Some remarks on the generalized Hertling conjecture for Tjurina spectrum

TL;DR

This paper investigates the generalized Hertling conjecture related to the variance of Tjurina spectral numbers, providing conditions for its failure and illustrating examples with computational tools.

Contribution

It offers a new sufficient condition for the conjecture's failure and demonstrates calculations using Singular code.

Findings

Identifies conditions under which the conjecture fails

Provides explicit computational examples

Utilizes Singular for spectral number calculations

Abstract

We study the generalized Hertling conjecture on the variance of the Tjurina spectral numbers, and provide a sufficient condition for the conjecture to fail, employing a theorem of Hertling in an essential way. We calculate certain examples using some codes in Singular.