Extendability of the $B_2$-arrangement
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912411066302464 |
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| author | Hoge, Torsten Maehara, Shota Wiesner, Sven |
| author_facet | Hoge, Torsten Maehara, Shota Wiesner, Sven |
| contents | Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always exists. In this work, we demonstrate that if $\mathscr{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathscr{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extendability of the $B_2$-arrangement Hoge, Torsten Maehara, Shota Wiesner, Sven Combinatorics 52C35, 32S22, 51F15 Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always exists. In this work, we demonstrate that if $\mathscr{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathscr{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank. |
| title | Extendability of the $B_2$-arrangement |
| topic | Combinatorics 52C35, 32S22, 51F15 |
| url | https://arxiv.org/abs/2506.02512 |