On perturbations of singular complex analytic curves

Suppose $V$ is a singular complex analytic curve inside ${\mathbb{C}}^2$. We investigate when a singular or non-singular complex analytic curve $W$ inside ${\mathbb{C}}^2$ with sufficiently small Hausdorff distance $d_H(V,W)$ from $V$ must intersect $V$. We obtain a sufficient condition on $W$ which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such $W$ that admits at most one non-normal crossing type discriminant point associated with some proper projection. As an application, we prove a special case of the higher-dimensional analog, and also a holomorphic multifunction analog of a result by Lyubich-Peters.