Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866915212115836928 |
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| author | Amram, Meirav Shwartz, Robert Sinichkin, Uriel Tan, Sheng-Li Tokunaga, Hiro-o |
| author_facet | Amram, Meirav Shwartz, Robert Sinichkin, Uriel Tan, Sheng-Li Tokunaga, Hiro-o |
| contents | We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to determine the fundamental groups.
Two examples of degree $7$ Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). We were able to provide alternative proof of the fact those are indeed Zariski pairs by our methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_03507 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups Amram, Meirav Shwartz, Robert Sinichkin, Uriel Tan, Sheng-Li Tokunaga, Hiro-o Algebraic Geometry 14H30 (Primary) 14H50, 14Q05, 20F55 (Secondary) We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to determine the fundamental groups. Two examples of degree $7$ Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). We were able to provide alternative proof of the fact those are indeed Zariski pairs by our methods. |
| title | Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups |
| topic | Algebraic Geometry 14H30 (Primary) 14H50, 14Q05, 20F55 (Secondary) |
| url | https://arxiv.org/abs/2106.03507 |