On the Alexander polynomials of conic-line arrangements
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908599629905920 |
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| author | Dimca, Alexandru Pokora, Piotr Sticlaru, Gabriel |
| author_facet | Dimca, Alexandru Pokora, Piotr Sticlaru, Gabriel |
| contents | In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01450 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Alexander polynomials of conic-line arrangements Dimca, Alexandru Pokora, Piotr Sticlaru, Gabriel Algebraic Geometry Algebraic Topology Combinatorics Primary: 14H50, Secondary: 14N20, 14B05, 13D02, 32S35, 32S40, 32S55 In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements. |
| title | On the Alexander polynomials of conic-line arrangements |
| topic | Algebraic Geometry Algebraic Topology Combinatorics Primary: 14H50, Secondary: 14N20, 14B05, 13D02, 32S35, 32S40, 32S55 |
| url | https://arxiv.org/abs/2305.01450 |