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A note on splitting numbers for Galois covers and $ \pi_1$-equivalent Zariski $ k$-plets


Author: Taketo Shirane
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
Published electronically: September 15, 2016
MathSciNet review: 3589301
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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$.


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Taketo Shirane
Affiliation: National Institute of Technology, Ube College, 2-14-1 Tokiwadai, Ube 755-8555, Yamaguchi Japan
Email: tshirane@ube-k.ac.jp

DOI: https://doi.org/10.1090/proc/13298
Keywords: Zariski pair, $\pi_1$-equivalent Zariski $k$-plet, Galois cover, splitting curve
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
Article copyright: © Copyright 2016 American Mathematical Society