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

Part of: Graph theory Curves

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).

References

Artal Bartolo, E., Sur les couples des Zariski. J. Algebraic Geom. 3(1994), no. 2, 223–247.Google Scholar
Artal Bartolo, E., Cogolludo, J.-I., and Tokunaga, H., A survey on Zariski pairs. In: Algebraic geometry in East Asia–Hanoi 2005. Adv. Stud. Pure Math., 50, Math. Soc. Japan, Tokyo, 2008, pp. 1–100. https://doi.org/10.2969/aspm/05010001Google Scholar
Artal Bartolo, E. and Tokunaga, H., Zariski k-plets of rational curve arrangements and dihedral covers. Topology Appl. 142(2004), no. 1–3, 227–233. https://doi.org/10.1016/j.topol.2004.02.003CrossRefGoogle Scholar
Bannai, S., A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces. Topology Appl. 202(2016), 423–439. https://doi.org/10.1016/j.topol.2016.02.005CrossRefGoogle Scholar
Bannai, S., Guerville-Ballé, B., Shirane, T., and Tokunaga, H., On the topology of arrangements of a cubic and its inflectional tangents. Proc. Japan Acad. 93(2017), 50–53. https://doi.org/10.3792/pjaa.93.50CrossRefGoogle Scholar
Bannai, S. and Tokunaga, H., Geometry of bisections of elliptic surfaces and Zariski N-plets for conic arrangements. Geom. Dedicata 178(2015), 219–237. https://doi.org/10.1007/s10711-015-0054-zCrossRefGoogle Scholar
Bannai, S., Tokunaga, H., and Yamamoto, M., A note on the topology of arrangements for a smooth plane quartic and its bitangent lines. Hiroshima Math. J. 49(2019), 289–302. https://doi.org/10.32917/hmj/1564106549CrossRefGoogle Scholar
Dolgachev, I. V., Classical algebraic geometry. A modern view. Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139084437CrossRefGoogle Scholar
Guerville-Ballé, B. and Meilhan, J.-B., A linking invariant for algebraic curves. arxiv:1602.04916Google Scholar
Mallows, C. L. and Sloane, N. J. A., Two-graphs, switching classes and Euler graphs are equal in number. SIAM J. Appl. Math. 28(1975), 876–880. https://doi.org/10.1137/0128070CrossRefGoogle Scholar
Seidel, J. J., A survey of two-graphs. Proc. Int. Coll. Theorie Combinatorie, Acc. Naz. Lincei, Rome, 1973.Google Scholar
Shioda, T., On the Mordell–Weil lattices. Comment. Math. Univ. St. Paul. 39(1990), 211–240.Google Scholar
Shioda, T., Plane Quartics and Mordell-Weil Lattices of Type E 7∗. Comment. Math. Univ. St. Paul. 42(1993), 61–79.Google Scholar
Shirane, T., A note on splitting numbers for Galois covers and 𝜋1-equivalent Zariski k-plets. Proc. Amer. Math. Soc. 145(2017), no. 3, 1009–1017. https://doi.org/10.1090/proc/13298CrossRefGoogle Scholar
Shirane, T., Connected numbers and the embedded topology of plane curves. Canad. Math. Bull. 61(2018), no. 3, 650–658. https://doi.org/10.4153/CMB-2017-066-5CrossRefGoogle Scholar
Tokunaga, H., Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers. J. Math. Soc. Japan 66(2014), 613–640. https://doi.org/10.2969/jmsj/06620613CrossRefGoogle Scholar
van Lint, J. H. and Seidel, J. J., Equilateral point sets in elliptic geometry. Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28(1966), 335–348.Google Scholar
Zariski, O., On the problem of existence of algebraic functions of two variables possessing a given branch curve. Amer. J. Math. 51(1929), 305–328. https://doi.org/10.2307/2370712CrossRefGoogle Scholar