Alternative invariants for the embedded resolution of purely inseparable surface singularities

Abstract

We propose two local invariants for the inductive proof of the embedded resolution of purely inseparable surface singularities of order equal to the characteristic. The invariants are built on a detailed analysis of the so called “kangaroo phenomenon” in positive characteristic. They thus measure accurately the algebraic complexity of an equation defining a surface singularity in characteristic $p$. As the invariants are shown to drop after each blowup, induction applies.