Poncelet’s closure theorem and the embedded topology of conic-line arrangements
Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 109-123
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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.
Mots-clés :
conic-line arrangements, embedded topology, Poncelet’s closure theorem, Zariski pairs, splitting invariants
Bannai, Shinzo; Masuya, Ryosuke; Shirane, Taketo; Tokunaga, Hiro-o; Yorisaki, Emiko. Poncelet’s closure theorem and the embedded topology of conic-line arrangements. Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 109-123. doi: 10.4153/S0008439524000481
@article{10_4153_S0008439524000481,
author = {Bannai, Shinzo and Masuya, Ryosuke and Shirane, Taketo and Tokunaga, Hiro-o and Yorisaki, Emiko},
title = {Poncelet{\textquoteright}s closure theorem and the embedded topology of conic-line arrangements},
journal = {Canadian mathematical bulletin},
pages = {109--123},
year = {2025},
volume = {68},
number = {1},
doi = {10.4153/S0008439524000481},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000481/}
}
TY - JOUR AU - Bannai, Shinzo AU - Masuya, Ryosuke AU - Shirane, Taketo AU - Tokunaga, Hiro-o AU - Yorisaki, Emiko TI - Poncelet’s closure theorem and the embedded topology of conic-line arrangements JO - Canadian mathematical bulletin PY - 2025 SP - 109 EP - 123 VL - 68 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000481/ DO - 10.4153/S0008439524000481 ID - 10_4153_S0008439524000481 ER -
%0 Journal Article %A Bannai, Shinzo %A Masuya, Ryosuke %A Shirane, Taketo %A Tokunaga, Hiro-o %A Yorisaki, Emiko %T Poncelet’s closure theorem and the embedded topology of conic-line arrangements %J Canadian mathematical bulletin %D 2025 %P 109-123 %V 68 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000481/ %R 10.4153/S0008439524000481 %F 10_4153_S0008439524000481
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