Global resolution of singularities in $1$-dimensional foliated spaces

TL;DR

This paper proves a resolution of singularities for 1-dimensional singular distributions on analytic manifolds that preserves their Log-Canonical or monomial nature, with applications to parameterized families.

Contribution

It introduces a method to resolve singularities in 1D foliated spaces while maintaining their specific singularity types, extending existing resolution techniques.

Findings

Resolution preserves Log-Canonical and monomial singularities.

Applicable to families of ideal sheaves with parameter space dimension one less than ambient.

Provides a constructive approach for singularity resolution in foliated spaces.

Abstract

Let $M$ be an analytic manifold over $\mathbb{R}$ or $\mathbb{C}$, $\theta$ a $1$-dimensional Log-Canonical (resp. monomial) singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a resolution of singularities for $\mathcal{I}$ that preserves the Log-Canonicity (resp. monomiality) of the singularities of $\theta$. 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.