The Nakai Conjecture for isolated hypersurface singularities of modality $\le 2$

TL;DR

This paper verifies the Nakai Conjecture for isolated hypersurface singularities with modality less than or equal to 2, extending previous results and exploring the conditions under which the algebra's differential operators generate the algebra.

Contribution

It extends the verification of the Nakai Conjecture to a broader class of singularities with modality up to 2, advancing understanding of the conjecture's scope.

Findings

Confirmed the Nakai Conjecture for isolated hypersurface singularities of modality ≤ 2

Extended previous results to include more complex singularities

Provided new insights into the structure of differential operators in these cases

Abstract

The well-known Nakai Conjecture concerns a very natural question: For an algebra of finite type over a characteristic zero field, if the ring of its differential operators is generated by the first order derivations, is the algebra regular? And it is natural to extend the Nakai Conjecture to local domains, in this paper, we verify it for isolated hypersurface singularities of modality $\le 2$, this extends the existing works.