Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

TL;DR

This paper provides a complete classification of unimodal plane curve singularities over algebraically closed fields of positive characteristic, extending previous results from characteristic zero and employing characteristic-independent methods.

Contribution

It introduces a new approach for classifying unimodal plane branches in positive characteristic, applicable where previous methods fail, and proves the semicontinuity of key invariants.

Findings

Classification holds in large characteristic

Methods are characteristic-independent

Semicontinuity of semigroup and modality proven

Abstract

In 2011, Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a certain equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by constructing lists of normal forms. The goal of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes are not applicable in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with SINGULAR. Our methods are characteristic-independent and provide a different proof for the classification in characteristic 0, showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and the modality, which we prove.