Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

Muhammad Ahsan Binyamin, Gert-Martin Greuel, Khawar Mehmood, Gerhard Pfister

TL;DR

This work delivers a complete classification of unimodal parametric plane curve singularities over algebraically closed fields of positive characteristic. It develops a characteristic-free framework built around ${\\mathcal{A}}$-modality, finite determinacy, and the semicontinuity of both the modality and the semigroup, culminating in explicit normal-form lists across characteristic ranges: $p\ge 5$, $p=2$, and $p=3$. A key innovation is the extension of Zariski’s short parameterization to arbitrary characteristic, combined with modular/full families to certify unimodality, and selective use of SINGULAR computations for sporadic small-characteristic cases. The paper also furnishes new, purely theoretical proofs of Hefez–Hernandes results in large characteristic, showing that their characteristic-0 classification remains valid for $p>c(\Gamma)$ and, consequently, for large enough characteristic. Overall, the results provide a comprehensive, algorithmically verifiable catalog of unimodal plane branches in positive characteristic, with direct connections to the characteristic-0 classification.

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.

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

TL;DR

This work delivers a complete classification of unimodal parametric plane curve singularities over algebraically closed fields of positive characteristic. It develops a characteristic-free framework built around -modality, finite determinacy, and the semicontinuity of both the modality and the semigroup, culminating in explicit normal-form lists across characteristic ranges: , , and . A key innovation is the extension of Zariski’s short parameterization to arbitrary characteristic, combined with modular/full families to certify unimodality, and selective use of SINGULAR computations for sporadic small-characteristic cases. The paper also furnishes new, purely theoretical proofs of Hefez–Hernandes results in large characteristic, showing that their characteristic-0 classification remains valid for and, consequently, for large enough characteristic. Overall, the results provide a comprehensive, algorithmically verifiable catalog of unimodal plane branches in positive characteristic, with direct connections to the characteristic-0 classification.

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.
Paper Structure (19 sections, 36 theorems, 138 equations, 5 tables)

This paper contains 19 sections, 36 theorems, 138 equations, 5 tables.

Table of Contents

  1. Introduction and Preliminaries
  2. Semicontinuity of the Modality
  3. Methods of proof and overview of results
  4. Classification in characteristic $\geq 5$
  5. Multiplicity 3
  6. Multiplicity 4
  7. parameterizations with semigroup $\langle 4,6,2k+7\rangle$, $k\geq 1$.
  8. parameterizations with semigroup $\langle 4, 9\rangle$.
  9. parameterizations with semigroup $\langle 4,10,2k+11\rangle$.
  10. parameterizations with semigroup $\langle 4, 11\rangle$.
  11. Multiplicity 5
  12. Parameterizations with semigroup $\langle 5, 6\rangle$
  13. parameterizations with semigroup $\langle 5, 7\rangle$.
  14. parameterizations with semigroup $\langle 5, 8\rangle$
  15. Classification in characteristic $2,3$
  16. ...and 4 more sections

Key Result

Proposition 2.2

The parameterization $\psi$ is $(4\cdot\delta(\psi)-1)$-determined.

Theorems & Definitions (76)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1: ${\mathcal{A}}$-determinacy
  • Proposition 2.2: GP, Proposition 11
  • Definition 2.3: Family of parameterizations
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6: Jet space family
  • Proposition 2.7
  • ...and 66 more