Resolutions of differential operators of low order for an isolated hypersurface singularity

TL;DR

This paper introduces a new approach to study differential operators on isolated hypersurface singularities, providing explicit generators and resolutions for low-order operators, expanding prior algebraic results.

Contribution

It develops a novel method to explicitly construct minimal generators and free resolutions of differential operator modules for hypersurface singularities.

Findings

Constructed explicit minimal generating sets for differential operators of order two and three.

Provided minimal free resolutions for these modules.

Extended previous algebraic results to a broader class of singularities.

Abstract

In this paper we develop a new approach for studying differential operators of an isolated singularity graded hypersurface ring $R$ defining a surface in affine three-space over a field of characteristic zero. With this method, we construct an explicit minimal generating set for the modules of differential operators of order two and three, as well as their minimal free resolutions; this expands results of Bernstein, Gel'fand, and Gel'fand and of Vigu\'e. Our construction relies, in part, on a description of these modules that we derive in the singularity category of $R$. Namely, we build explicit matrix factorizations starting from that of the residue field.