Global resolution of singularities subordinated to a 1-dimensional foliation

Abstract

Let M be an analytic manifold over R or C, θ a 1-dimensional Log-Canonical (resp. monomial) singular distribution and I a coherent ideal sheaf defined on M. We prove the existence of a resolution of singularities for I that preserves the Log-Canonicity (resp. monomiality) of the singularities of θ. Furthermore, we apply this result to provide a resolution of a family of ideal sheaves when the dimension of the parameter space is equal to the dimension of the ambient space minus one.