Homology inclusion of complex line arrangements
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917543507132416 |
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| author | Rodau, Adrien |
| author_facet | Rodau, Adrien |
| contents | We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ballé and W. Cadiegan-Schlieper, we consider the inclusion map of the boundary manifold to the exterior and its effect on homology classes. A careful study of the graph Waldhausen structure of the boundary manifold allows to identify specific generators of the homology. Their potential images are encoded in a group, the graph stabiliser, with a nice combinatorial presentation. The invariant related to the inclusion map is an element of this group. Using a computer implementation in Sage, we compute the invariant for some examples and exhibit new Zariski pairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homology inclusion of complex line arrangements Rodau, Adrien Geometric Topology 32S22, 52C35, 57M05 We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ballé and W. Cadiegan-Schlieper, we consider the inclusion map of the boundary manifold to the exterior and its effect on homology classes. A careful study of the graph Waldhausen structure of the boundary manifold allows to identify specific generators of the homology. Their potential images are encoded in a group, the graph stabiliser, with a nice combinatorial presentation. The invariant related to the inclusion map is an element of this group. Using a computer implementation in Sage, we compute the invariant for some examples and exhibit new Zariski pairs. |
| title | Homology inclusion of complex line arrangements |
| topic | Geometric Topology 32S22, 52C35, 57M05 |
| url | https://arxiv.org/abs/2501.10558 |