Equisingularity of families of map germs from the plane to 3-space

In this work, we consider a finitely determined map germ $f$ from $({\mathbb{C}}^2,0)$ to $({\mathbb{C}}^3,0)$. We characterize the Whitney equisingularity of an unfolding $F=(f_t,t)$ of $f$ through the constancy of a single invariant in the source. Namely, the Milnor number of the curve $W_t(f)=D(f_t)\cup f_t^{-1}(\gamma )$, where $D(f)$ denotes the double point curve of $f_t$. This gives an answer to a question by Ruas in 1994.