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Tjurina Number Jumps and Unimodal Hypersurface Singularities in Positive Characteristic

Hongrui Ma, Aoyu Ying, Huaiqing Zuo

TL;DR

The work addresses unimodal hypersurface singularities in positive characteristic and develops stronger modality bounds linked to Tjurina jumps. By generalizing the modality framework to connect orbit dimensions with extended Tjurina algebras and employing Newton-diagram and complete-transversal methods, the authors obtain a sharp, characteristic-dependent classification for $p>3$. A key phenomenon is that sudden jumps in the extended Tjurina number can force higher modality, distinguishing positive characteristic behavior from the complex case. The paper delivers a finite, explicit classification of unimodal isolated hypersurface singularities under contact equivalence in characteristic $p>3$ and provides a practical deformation-theoretic criterion to verify modality, offering tools for future work in small characteristic as well.

Abstract

This paper generalizes existing methods to derive stronger bounds on the modality of hypersurface singularities. Our results demonstrate that each sudden jump in the Tjurina number necessarily increases the modality. Furthermore, we provide a full classification of unimodal isolated hypersurface singularities in characteristic p > 3 under contact equivalence. Keywords. isolated singularity, classification, modality, positive characteristic.

Tjurina Number Jumps and Unimodal Hypersurface Singularities in Positive Characteristic

TL;DR

The work addresses unimodal hypersurface singularities in positive characteristic and develops stronger modality bounds linked to Tjurina jumps. By generalizing the modality framework to connect orbit dimensions with extended Tjurina algebras and employing Newton-diagram and complete-transversal methods, the authors obtain a sharp, characteristic-dependent classification for . A key phenomenon is that sudden jumps in the extended Tjurina number can force higher modality, distinguishing positive characteristic behavior from the complex case. The paper delivers a finite, explicit classification of unimodal isolated hypersurface singularities under contact equivalence in characteristic and provides a practical deformation-theoretic criterion to verify modality, offering tools for future work in small characteristic as well.

Abstract

This paper generalizes existing methods to derive stronger bounds on the modality of hypersurface singularities. Our results demonstrate that each sudden jump in the Tjurina number necessarily increases the modality. Furthermore, we provide a full classification of unimodal isolated hypersurface singularities in characteristic p > 3 under contact equivalence. Keywords. isolated singularity, classification, modality, positive characteristic.

Paper Structure

This paper contains 9 sections, 40 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contact equivalence and modality
  3. Classification methods
  4. A new criterion of modality of hypersurface singularity
  5. The classification in characteristic $p >3$
  6. Unimodal hypersurface singularities in $K[[x,y]]$
  7. Unimodal hypersurface singularities in $K[[x_1,\dots,x_n]]$ with order 2
  8. Unimodal hypersurface singularities in $K[[x,y,z]]$ with order 3
  9. Check the modality

Key Result

Theorem 1.1

Assume $X$ is irreducible, for every $x \in X$, let $G\text{-modality}(x)$ be the modality of $x$ under $G$ (see Definition def-mod). Then

Figures (1)

  • Figure 1: The Newton diagram of $x^3+xy^3+y^5+yz^2+z^3$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1: finitedeter
  • Remark 3.2
  • Example 3.3
  • ...and 60 more