A new Alexander-equivalent Zariski pair

Mutsuo Oka

Abstract

     The purpose of this note is to construct an Alexander-equivalent Zariski pair $(D,D')$ of irreducible curves of  degree 8 with  12 cusps. We give a brief recipe for the construction. Consider the moduli space $\cM(cA_2;n)$ plane curves of degree n with c cusps of type $y^2-x^3=0$. As we only consider cuspidal curves in this note, we simply denote $\cM(c;n)$ in stead of  $\cM(cA_2;n)$. It is well-known that  $\cM(3 ;4)$ is irreducible. In fact, its dual is the moduli of plane curves of degree 3 with one node by the Pl\"ucker's formula (see [N]). The fundamental group of the complement  $\bfP^2-C,~C\in \cM(3 ;4)$, is a finite non-abelian group of order 12 ([8,3]).