Combinatorial Aspects of Classical Resolution of Singularities

TL;DR

This paper explores combinatorial methods for classical resolution of singularities, extending to foliations, vector fields, functions, and varieties, and introduces a combinatorial version of Hironaka's maximal contact theory.

Contribution

It provides a combinatorial framework for resolution of singularities and proves the global existence of maximal contact within this approach.

Findings

Develops a combinatorial version of Hironaka's maximal contact theory.

Establishes global existence of maximal contact in the combinatorial setting.

Applies combinatorial methods to singular foliations, vector fields, functions, and varieties.

Abstract

We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.