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Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines

Part of: Curves Graph theory

Published online by Cambridge University Press:  24 January 2020

Shinzo Bannai  and
Momoko Ohno
Shinzo Bannai
Affiliation:
Department of Natural Sciences, National Institute of Technology, Ibaraki College, 866 Nakane, Hitachinaka-shi, Ibaraki-Ken 312-8508, Japan Email: sbannai@ge.ibaraki-ct.ac.jp
Momoko Ohno
Affiliation:
Department of Mathematics and Information Sciences, Graduate School of Science and Engineering, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachiohji 192-0397, Japan Email: yamamoto-momoko@ed.tmu.ac.jp
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Abstract

In this paper, we study how to distinguish the embedded topology of a smooth quartic and its bitangent lines. In order to do this, we introduce the concept of two-graphs and switching classes from graph theory. This new method improves previous results about a quartic and three bitangent lines considered by E. Artal Bartolo and J. Vallès, four bitangent lines considered by the authors and H. Tokunaga, and enables us to distinguish the embedded topology of a smooth quartic and five or more bitangent lines. As an application, we obtain a new Zariski 5-tuple and a Zariski 9-tuple for arrangements consisting of a smooth quartic and five of its bitangent lines and six of its bitangent lines, respectively.

Keywords

Zariski pairembedded topologytwo-graphsmooth quarticbitangent line

MSC classification

Primary: 14H30: Coverings, fundamental group
Secondary: 05C65: Hypergraphs
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 802 - 812
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The first author is partially supported by Grant-in-Aid for Scientific Research C (18K03263).

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