A new strategy for resolution of singularities in the monomial case in positive characteristic

TL;DR

This paper introduces a new strategy and invariant for resolving singularities in the monomial case in positive characteristic, aiming to improve the process especially in higher dimensions.

Contribution

It proposes a novel approach and invariant for Step 2 of the resolution process in positive characteristic, building on and differing from previous methods.

Findings

The new invariant increases and then decreases, ensuring progress in resolution.

The strategy is more aligned with Villamayor's philosophy and potentially more effective in higher dimensions.

The approach is logically independent of Moh and Hauser's classical analysis.

Abstract

According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P. for short) the algorithm is devided into the following two steps: Step 1. Reduction of the general case to the monomial case. Step 2. Solution in the monomial case. While we have established Step 1 in arbitrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy. In dimension 3, we provided an invariant in the previous paper, inspired by the work of Benito-Villamayor, which establishes Step 2. In this paper, we propose a new strategy to approach Step 2, and provide a different invariant in dimension 3 based upon this strategy. The new invariant increases from time to time (the well-known Moh-Hauser jumping phoenomena), while it is then shown to eventually decrease. (The analysis of the jumping phoenomena and eventual decrease is done in the monomial case in our setting, while the classical analysis by Moh or Hauser is done in a different setting without any reference to the monomial case. Therefore, even though we owe most of the ideas to Moh and Hauser, our argument is carried out logically independent of their papers.) Since the old invariant in our previous paper strictly decreases after each transformation, this may look like a step backward rather than forward. However, the construction of the new invariant is more faithful to the original philosophy of Villamayor, and we believe that the new strategy has a better fighting chance in higher dimensions.