A note on splitting numbers for Galois covers and $\pi _1$-equivalent Zariski $k$-plets
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Abstract:
In this paper, we introduce splitting numbers of subvarieties in a smooth complex variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. In particular cases, we show that splitting numbers enable us to distinguish the topology of complex plane curves even if the fundamental groups of the complements of plane curves are isomorphic. Consequently, we prove that there are $\pi _1$-equivalent Zariski $k$-plets for any integer $k\geq 2$.References
- Enrique Artal-Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), no. 2, 223–247 (French). MR 1257321
- Enrique Artal Bartolo, Jorge Carmona Ruber, and José Ignacio Cogolludo Agustín, Braid monodromy and topology of plane curves, Duke Math. J. 118 (2003), no. 2, 261–278. MR 1980995, DOI 10.1215/S0012-7094-03-11823-2
- Enrique Artal Bartolo, José Ignacio Cogolludo, and Hiro-o Tokunaga, A survey on Zariski pairs, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 1–100. MR 2409555, DOI 10.2969/aspm/05010001
- E. Artal Bartolo, V. Florens, and B. Guerville-Ballé, A topological invariant of line arrangements, Preprint available at arXiv : 1407.3387v1. 2014.
- Enrique Artal Bartolo and Hiro-o Tokunaga, Zariski $k$-plets of rational curve arrangements and dihedral covers, Topology Appl. 142 (2004), no. 1-3, 227–233. MR 2071304, DOI 10.1016/j.topol.2004.02.003
- Shinzo Bannai, A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces, Topology Appl. 202 (2016), 428–439. MR 3464177, DOI 10.1016/j.topol.2016.02.005
- S. Bannai and T. Shirane, Nodal curves with a contact-conic and Zariski pairs, preprint available at arXiv: 1608.03760.
- Shinzo Bannai and Hiro-o Tokunaga, Geometry of bisections of elliptic surfaces and Zariski $N$-plets for conic arrangements, Geom. Dedicata 178 (2015), 219–237. MR 3397492, DOI 10.1007/s10711-015-0054-z
- Alex Degtyarev, On deformations of singular plane sextics, J. Algebraic Geom. 17 (2008), no. 1, 101–135. MR 2357681, DOI 10.1090/S1056-3911-07-00469-9
- Alex Degtyarev, On the Artal-Carmona-Cogolludo construction, J. Knot Theory Ramifications 23 (2014), no. 5, 1450028, 35. MR 3233625, DOI 10.1142/S021821651450028X
- Benoît Guerville-Ballé, An arithmetic Zariski 4-tuple of twelve lines, Geom. Topol. 20 (2016), no. 1, 537–553. MR 3470721, DOI 10.2140/gt.2016.20.537
- B. Guerville-Ballé and J.-B. Meilhan, A linking invariant for algebraic curves, Available at arXiv:1602.04916.
- B. Guerville-Ballé and T. Shirane, Equivalence between splitting number and linking invariant, preprint available at arXiv: 1607.04951.
- Mutsuo Oka, Symmetric plane curves with nodes and cusps, J. Math. Soc. Japan 44 (1992), no. 3, 375–414. MR 1167373, DOI 10.2969/jmsj/04430375
- Ichiro Shimada, Equisingular families of plane curves with many connected components, Vietnam J. Math. 31 (2003), no. 2, 193–205. MR 2011641
- Ichiro Shimada, Lattice Zariski $k$-ples of plane sextic curves and $Z$-splitting curves for double plane sextics, Michigan Math. J. 59 (2010), no. 3, 621–665. MR 2745755, DOI 10.1307/mmj/1291213959
- Hiro-o Tokunaga, Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers, J. Math. Soc. Japan 66 (2014), no. 2, 613–640. MR 3201828, DOI 10.2969/jmsj/06620613
- Oscar Zariski, On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve, Amer. J. Math. 51 (1929), no. 2, 305–328. MR 1506719, DOI 10.2307/2370712
Additional Information
- Taketo Shirane
- Affiliation: National Institute of Technology, Ube College, 2-14-1 Tokiwadai, Ube 755-8555, Yamaguchi Japan
- Email: tshirane@ube-k.ac.jp
- Received by editor(s): February 24, 2016
- Received by editor(s) in revised form: May 12, 2016
- Published electronically: September 15, 2016
- Communicated by: Lev Borisov
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1009-1017
- MSC (2010): Primary 14E20, 14F45, 14H30, 14H50, 14N20, 32S22, 57M12
- DOI: https://doi.org/10.1090/proc/13298
- MathSciNet review: 3589301