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 is an important invariant for isolated hypersurface singularities. This invariant is first defined by Steenbrink (see [Ste77b] or [Str20]). For a holomorphic function germ f at 0 ∈ C n+1 that defines an isolated singularity, the spectrum of f is µ rational numbers sp(f ) = {α 1 ≤ α 2 ≤ ... ≤ α µ } (called exponents or spectrum numbers) or a polynomial Sp(f ) = µ i=1 t α i , where µ = µ(f ) is the Milnor number of f . In Steenbrink's original definition, those rational numbers come from the mixed Hodge structure of Milnor fibration. There are many beautiful properties of spectrum, for example, symmetry and invariant under µ-constant deformation. We review this part in Subsection 2.2.
The computation for spectrum is largely credited to M. Saito’s theorem that when f is Newton non-degenerated, then its exponents are exactly the jumping numbers of Newton filtration on Ω n+1 f (see [Sai88]). We review this part in Subsection 2.5 and give a detailed explanation on computation.
In 2000, Hertling proposed a well-known conjecture when studying F -manifolds. It says the variance of all spectrum numbers are no greater than a twelfth of their range (see [Her01]) i.e.
Conjecture 1.1 (Hertling). Let f ∈ O n+1 be a germ which defines an isolated singularity at the origin, with Milnor number µ. Suppose α 1 ≤ … ≤ α µ are spectrum numbers of f , then the following inequality holds:
The 25-year-old problem has witnessed some progresses (see Subsection 2.2). In this paper, we first push it a step forward, proving the conjecture for trimodal singularities.
Theorem A (Theorem 3.8). Hertling Conjecture holds for singularities of modality 3.
Most recently, S. Jung, I. Kim, M. Saito, and Y. Yoon defined Hodge ideal spectrum and Tjurina spectrum for isolated singularities (see [JKSY22]). Those new spectra come from Hodge ideals and are highly related to the original object. It is rather hard to describe Hodge ideals in a few lines. Roughly speaking, let Z = div(f ) be principal divisor defined by f , then Hodge ideal I k (αZ) is an ideal of O n+1 indexed by k ∈ N and α ∈ Q ∩ (0, 1]. Hodge ideals induce a filtration on O n+1 by V β HI := α+p≥β I p (αZ).
Naturally, we have induced filtrations on Milnor algebra O n+1 /( ∂f ∂x 0 , …, ∂f ∂xn ) and Tjurina algebra O n+1 /(f, ∂f ∂x 0 , …, ∂f ∂xn ) by canonical surjections. We mainly focus on the filtration V • HI on Tjurina algebra. The Poincaré polynomial with respsect to this filtered vector space is called Tjurina spectrum, denoted as Sp τ (f ). Or equivalently, written as τ rational numbers sp τ (f ) = {β 1 ≤ β 2 ≤ … ≤ β τ }, where τ = τ (f ) is the Tjurina number of f . We call β 1 , …, β τ exponents of Tjurina spectrum. We will review in Subsection 2.4 that Tjurina spectrum is a subset of spectrum.
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The study of Tjurina spectrum is a brand new direction in mathematics, which may play a significant role in understanding of modern D-module theory, especially Hodge ideal and mixed Hodge module.
Besides, to obtain data for concrete examples, we propose a method and a code to compute the Tjurina spectrum through V -filtration, and complete the computation for singularities of modality ≤ 3 except for four cases (see Subsection 4.1). Results of the four cases are left unkown since these cases are Newton degenerate, and so far there is not yet an effective method to compute V -filtration for Newton degenerate singularities.
The study of Tjurina spectrum is equivalent to the estimation of spectrum numbers not in Tjurina spectrum. We hence provide a way to estimate them. The following is the estimation.
Theorem B (Theorem 4.3). Given an isolated hypersurface (V (f ), 0), Let µ and τ be its Milnor and Tjurina number respectively. Let d = µ -τ , α 1 ≤ α 2 ≤ … ≤ α µ be all spectrum numbers, α j 1 < … < α jt be all jumpings i.e. α j i < α j i +1 and h l = Gr
V Ω n+1 f = j l -j l-1 (j 0 = 0) be the multiplicity of α j l . Besides, let α i 1 ≤ α i 2 ≤ … ≤ α i d be all spectrum numbers not in Tjurina spectrum.
For each 1 ≤ k ≤ d, let r k be the maximal integer such that #{j | α j ≤ α r k } ≤ k, then α i k ≥ α jr k + 1. In particular, we have
In the first version of this paper, we conjectured the greatest spectrum number is not in the Tjurina spectrum and proved it for normal forms of singularities of modality ≤ 3 (see Conjecture 1.2 below). Recently, Jung, Kim, Saito, and Yoon gave a proof (see Subsection 4.2). Conjecture 1.2. Suppose f ∈ O n+1 defines an isolated singularity at the origin and the difference between its Milnor number and Tjurina number is positive i.e. µ -τ ≥ 1, then the maximal spectrum number of f is not contained in the Tjurina spectrum of f . Furthermore, we propose a conjecture for Tjurina spectrum parallel to Hertling conjecture, called Generalized Hertling Conjecture for Tjurina Spectrum.
Conjecture 1.3. Suppose f is itself a τ -max form. Let β 1 , …, β τ be exponents of Tjurina spectrum. Then the following inequality holds:
where β
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