Resolution of $1$-foliations singularities on surfaces and threefolds

TL;DR

This paper studies the resolution of singularities for 1-foliations on surfaces and threefolds in positive characteristic, demonstrating that singularities can be fully resolved using tame stacks or simplified to multiplicative types on varieties.

Contribution

It introduces methods to resolve or simplify 1-foliation singularities on low-dimensional varieties in positive characteristic, utilizing tame stacks and reduction techniques.

Findings

Singularities can be fully resolved with tame stacks.

Singularities can be simplified to multiplicative types.

Resolution methods are effective in positive characteristic.

Abstract

We consider resolution of singularities for $1$-foliations on varieties of dimension at most three in positive characteristic. We prove that such singularities can be completely resolved if we allow tame regular Deligne--Mumford stacks as underlying spaces. If one restricts to underlying varieties, we show that $1$-foliations singularities can be simplified into multiplicative ones.