Diffeomorphisms, Isotopoies, and Braid Monodromy Factorizations of Plane Cuspidal Curves

Abstract

We prove that there is an infinite sequence of pairs of plane cuspidal curves $C_{m,1}$ and $C_{m,2}$, such that the pairs $(\mathbb{C}{P}^2,C_{m,1})$ and $(\mathbb{C}{P}^2,C_{m,2})$ are diffeomorphic, but $C_{m,1}$ and $C_{m,2}$ have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of "Dif=Def" and "Dif=Iso" problems for plane irreducible cuspidal curves. In our examples, $C_{m,1}$ and $C_{m,2}$ are complex conjugated.

Principal changement concerns the calculation of the number of double points and cups