# Characterizing the increase of the residual order under blowup in positive characteristic

## TL;DR

This paper investigates the behavior of residual order during blowups in positive characteristic, highlighting circumstances of increase and exploring potential modifications to improve its stability in singularity resolution.

## Contribution

It provides a detailed analysis of when residual order increases under blowup in positive characteristic, aiming to improve its utility in singularity resolution.

## Key findings

- Residual order can increase under blowup in positive characteristic.
- Conditions leading to residual order increase are characterized.
- Potential modifications to residual order are discussed.

## Abstract

In characteristic zero, the residual order constitutes, after the local multiplicity, the second key invariant for the resolution of singularities. It is defined as the order of the coefficient ideal in a local hypersurface of maximal contact, minus the exceptional multiplicities. It does not increase under blowup in permissible centers as long as the local multiplicity remains constant. In positive characteristic, however, the residual order (defined now as the maximum over all smooth local hypersurfaces) may increase under blowup. In the article we analyze in detail the circumstances when this happens. This may help to develop a modification of the residual order which does work in positive characteristic.