Ideals of Rings of Differential Operators on Algebraic Curves (With an Appendix by George Wilson)

TL;DR

This paper classifies left ideals in the ring of differential operators on algebraic curves using geometric methods, introducing Calogero-Moser spaces as a key tool for understanding their structure.

Contribution

It provides a refined geometric classification of ideals in D(X) by describing fibers via Calogero-Moser spaces, extending classical results to algebraic curves.

Findings

Calogero-Moser spaces are smooth affine irreducible varieties of dimension 2n.

The classification relates ideals to representation varieties of deformed preprojective algebras.

Generalizes the description of ideals in the Weyl algebra to algebraic curves.

Abstract

Let X be a complex smooth affine irreducible curve, and let D = D(X) be the ring of global differential operators on X. In this paper, we give a geometric classification of left ideals in $ D $ and study the natural action of the Picard group of D on the space J(D) of isomorphism classes of such ideals. We recall that, up to isomorphism in the Grothendieck group K_0(D), the ideals of D are classified by the Picard group of X: there is a natural fibration \gamma: J(D) \to Pic(X), whose fibres are the stable isomorphism classes of ideals of D (see \cite{BW}). In this paper, we refine this classification by describing the fibres of \gamma in terms of finite-dimensional algebraic varieties C_n(X, I), which we call the (generalized) Calogero-Moser spaces. We define these varieties as representation varieties of deformed preprojective algebras over a certain extension of the ring of regular functions on $ X $. As in the classical case (see \cite{Wi}), we prove that C_n(X, I) are smooth affine irreducible varieties of dimension 2n. Our results generalize the description of left ideals of the first Weyl algebra A_1(C) in \cite{BW1, BW2}; however, our methods are quite different.