Inductive Freeness of Ziegler's Canonical Multiderivations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909512052506624 |
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| author | Hoge, Torsten Roehrle, Gerhard |
| author_facet | Hoge, Torsten Roehrle, Gerhard |
| contents | Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $\mathcal A''$ of $\mathcal A$ to any hyperplane endowed with the natural multiplicity $κ$ is then a free multiarrangement $(\mathcal A'',κ)$. The aim of this paper is to prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if $\mathcal A$ is inductively free, then so is the multiarrangement $(\mathcal A'',κ)$. In a related result we derive that if a deletion $\mathcal A'$ of $\mathcal A$ is free and the corresponding restriction $\mathcal A''$ is inductively free, then so is $(\mathcal A'',κ)$ -- irrespective of the freeness of $\mathcal A$. In addition, we show counterparts of the latter kind for additive and recursive freeness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_09540 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Inductive Freeness of Ziegler's Canonical Multiderivations Hoge, Torsten Roehrle, Gerhard Combinatorics 52C35, 14N20, 32S22, 51D20 Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $\mathcal A''$ of $\mathcal A$ to any hyperplane endowed with the natural multiplicity $κ$ is then a free multiarrangement $(\mathcal A'',κ)$. The aim of this paper is to prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if $\mathcal A$ is inductively free, then so is the multiarrangement $(\mathcal A'',κ)$. In a related result we derive that if a deletion $\mathcal A'$ of $\mathcal A$ is free and the corresponding restriction $\mathcal A''$ is inductively free, then so is $(\mathcal A'',κ)$ -- irrespective of the freeness of $\mathcal A$. In addition, we show counterparts of the latter kind for additive and recursive freeness. |
| title | Inductive Freeness of Ziegler's Canonical Multiderivations |
| topic | Combinatorics 52C35, 14N20, 32S22, 51D20 |
| url | https://arxiv.org/abs/2204.09540 |