Resolution of Corank $1$ Singularities of a Generic Front

We construct a resolution of singularities for wave fronts having only stable singularities of corank $1$. It is based on a transformation that takes a given front to a new front with singularities of the same type in a space of smaller dimension. This transformation is defined by the class $A_{\mu}$ of Legendre singularities. The front and the ambient space obtained by the $A_{\mu}$-transformation inherit topological information on the closure of the manifold of singularities $A_{\mu}$ of the original front. The resolution of every (reducible) singularity of a front is determined by a suitable iteration of $A_{\mu}$-transformations. As a corollary, we obtain new conditions for the coexistence of singularities of generic fronts.