On Zariski multiplicity conjecture for quasihomogeneous surfaces with non-isolated singularities

Abstract: We consider a finitely determined map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. In this case, the double point curve $D(f)$ plays a fundamental role in studying the topology of the image of $f$. When $f$ is quasihomogeneous and has corank $1$, we present a characterization of the fold components of the double point curve $D(f)$. In our setting, we also consider Zariski multiplicity question for a pair of germs of (parametrized) quasihomogeneous surfaces $(\textbf{X},0)$ and $(\textbf{Y},0)$ in $(\mathbb{C}^3,0)$ with one dimensional singular set. Finally, we characterize Whitney equisingularity of an unfolding $\mathcal{F} =(f_t,t)$ of $f$ through the constancy of the Milnor number of a single plane curve in the source. This gives in some sense an answer to a question by Ruas in $1994$.