Algebraic geometry of center curves in the moduli space of the cubic maps (Q1340346)

The authors consider the family of cubic maps \(f(x) = x^3 - 3Ax \pm \sqrt B\) parametrized by \((1 : A : B) \in \mathbb{P}^2_\mathbb{C}\). For the family, they define special curves \(BC_p : (f^p (\sqrt A) - (- \sqrt A))\) \((f^p (-\sqrt A) - \sqrt A) = 0\) and similar ones \(CD_p\), that they call center curves, as in their previous paper [Proc. Japan Acad., Ser. A 69, No. 6, 179-184 (1993; Zbl 0804.58028)]. \textit{J. Milnor} considered the moduli space of real cubics, and defined the concept of centers for the cubics. In this paper, the authors show that these center curves are singular at \((0 : 0 : 1)\) and for \(p = 1,2\), these are irreducible. Moreover, they conjecture that every center curve is singular and has cuspidal singularity only at \((0 : 0 : 1)\).