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

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 $\leq 3$.