Commuting differential operators and higher-dimensional algebraic varieties

TL;DR

This paper explores the algebro-geometric properties of commutative rings of partial differential operators, establishing a geometric data association and comparing different theoretical approaches in the context of higher-dimensional algebraic varieties.

Contribution

It introduces a new method to associate geometric data with commutative rings of PDEs and compares this with existing theories, advancing understanding of their algebraic and geometric structures.

Findings

Established a correspondence between commutative rings of PDEs and geometric data.

Compared different approaches to PDE rings in two variables.

Analyzed properties of geometric data associated with PDE rings.

Abstract

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of partial differential operators, and we investigate the properties of these geometric data. This construction is similar to the construction of a formal module of Baker-Akhieser functions. On the other hand, there is a recent generalization of Sato's theory which belongs to the third author of this paper. We compare both approaches to the commutative rings of partial differential operators in two variables.