily Halacheva, Iva Liu, Haggai Ross, Dustin |
| author_facet | Blankers, Vance Clader, Emily Halacheva, Iva Liu, Haggai Ross, Dustin |
| contents | We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful compactification of the braid arrangement (with respect to its minimal building set) is well-known to yield the moduli space $\overline{\mathcal{M}}_{0,n}$, and, in this work, we generalize this result, constructing a moduli space $\overline{\mathcal{M}}^r_{n}$ of certain genus-zero curves with an order-$r$ involution that we identify with the corresponding wonderful compactification of the $r$-braid arrangement. The resulting space is a variant of the previously studied moduli space $\overline{\mathcal{L}}^r_n$ [arXiv:2104.06526], related via a change of weights on the markings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moduli Theory of the $r$-Braid Arrangement Blankers, Vance Clader, Emily Halacheva, Iva Liu, Haggai Ross, Dustin Algebraic Geometry Combinatorics 14H10 (primary), 14N20 (secondary) We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful compactification of the braid arrangement (with respect to its minimal building set) is well-known to yield the moduli space $\overline{\mathcal{M}}_{0,n}$, and, in this work, we generalize this result, constructing a moduli space $\overline{\mathcal{M}}^r_{n}$ of certain genus-zero curves with an order-$r$ involution that we identify with the corresponding wonderful compactification of the $r$-braid arrangement. The resulting space is a variant of the previously studied moduli space $\overline{\mathcal{L}}^r_n$ [arXiv:2104.06526], related via a change of weights on the markings. |
| title | Moduli Theory of the $r$-Braid Arrangement |
| topic | Algebraic Geometry Combinatorics 14H10 (primary), 14N20 (secondary) |
| url | https://arxiv.org/abs/2506.18205 |