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Fundamental groups of join‐type sextics via dessins d'enfants
Author(s) -
Eyral Christophe,
Oka Mutsuo
Publication year - 2013
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pds085
Subject(s) - mathematics , join (topology) , type (biology) , conjecture , degree (music) , combinatorics , group (periodic table) , projective plane , plane (geometry) , plane curve , pure mathematics , geometry , ecology , biology , chemistry , physics , organic chemistry , acoustics , correlation
An ℝ‐join‐type curve is a plane complex projective curve defined by an equation of the form∏ j = 1 l( Y − β j Z )ν j= c ⋅ ∏ i = 1 m( X − α i Z )λ i,where∑ j = 1 lν j= ∑ i = 1 mλ iis the degree of the curve, c ∈ ℝ\ {0}, and β 1 , …, β l (respectively, α 1 , …, α m ) mutually distinct real numbers. In this paper, we determine the fundamental group π 1 (ℙ 2 \ C ) for every ℝ‐join‐type curve C of degree 6. For higher degrees and non‐real coefficients, we propose a conjecture.