Commuting differential operators and higher-dimensional algebraic varieties

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 in some sense 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. As a by-product we get several necessary conditions on geometric data describing commutative rings of partial differential operators.

Fourier–Mukai transform on Weierstrass cubics and commuting differential operators

“…The statements about kr. dim(B) and coherence of F B can be proven exactly in the same way as in [34].…”

A Class of Baker–Akhiezer Arrangements

“…Finally, papers [20,21] deal with the correspondence between 'geometric data' and commutative rings of partial differential operators, particularly, in two variables. Thus it would be interesting to describe the geometric data corresponding to the rings of quasi-invariants considered in this paper.…”

On rings of commuting partial differential operators

“…To formulate this statement recall one construction (without details) given in section 3.2 of [18]. For a given integral two-dimensional scheme X of finite type over a field k (or over the integers) there is a "minimal" Cohen-Macaulay scheme CM (X) and a finite morphism CM (X) → X (and a finite morphism from the normalization of X to CM (X) ).…”