Differential operators of low order for an isolated hypersurface singularity

We study the ring D differential operators of low order for an isolated singularity hypersurface ring of the form R=k[x,y,z]/(f) for a field k of characteristic zero and homogeneous polynomial of of degree at least 3 (that is, the cone of a smooth projective plane curve).  For such hypersurfaces, it was long known that there are no negative degree operators by results of Vigué building on the work of Bernstein, Gel’fand, and Gel’fand, the proofs going via investigations of certain sheaf cohomologies, but the explicit operators were not known as far as we know (and further recent results by Mallory provide similar behavior in 4 variables). We find the differential operators of order up to 3 for such hypersurfaces in 3 variables by completely new, homological methods, as well as minimal resolutions of the R-module that they form. 

This result highlights further the contrasting and yet interactive behaviors in the smooth and non-smooth cases. In the smooth case, such rings of differential operators tend to have many operators of negative degree, yielding that certain important D-modules are simple. In addition, when R is a polynomial ring, for example, many important, but infinitely generated, R-modules such as local cohomology become of finite length as D-modules, which has been an important tool for studying them. 

This is joint work with Rachel Diethorn, Jack Jeffries, Nick Packauskas, Josh Pollitz, Hamid Rahmati, and Sophia Vassiliadou.