Fundamental groups of join‐type sextics via dessins d'enfants

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.