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

TL;DR

This paper proves that for weighted homogeneous polynomials with isolated singularities, their higher Nash blowup local algebras do not admit negative weighted derivations for all n≥2, confirming a conjecture.

Contribution

It establishes the non-existence of negative derivations on higher Nash blowup local algebras of weighted homogeneous isolated singularities, resolving a conjecture.

Findings

No negative weighted derivations exist for n≥2

Confirms the conjecture of Hussain-Ma-Yau-Zuo

Advances understanding of algebraic structures in singularity theory

Abstract

Let $f\in\mathbb{C}[x_1,\ldots,x_s]$ 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 $n\geq2$. This answers affirmatively a conjecture of Hussain-Ma-Yau-Zuo.