New Ideas for Resolution of Singularities in Arbitrary Characteristics

TL;DR

This paper introduces a new method based on torus embeddings to replace maximal contact, aiming to develop an effective resolution of singularities applicable in all characteristics.

Contribution

It proposes a substitute for maximal contact using torus embeddings, enabling resolution of singularities in arbitrary characteristics.

Findings

Replaces maximal contact with a torus embedding-based theorem

Provides a foundation for global resolution of singularities in all characteristics

Lays groundwork for future comprehensive resolution theories

Abstract

The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global sense in the forthcoming articles.