Differential operators on an affine curve: ideal classes and Picard groups

TL;DR

This paper explores the structure of ideal classes and Picard groups related to differential operators on affine curves, introducing a fibration and linking it to classical limits and Calogero-Moser spaces.

Contribution

It introduces a new fibration  hat connects ideal classes with Picard groups and relates it to classical limits and Calogero-Moser spaces, advancing understanding of differential operators on curves.

Findings

Defined a fibration etween ideal classes and Picard groups.

Related the fibration to classical limits and Calogero-Moser spaces.

Analyzed the action of Pic(D) on the fibration.

Abstract

Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration \gamma : R \to Pic(X). We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant $ n $ which indexes the decomposition of the fibres of \gamma into Calogero-Moser spaces (see [BC]). We also study the action of the group Pic(D) on our fibration; and we explain how to define \gamma in the framework of the Grassmannian description of R due to Cannings and Holland.