Geometric properties of commutative subalgebras of partial differential operators

TL;DR

This paper explores the geometric and algebraic properties of commutative rings of partial differential operators, providing new characterizations, explicit examples, and connections to spectral data and moduli spaces.

Contribution

It offers a complete spectral data characterization for trivial rings, introduces a restriction map for moduli spaces, and constructs novel examples of surfaces related to integrable systems.

Findings

Complete spectral data characterization for trivial rings

Explicit examples of spectral data and associated rings of operators

Identification of surfaces not isomorphic to spectral surfaces of rank one PDO rings

Abstract

We investigate further alebro-geometric properties of commutative rings of partial differential operators continuing our research started in previous articles. In particular, we start to explore the most evident examples and also certain known examples of algebraically integrable quantum completely integrable systems from the point of view of a recent generalization of Sato's theory which belongs to the second author. We give a complete characterisation of the spectral data for a class of "trivial" rings and strengthen geometric properties known earlier for a class of known examples. We also define a kind of a restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to analogous moduli space on a divisor (both the surface and divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding rings of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any commutative ring of PDOs of rank one. At last, we prove that any commutative ring of PDOs, whose normalisation is isomorphic to the ring of polynomials $k[u,t]$, is a Darboux transformation of a ring of operators with constant coefficients.