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Poncelet’s closure theorem and the embedded topology of conic-line arrangements

Part of: Curves Birational geometry Low-dimensional topology

Published online by Cambridge University Press:  25 November 2024

Shinzo Bannai Open the ORCID record for Shinzo Bannai [Opens in a new window] ,
Ryosuke Masuya ,
Taketo Shirane Open the ORCID record for Taketo Shirane [Opens in a new window] ,
Hiro-o Tokunaga  and
Emiko Yorisaki
Shinzo Bannai*
Affiliation:
Department of Applied Mathematics, Faculty of Science, Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama 700-0005, Japan
Ryosuke Masuya
Affiliation:
Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachiohji 192-0397, Japan e-mail: tokunaga@tmu.ac.jp masuya-ryosuke@ed.tmu.ac.jp sagawa-emiko@ed.tmu.ac.jp
Taketo Shirane
Affiliation:
Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachiohji 192-0397, Japan e-mail: tokunaga@tmu.ac.jp masuya-ryosuke@ed.tmu.ac.jp sagawa-emiko@ed.tmu.ac.jp
Hiro-o Tokunaga
Affiliation:
Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachiohji 192-0397, Japan e-mail: tokunaga@tmu.ac.jp masuya-ryosuke@ed.tmu.ac.jp sagawa-emiko@ed.tmu.ac.jp
Emiko Yorisaki
Affiliation:
Department of Mathematical Science, Faculty of Science and Technology, Tokushima University, 2-1 Minamijyousanjima-cho, Tokushima 770-8506, Japan e-mail: shirane@tokushima-u.ac.jp
*
e-mail: bannai@ous.ac.jp
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Abstract

In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet’s closure theorem. We study unramified double covers of the union of two conics that are induced by a $2m$-sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree $2m+6$ for $m\geq 2$ that consist of reducible curves having two conics and $2m+2$ lines as irreducible components.

Keywords

conic-line arrangementsembedded topologyPoncelet’s closure theoremZariski pairssplitting invariants

MSC classification

Primary: 14E20: Coverings 14H30: Coverings, fundamental group 14H50: Plane and space curves 57M10: Covering spaces

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 1 , March 2025 , pp. 109 - 123
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author was partially supported by JSPS KAKENHI Grant Numbers JP18K03263, JP23K03042. The third author was partially supported by JSPS KAKENHI Grant Number JP21K03182. The fourth author was partially supported by JSPS KAKENHI Grant Number JP20K03561, JP24K06673.

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