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On the $k$-th Tjurina number of weighted homogeneous singularities

Chuangqiang Hu, Stephen S. -T. Yau, Huaiqing Zuo

TL;DR

The paper extends the theory of singularities by introducing and computing the full family of k-th Tjurina numbers for isolated weighted homogeneous hypersurface singularities. It develops a new Koszul-type complex to resolve the graded Jacobian filtration, enabling explicit formulas for the Hilbert-Poincaré series $\mathbb{A}_f(t)$ and $\mathbb{M}_f(t)$ that encode $\tau_k$ and $\mu_k$ for all $k\ge0$, and it relates deformation theory to these algebras through a tangent-space description of $k$-pointed deformations. A key outcome is the explicit computation of these invariants for all weighted homogeneous singularities in three variables, including a complete treatment of the seven canonical 3-variable types and a detailed analysis of the Case $f=f^{(5)}$ with its gap structure. The results generalize Milnor-Orlik-type formulas to the $k$-th level, linking weights, multiplicities, and filtrations to deformation theory, and they provide a computable framework with direct applications to the study of three-dimensional weighted homogeneous singularities.

Abstract

Let $ (X,0) $ denote an isolated singularity defined by a weighted homogeneous polynomial $ f $. Let $ \mathcal{O}$ be the local algebra of holomorphic function germs at the origin, with the maximal ideal $m $. We study the $k$-th Tjurina algebra, defined by $ A_k(f): = \mathcal{O} / \left( f , m^k J(f) \right) $, where $J(f)$ denotes the Jacobian ideal of $ f $. The zeroth Tjurina algebra is well known to represent the tangent space of the base space of the semi-universal deformation of $(X, 0)$. Motivated by this observation, we explore the deformation of $(X,0)$ with respect to a fixed $k$-residue point. We show that the tangent space of the corresponding deformation functor is a subspace of the $k$-th Tjurina algebra. Explicit calculation of the $k$-th Tjurina numbers, which correspond to the dimensions of the $k$-th Tjurina algebras, plays a crucial role in understanding these deformations. According to the results of Milnor and Orlik, the zeroth Tjurina number can be expressed explicitly in terms of the weights of the variables in $f$. However, we observe that for values of $k$ exceeding the multiplicity of $X$, the $k$-th Tjurina number becomes more intricate and is not solely determined by the weights of the variables. In this paper, we introduce a novel complex derived from the classical Koszul complex and obtain a computable formula for the $k$-th Tjurina numbers for all $ k \geqslant 0 $. As an application, we calculate the $k$-th Tjurina numbers for all weighted homogeneous singularities in three variables.

On the $k$-th Tjurina number of weighted homogeneous singularities

TL;DR

The paper extends the theory of singularities by introducing and computing the full family of k-th Tjurina numbers for isolated weighted homogeneous hypersurface singularities. It develops a new Koszul-type complex to resolve the graded Jacobian filtration, enabling explicit formulas for the Hilbert-Poincaré series and that encode and for all , and it relates deformation theory to these algebras through a tangent-space description of -pointed deformations. A key outcome is the explicit computation of these invariants for all weighted homogeneous singularities in three variables, including a complete treatment of the seven canonical 3-variable types and a detailed analysis of the Case with its gap structure. The results generalize Milnor-Orlik-type formulas to the -th level, linking weights, multiplicities, and filtrations to deformation theory, and they provide a computable framework with direct applications to the study of three-dimensional weighted homogeneous singularities.

Abstract

Let denote an isolated singularity defined by a weighted homogeneous polynomial . Let be the local algebra of holomorphic function germs at the origin, with the maximal ideal . We study the -th Tjurina algebra, defined by , where denotes the Jacobian ideal of . The zeroth Tjurina algebra is well known to represent the tangent space of the base space of the semi-universal deformation of . Motivated by this observation, we explore the deformation of with respect to a fixed -residue point. We show that the tangent space of the corresponding deformation functor is a subspace of the -th Tjurina algebra. Explicit calculation of the -th Tjurina numbers, which correspond to the dimensions of the -th Tjurina algebras, plays a crucial role in understanding these deformations. According to the results of Milnor and Orlik, the zeroth Tjurina number can be expressed explicitly in terms of the weights of the variables in . However, we observe that for values of exceeding the multiplicity of , the -th Tjurina number becomes more intricate and is not solely determined by the weights of the variables. In this paper, we introduce a novel complex derived from the classical Koszul complex and obtain a computable formula for the -th Tjurina numbers for all . As an application, we calculate the -th Tjurina numbers for all weighted homogeneous singularities in three variables.
Paper Structure (28 sections, 31 theorems, 251 equations)

This paper contains 28 sections, 31 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Geometric Perspective of $k$-th Tjurina Numbers
  3. Deformation of Singularity
  4. Pointed Deformations
  5. Deformation Functor
  6. Connections of Hilbert-Poincaré Series $\mathbb{M}(t) , \mathbb{A}(t), \mathbb{J}(t) , \mathbb{K}(t)$
  7. The Relation between $\tau_k$ and $\mu_k$
  8. Graded Glgebra of $m$-Filtration of Jacobian ideal
  9. Representation of $\operatorname{gr} J(f)$
  10. Representation of $K (f)$
  11. New Koszul Type Complex
  12. Filtration of $K^{(r)}$
  13. Filtration of $Z^{(r)}$
  14. Main Results
  15. Bigraded Structure
  16. ...and 13 more sections

Key Result

Theorem 1

Let $f (x_1, \cdots, x_n )$ be a weighted homogeneous polynomial of type $(w_1,\ldots, w_n)$ along with a total weight $W$, and under the assumption that $V(f)$ represents an isolated singularity at the origin. In this context, the Milnor (and equivalently, the Tjurina) number is given by the expres

Theorems & Definitions (63)

  • Theorem 1: Milnor-Orlik
  • Definition 2
  • Theorem 3
  • proof
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Definition 8
  • Lemma 9
  • ...and 53 more