Cycles of Singularities appearing in the Resolution Problem in positive Characteristic

TL;DR

This paper constructs a specific hypersurface singularity in positive characteristic that exhibits a cyclical pattern under point blowups, challenging previous assumptions about the stability of residual order during resolution.

Contribution

It introduces a new example of a singularity cycle in positive characteristic, disproving a claimed theorem about residual order stability in resolution processes.

Findings

Constructs a hypersurface singularity with a cyclical pattern under blowups.

Demonstrates that residual order can increase infinitely through iteration.

Disproves Moh's theorem on residual order stability in positive characteristic.

Abstract

We present a hypersurface singularity in positive characteristic which is defined by a purely inseparable power series, and a sequence of point blowups so that, after applying the blowups to the singularity, the same type of singularity reappears after the last blowup, with just certain exponents of the defining power series shifted upwards. The construction hence yields a cycle. Iterating this cycle leads to an infinite increase of the residual order of the defining power series. This disproves a theorem claimed by Moh about the stability of the residual order under sequences of blowups. It is not a counter-example to the resolution in positive characteristic since larger centers are also permissible and prevent the phenomenon from happening.