Rings of differential operators over projective rational curves.

Abstract

We describe the ring of global differential operators over a singular, rational, projective curve over an algebraically closed field of characteristic zero. For a affine curves, it is well-known that the ring of differential operators has a unique minimal non-zero ideal, and so the ring structure is determined by factoring out this ideal. In Chapter I we consider the twisted ring of differential operators for an invertible sheaf over a fixed curve. A sufficiently twisted ring of differential operators has a unique minimal non-zero ideal, and the factor ring obtained by factoring the minimal ideal is finite-dimensional. This factor ring can be completely described in terms of endomorphisms of the zeroth cohomology group of the invertible sheaf which fix certain subspaces. Moreover, an analog of Beilinson and Bernstein's equivalence of categories holds. Namely, if the ring of differential operators is sufficiently twisted, then the category of coherent sheaves of modules over the twisted sheaf of differential operators is equivalent to the category of finitely generated left modules over the twisted ring of global differential operators. The untwisted case is significantly more difficult. In Chapter II we discuss several aspects of the structure of the standard ring of global differential operators. We can obtain information from the sufficiently twisted case. In particular, the untwisted ring of differential operators arises as the endomorphism ring of a specific module over the sufficiently twisted ring of differential operators. For a more in depth look at the ideal structure of the untwisted case, we study certain finite-dimensional factor rings of the ring of differential operators. For example, the differential operators from the unramified closure of the curve to the curve itself forms an ideal. Factoring out this ideal gives us a finite-dimensional ring which bears a striking resemblance to the factor ring of the ring of differential operators of an open affine subset of the curve which contains all the singularities.