Weighted Homogeneous Isolated Singularities

• Weighted homogeneous isolated singularities are hypersurface singularities defined by polynomials with prescribed weights that yield an isolated singular point at the origin.
• Key invariants such as Milnor and Tjurina numbers, along with the Łojasiewicz exponent and Hodge filtration, enable precise classification and deformation analysis.
• Their computable combinatorial structure and connections to syzygy, Bernstein-Sato theory, and smoothing mechanisms make them pivotal in algebraic geometry and mathematical physics.

Weighted homogeneous isolated singularities are hypersurface singularities defined by polynomials whose monomials possess prescribed weights and degrees, such that the singularity at the origin is isolated. These singularities are central objects in singularity theory, Hodge theory, deformation theory, and algebraic geometry due to their rigidity, rich combinatorics, computable invariants (Milnor number, Tjurina number, spectrum), and their role as test cases for broader conjectures and invariants. Their structure interacts deeply with syzygy theory, Bernstein-Sato theory, mixed Hodge modules, deformation and smoothing theory, and classification via numerical semigroups and plumbing graphs.

1. Definition and Characterization

Let $f(x_1,\dots,x_n)\in\C[x_1,\dots,x_n]$ be a polynomial. $f$ is weighted homogeneous of weights $(w_1,\ldots,w_n)\in\Q_{>0}^n$ and degree $d\in\Q_{>0}$ if every monomial $x_1^{a_1}\cdots x_n^{a_n}$ in $f$ satisfies $\sum_{i=1}^n w_i a_i = d$ (Zhang, 2018, Abderrahmane, 2015, Hertling et al., 2010). The hypersurface $Z = \{f=0\}$ has an isolated singularity at the origin if the Jacobian ideal $(\partial_{x_1}f,\ldots,\partial_{x_n}f)$ defines the origin as its only zero (Zhang, 2018, Abderrahmane, 2015).

These singularities admit several equivalent characterizations:

• Saito's criterion: in local analytic coordinates, $g \in (g_{y_1},\dots,g_{y_n})$ defines weighted homogeneity at $0$ (Dimca et al., 2014).
• Combinatorial graphs encoding monomial support; types include chain, cycle, Fermat, and sums thereof (Hertling et al., 2010).
• Existence of analytic weights is a topological invariant for $n\leq 3$; for Arc-analytic (Blow-Nash) equivalence, weights are preserved (Campesato, 2016).

2. Invariants: Milnor Number, Tjurina Number, Łojasiewicz Exponent

The Milnor number $\mu(f)$ is the dimension of the Milnor algebra, $\mu(f) = \dim_\C \C[x_1,\dots,x_n]/(\partial_{x_1}f,\dots,\partial_{x_n}f)$ (Abderrahmane, 2015, Hertling et al., 2010).

The Tjurina number $\tau(f)$ is the dimension of the Tjurina algebra, $\tau(f) = \dim_\C \C[x_1, \dots, x_n]/(f, \partial_{x_1}f, ..., \partial_{x_n}f)$ (Hu et al., 2024). Refined invariants $\tau_k(f)$ stratify the deformation space by jets, giving the "k-th Tjurina numbers" with explicit combinatorial formulas for all $k$ (Hu et al., 2024).

The local Łojasiewicz exponent $\mathcal{L}(f)$ governs the order of vanishing of the gradient: $$\mathcal{L}(f) = \max_{i} \left( \frac{d}{w_i} - 1 \right)$$ for $f$ weighted homogeneous of type $(d; w)$ with $d > 2w_i$ and isolated singularity (Abderrahmane, 2015, Brzostowski, 2014). This exponent is invariant in topologically trivial (Milnor number $\mu$-constant) families, answering a case of Teissier's conjecture (Brzostowski, 2014).

3. Hodge Theory and Bernstein-Sato Invariants

Weighted homogeneous isolated singularities allow explicit computation of deep Hodge-theoretic data:

• The Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-\alpha}$ is explictly given by

$F_p\,\M(f^{1-\alpha}) = \sum_{i=0}^p F_{p-i}\D_X\left(\O_X^{\ge \alpha+i} f^{1-\alpha}\right)$

with the order filtration $F_k\D_X$, and weight filtration $\O_X^{\ge\beta}$ (Zhang, 2018, Jung et al., 2018). The corresponding Hodge ideals $I_k(D)$ satisfy

$I_{k+1}(D) = \sum_{j\,:\,p(v_j)\ge\alpha+k+1} \O_X v_j + \sum_{i=1}^n \partial_{x_i} I_k(D)$

where $v_j$ are Milnor algebra monomials, with generating level given by $$\ell = \lfloor n - \widetilde\alpha_f - \alpha \rfloor$$ (Zhang, 2018).

• The Steenbrink spectrum and the Hodge ideal spectrum coincide for weighted homogeneous isolated singularities. Both are given by

$$Sp(f)(t) = \prod_{i=1}^n \frac{t^{w_i} - t^d}{1 - t^{w_i}}$$

and their "jumps" coincide with the Hodge ideal filtration (Jung et al., 2019, Jung et al., 2018). Additionally, the coincidence $Sp_\text{Hodge}(f) = Sp(f)$ is proved via the Mustaţă–Popa formula.

