Nakai conjectures for isolated homogeneous hypersurface singularities

TL;DR

This paper verifies Nakai's conjecture for isolated homogeneous hypersurface singularities, exploring whether differential operators generated by derivations imply smoothness of the variety.

Contribution

The paper confirms Nakai's conjecture specifically for isolated homogeneous hypersurface singularities, advancing understanding of the relationship between differential operators and singularity detection.

Findings

Nakai Conjecture holds for isolated homogeneous hypersurface singularities.

Differential operators generated by derivations imply smoothness in this case.

Provides new insights into the algebraic structure of singularities.

Abstract

The long-standing Nakai Conjecture concerns a very natural question: can differential operators detect singularities on algebraic varieties? On a smooth complex variety, it is well known that the ring of differential operators is generated by derivations. Nakai asked whether the converse holds: if the ring of differential operators is generated by derivations, is the variety smooth? In this paper, we verify the Nakai Conjecture for isolated homogeneous hypersurface singularities.