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The Nakai Conjecture for isolated hypersurface singularities of modality $\le 2$

Rui Li, Zida Xiao, Huaiqing Zuo

TL;DR

The paper proves Nakai Conjecture in the local analytic setting for isolated hypersurface singularities of modality at most two by a detailed, case-by-case analysis across Arnol'd's unimodal and bimodal classifications. It develops and uses high-order derivation theory, Hessian/cofactor techniques, and Jacobian-based descent to Artinian quotients, constructing explicit liftings via symmetric matrices $(\beta_{ij})$ to certify the needed $der^2(A) \neq Der^2(A)$. The principal contribution is a complete verification of Nakai's statement for all such singularities, strengthening links to the Zariski–Lipman conjecture and extending known results from homogeneous/fewnomial cases to the local analytic setting. The results provide a comprehensive framework and computational toolkit for handling high-order differential operators in singularity theory with concrete, verifiable steps.

Abstract

The well-known Nakai Conjecture concerns a very natural question: For an algebra of finite type over a characteristic zero field, if the ring of its differential operators is generated by the first order derivations, is the algebra regular? And it is natural to extend the Nakai Conjecture to local domains, in this paper, we verify it for isolated hypersurface singularities of modality $\le 2$, this extends the existing works.

The Nakai Conjecture for isolated hypersurface singularities of modality $\le 2$

TL;DR

The paper proves Nakai Conjecture in the local analytic setting for isolated hypersurface singularities of modality at most two by a detailed, case-by-case analysis across Arnol'd's unimodal and bimodal classifications. It develops and uses high-order derivation theory, Hessian/cofactor techniques, and Jacobian-based descent to Artinian quotients, constructing explicit liftings via symmetric matrices to certify the needed . The principal contribution is a complete verification of Nakai's statement for all such singularities, strengthening links to the Zariski–Lipman conjecture and extending known results from homogeneous/fewnomial cases to the local analytic setting. The results provide a comprehensive framework and computational toolkit for handling high-order differential operators in singularity theory with concrete, verifiable steps.

Abstract

The well-known Nakai Conjecture concerns a very natural question: For an algebra of finite type over a characteristic zero field, if the ring of its differential operators is generated by the first order derivations, is the algebra regular? And it is natural to extend the Nakai Conjecture to local domains, in this paper, we verify it for isolated hypersurface singularities of modality , this extends the existing works.

Paper Structure

This paper contains 9 sections, 12 theorems, 452 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The unimodal isolated hypersurface singularity case
  4. Cases of simple elliptic singularities
  5. Cases of exceptional families of unimodal isolated hypersurface singularities
  6. The bimodal isolated hypersurfce singularity case
  7. Cases of corank 2 bimodal singularities
  8. Cases of corank 3 bimodal singularities
  9. Cases of exceptional families of bimodal isolated hypersurface singularities

Key Result

Theorem 2.1

Let $P=k\left[x_1, x_2, \cdots, x_n\right]$ or $k\{x_1,\cdots,x_n\}$, $I$ be a proper ideal of $P$ and $A=P/I$. Then

Theorems & Definitions (26)

  • Remark 1.1
  • Conjecture 1.2: Nakai BLLS Nakai
  • Conjecture 1.3: Zariski-Lipman Lipman
  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Theorem 2.7
  • ...and 16 more