Geodesic
Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines
Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 802-812

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008439520000053
Mots-clés : Zariski pair, embedded topology, two-graph, smooth quartic, bitangent line 
Bannai, Shinzo; Ohno, Momoko. Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines. Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 802-812. doi: 10.4153/S0008439520000053
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[1] Artal Bartolo, E., Sur les couples des Zariski. J. Algebraic Geom. 3(1994), no. 2, 223–247. Google Scholar

[2] 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/05010001 Google Scholar

[3] 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.2004.02.003 Google Scholar | DOI

[4] 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.2016.02.005 Google Scholar | DOI

[5] 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.93.50 Google Scholar | DOI

[6] 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-z Google Scholar | DOI

[7] 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/1564106549 Google Scholar | DOI

[8] Dolgachev, I. V., Classical algebraic geometry. A modern view. Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139084437 Google Scholar | DOI

[9] Guerville-Ballé, B. and Meilhan, J.-B., A linking invariant for algebraic curves. Google Scholar

[10] 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/10.1137/0128070 Google Scholar | DOI

[11] Seidel, J. J., A survey of two-graphs. Proc. Int. Coll. Theorie Combinatorie, Acc. Naz. Lincei, Rome, 1973. Google Scholar

[12] Shioda, T., On the Mordell–Weil lattices. Comment. Math. Univ. St. Paul. 39(1990), 211–240. Google Scholar

[13] Shioda, T., Plane Quartics and Mordell-Weil Lattices of Type E ∗. Comment. Math. Univ. St. Paul. 42(1993), 61–79. Google Scholar

[14] Shirane, T., A note on splitting numbers for Galois covers and 𝜋-equivalent Zariski k-plets. Proc. Amer. Math. Soc. 145(2017), no. 3, 1009–1017. https://doi.org/10.1090/proc/13298 Google Scholar | DOI

[15] 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-2017-066-5 Google Scholar | DOI

[16] 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/06620613 Google Scholar | DOI

[17] 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

[18] 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/10.2307/2370712 Google Scholar | DOI

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