The fundamental group of the complement of a generic fiber-type curve
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908459176296448 |
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| author | Cogolludo-Agustín, José I. Elduque, Eva |
| author_facet | Cogolludo-Agustín, José I. Elduque, Eva |
| contents | In this paper we describe and characterize the fundamental group of the complement to generic fiber-type curves, i.e. unions of (the closure of) finitely many generic fibers of a component-free pencil $F=[f:g]:\mathbb C\mathbb P^2\dashrightarrow \mathbb C\mathbb P^1$. We also observe that these groups have already appeared in the literature in examples by Eyral and Oka associated with curves of fiber-type. As a byproduct of our techniques we obtain a criterion to address Zariski's Problem on the commutativity of the fundamental group of the complement of a plane curve. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The fundamental group of the complement of a generic fiber-type curve Cogolludo-Agustín, José I. Elduque, Eva Algebraic Geometry In this paper we describe and characterize the fundamental group of the complement to generic fiber-type curves, i.e. unions of (the closure of) finitely many generic fibers of a component-free pencil $F=[f:g]:\mathbb C\mathbb P^2\dashrightarrow \mathbb C\mathbb P^1$. We also observe that these groups have already appeared in the literature in examples by Eyral and Oka associated with curves of fiber-type. As a byproduct of our techniques we obtain a criterion to address Zariski's Problem on the commutativity of the fundamental group of the complement of a plane curve. |
| title | The fundamental group of the complement of a generic fiber-type curve |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.15814 |