Non-existence of negative derivations on the higher Nash blowup local algebra

Abstract

Let $f\in \mathbb{C}[x_1,\dots ,x_s]$ be a weighted homogeneous polynomial having an isolated singularity and ${\mathcal{T}}_n(f)$ be its higher Nash blowup local algebra. We show that ${\mathcal{T}}_n(f)$ does not admit negative weighted derivations for $n\ge 2$. This answers affirmatively a conjecture of Hussain-Ma-Yau-Zuo.