A short note on Hauser's Kangaroo phenomena and weak maximal contact in higher dimensions

TL;DR

This paper investigates the failure of maximal contact in resolution of singularities in positive characteristic, aiming to understand and find alternatives by studying specific examples where characteristic zero methods fail.

Contribution

It provides an example-driven analysis of Hauser's Kangaroo phenomena and explores potential replacements for maximal contact in higher dimensions.

Findings

Identification of cases where maximal contact fails in positive characteristic.

Insights into Hauser's Kangaroo phenomena in higher dimensions.

Proposals for alternative approaches to resolution of singularities.

Abstract

Currently there are several approaches to resolution of singularities in positive characteristic all of which have hit some obstruction. One natural idea is to try to construct new meaningful examples at this point to gain a wider range of experience. To produce such examples we mimic the characteristic zero approach and focus on cases where it fails. In particular, this short note deals with an example-driven study of failure of maximal contact and the search for an appropriate replacement.