Resolving singularities of curves with one toric morphism

Abstract

We give an explicit positive answer, in the case of reduced curve singularities, to a question of B. Teissier about the existence of a toric embedded resolution after reembedding. In the case of a curve singularity $(C,O)$ contained in a non singular surface $S$ such a reembedding may be defined in terms of a sequence of maximal contact curves of the minimal embedded resolution of $C$. We prove that there exists a toric modification, after reembedding, which provides an embedded resolution of $C$. We use properties of the semivaluation space of $S$ at $O$ to describe how the dual graph of the minimal embedded resolution of $C$ may be seen on the local tropicalization of $S$ associated to this reembedding.