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Spectrum, Tjurina spectrum, and Hertling conjecture for singularities of modality $\leq 3$

Quan Shi, Yang Wang, Huaiqing Zuo

TL;DR

The work analyzes spectrum invariants of isolated hypersurface singularities, proving Hertling's variance-range inequality for modality $3$ and introducing the Tjurina spectrum $\mathrm{Sp}^{\tau}$ arising from Hodge ideals. It proves a Generalized Hertling Conjecture for $\mathrm{Sp}^{\tau}$ and establishes its validity for all singularities of modality $\le 3$, while providing a practical $V$-filtration–based computation framework to determine $\mathrm{Sp}^{\tau}$ and its relation to the classical spectrum. The paper also gives a comprehensive computation of spectra for non-quasihomogeneous trimodal families, discusses four Newton-degenerate exceptions, and proves the conjecture for unimodal, bimodal, and trimodal cases (contingent on a conjecture for the exceptional four). Taken together, the results connect Hodge-ideal–driven invariants with Steenbrink spectra across low-modality singularities, offering both theoretical insights and explicit computational tools for singularity classification and deformation theory.

Abstract

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells the relation of range and variance of exponents i.e. elements of spectrum. For trimodal singularities, we compute their spectra and verify Hertling conjecture for them. Jung, Kim, Saito and Yoon recently defined Tjurina spectrum, stemming from Hodge ideals. This set of numerical invariants is a subset of spectrum in Steenbrink's sense. We give an estimation of exponents not in Tjurina spectrum and propose a similar Generalized Hertling Conjecture for Tjurina Spectrum. Moreover, we prove the conjecture for singularities of modality $\leq 3$.

Spectrum, Tjurina spectrum, and Hertling conjecture for singularities of modality $\leq 3$

TL;DR

The work analyzes spectrum invariants of isolated hypersurface singularities, proving Hertling's variance-range inequality for modality and introducing the Tjurina spectrum arising from Hodge ideals. It proves a Generalized Hertling Conjecture for and establishes its validity for all singularities of modality , while providing a practical -filtration–based computation framework to determine and its relation to the classical spectrum. The paper also gives a comprehensive computation of spectra for non-quasihomogeneous trimodal families, discusses four Newton-degenerate exceptions, and proves the conjecture for unimodal, bimodal, and trimodal cases (contingent on a conjecture for the exceptional four). Taken together, the results connect Hodge-ideal–driven invariants with Steenbrink spectra across low-modality singularities, offering both theoretical insights and explicit computational tools for singularity classification and deformation theory.

Abstract

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells the relation of range and variance of exponents i.e. elements of spectrum. For trimodal singularities, we compute their spectra and verify Hertling conjecture for them. Jung, Kim, Saito and Yoon recently defined Tjurina spectrum, stemming from Hodge ideals. This set of numerical invariants is a subset of spectrum in Steenbrink's sense. We give an estimation of exponents not in Tjurina spectrum and propose a similar Generalized Hertling Conjecture for Tjurina Spectrum. Moreover, we prove the conjecture for singularities of modality .
Paper Structure (16 sections, 37 theorems, 394 equations, 8 tables, 1 algorithm)

This paper contains 16 sections, 37 theorems, 394 equations, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rings of Local Analytic Germs, Hypersurface Singularities and Modalities
  4. Spectrum Numbers and Hertling Conjecture
  5. Hodge Ideal and Tjurina Spectrum
  6. Brieskorn Lattice and $V$-filtration
  7. Newton Filtration
  8. Spectrum and Hertling Conjecture for Trimodal Singularities
  9. Spectrum of Trimodal Singularities
  10. Hertling Conjecture for Singularities of Modality $3$
  11. Propoerties and Generalized Hertling Conjecture for Tjurina Spectrum
  12. Computation for Tjurina Spectrum
  13. Maximal Exponent and Generalized Hertling Conjecture for Tjurina Spectrum
  14. Generalized Hertling Conjecture for Tjurina Spectrum for Unimodal Singularities
  15. Generalized Hertling Conjecture for Tjurina Spectrum for Bimodal Singularities
  16. ...and 1 more sections

Key Result

Lemma 2.1

$U\subseteq \mathbb C^n$ is an open neighborhood of $0$. Let $f:U\to \mathbb C$ be holomorphic, then the following are equivalent: (a) $0$ is an isolated critical point of $f$. (b) $\mu(f,0) < \infty$. (c) $0$ is an isolated singularity of $f^{-1}(f(0)) = V(f-f(0))$. (d) $\tau(f-f(0),0) < \infty$.

Theorems & Definitions (61)

  • Conjecture 1.1: Hertling
  • Conjecture 1.2
  • Conjecture 1.3
  • Lemma 2.1: MR2290112, Lemma 2.3
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4: Finite Determinancy Theorem, MR2290112
  • Corollary 2.5
  • Theorem 2.6: MR2290112, Corollary 2.18
  • Definition 2.7
  • ...and 51 more