• The Bernstein-Sato polynomial $b_f(s)$ of $f$ has roots which, for weighted homogeneous isolated singularities, can be characterized by the Hilbert series of the Jacobian algebra, together with duality and Lefschetz properties. For projective hypersurfaces with weighted homogeneous isolated singularities, $R_f = \tfrac{1}{d}(\mathbb{Z}\cap[3,k']) \cup R_Z$, where $R_Z$ are roots from local singularities, and $k' = \max(2d-3, k_\max+3)$ (Saito, 2016). In positive characteristic, the Bernstein–Sato roots are given explicitly in terms of weights and F-thresholds (Tao et al., 2024).

4. Syzygies, Deformations, and Smoothing Theory

Weighted homogeneous isolated singularities have syzygy modules and deformation-theoretic properties that are highly rigid:

• Jacobian syzygies among the partial derivatives $(f_{x_1}, ..., f_{x_n})$ are characterized by the property that all Koszul syzygies whose first component vanishes on the singular locus are the only ones when all singularities are weighted homogeneous (Dimca et al., 2014). Saito’s criterion allows algorithmic and computer algebra recognition of weighted homogeneity.
• Deformation theory is stratified by the k-th Tjurina algebra, and the dimension $\tau_k(f)$ computes the tangent space to the pointed deformation functor $\mathrm{Def}_k(X)$, preserving fat points in the singularity (Hu et al., 2024). For $k$ less than the multiplicity, $\tau_k(f)$ and $\mu_k(f)$ are determined purely by weights; for $k > \mathrm{mult}(f)$, additional gap contributions appear (Hu et al., 2024).
• Rational homology disk smoothings: Only specific cyclic quotient and exceptional weighted homogeneous surface singularities admit $\mathbb{Q}$HD smoothings, characterized by their dual resolution graphs and configurations of rational curves on rational surfaces. The analytic classification is completed, and the uniqueness or multiplicity of smoothing components is combinatorially determined (Wahl, 2020).
• Numerical semigroups: For normal weighted homogeneous surface singularities with rational homology sphere links, the numerical semigroup $S(X,0)$ is representable if and only if it is a quotient of a flat semigroup, tightly connected to the plumbing graph and Seifert invariants (Baja et al., 2023).

5. Classification, Graph Types, and Invariants

The weight systems for weighted homogeneous isolated singularities are classified by combinatorial criteria involving support sets and directed graphs encoding dependencies among variables (Hertling et al., 2010):

• Graph types: Chain, cycle, invertible (fermats, sums thereof) constitute generic configurations yielding isolated singularities. Non-standard support can require additional monomials to achieve isolation.
• Milnor number bounds: For $f$ isolated quasihomogeneous in $n$ variables, $\mu = \prod_{i=1}^n(\frac{1}{w_i} - 1)$. The common denominator $d$ is bounded by $L(n)\mu$ for $L(n)$ the product of the first $n$ primes.
• Rigidity for prime $\mu$: If $\mu$ is prime, the only admissible graph is a unique chain type; thus, the singularity is rigidly determined (Hertling et al., 2010).

6. Generalizations, Topological and Analytic Invariance

Weighted homogeneity and isolatedness admit extensions:

• Non-degenerate Newton boundary germs: The explicit formulas for Hodge filtration and Hodge ideals extend to germs that are convenient and Newton non-degenerate, with Newton filtration substituting the weight filtration (Zhang, 2018).
• Arc-analytic invariance: For singular, non-degenerate convenient weighted homogeneous polynomials, the arc-analytic type rigidly determines the weight vector and degree; weights are analytic invariants for $n\geq 2$ and are topological invariants for $n = 2, 3$ (Campesato, 2016).
• Topological triviality: Weighted homogeneous singularities retain their Łojasiewicz exponent in topologically trivial (μ-constant) families, verifying particular cases of Teissier’s conjecture (Brzostowski, 2014).

7. Connections to Physics and Higher-Dimensional Geometry

Weighted homogeneous singularities underpin much of modern geometry and mathematical physics:

• Gorenstein compound Du Val (cDV) singularities are classified by their singularity types and weights; among weighted cE6, cE7, cE8 only special cases admit crepant resolutions and hence dual to quiver gauge theories, as established via symplectic cohomology, homological mirror symmetry, and toric/topological combinatorics (Fang et al., 2024).
• In type IIB string theory, placing branes at isolated Gorenstein singularities gives SCFTs; crepant resolutions are necessary for weakly coupled quiver gauge theories (Fang et al., 2024).

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Weighted homogeneous isolated singularities thus comprise a canonical class with explicit, highly computable invariants and powerful classification results, serving as rich examples across singularity theory, Hodge theory, combinatorial geometry, deformation theory, and mathematical physics.