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Classification of unimodal isolated complete intersection singularities in positive characteristic

Hongrui Ma, Stephen S. -T. Yau, Huaiqing Zuo

TL;DR

The paper addresses the problem of classifying unimodal isolated complete intersection singularities (ICIS) under contact equivalence across arbitrary characteristic. It extends the complete transversal method to positive characteristic, develops a robust framework based on tangent image and weighted filtrations, and provides explicit classifications for order $2$ and order $3$ ICIS, including characteristic $2$ and characteristic $3$ cases. The main contributions are the construction of a general positive-characteristic classification pipeline and the detailed tabulated unimodal forms (e.g., tables ord3, ord3-char2, ord3-char3) that mirror the characteristic-zero results while revealing characteristic-specific phenomena and conjectural refinements. The work offers new tools for singularity classification in algebraic geometry and can inform future investigations into semi-quasi homogeneous ICIS and related classification problems, with potential extensions to higher modalities and dimensions.

Abstract

In this paper we classify the unimodal isolated complete intersection singularities in arbitrary characteristic under contact equivalence. The classification over $\mathbb{C}$ has already done by A. Dimca and C.G. Gibson. We continue and generalize their work. To complete the classification, we generalized the complete transversal method into positive characteristic field, which is also useful in many other classification problem.

Classification of unimodal isolated complete intersection singularities in positive characteristic

TL;DR

The paper addresses the problem of classifying unimodal isolated complete intersection singularities (ICIS) under contact equivalence across arbitrary characteristic. It extends the complete transversal method to positive characteristic, develops a robust framework based on tangent image and weighted filtrations, and provides explicit classifications for order and order ICIS, including characteristic and characteristic cases. The main contributions are the construction of a general positive-characteristic classification pipeline and the detailed tabulated unimodal forms (e.g., tables ord3, ord3-char2, ord3-char3) that mirror the characteristic-zero results while revealing characteristic-specific phenomena and conjectural refinements. The work offers new tools for singularity classification in algebraic geometry and can inform future investigations into semi-quasi homogeneous ICIS and related classification problems, with potential extensions to higher modalities and dimensions.

Abstract

In this paper we classify the unimodal isolated complete intersection singularities in arbitrary characteristic under contact equivalence. The classification over has already done by A. Dimca and C.G. Gibson. We continue and generalize their work. To complete the classification, we generalized the complete transversal method into positive characteristic field, which is also useful in many other classification problem.

Paper Structure

This paper contains 19 sections, 55 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Basic settings
  3. Basic concepts
  4. Tangent image and finite determinacy
  5. Complete transversal
  6. Complete transversal and homogeneous filtrations
  7. Modality
  8. The classification of order 2
  9. The classification of order 2 in $charF= 2$
  10. The classification of order 3
  11. The classification of 3-jet
  12. The classification of unimodal
  13. The classification of order 3 when $char F=2$
  14. The classification of 3-jet
  15. The classification of unimodal
  16. ...and 4 more sections

Key Result

Proposition 2.5

The tangent image is identified with the submodule And the tangent image at $f$ to the orbit $\mathcal{K}f$ is the submodule where $\langle f_1,\dots,f_m\rangle$ is regarded as an ideal of $R$, and $\langle \frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n} \rangle$ is regarded as an ideal of $R^m$.

Theorems & Definitions (106)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • Remark 2.9
  • ...and 96 more