On the coordinate rings of Calogero-Moser spaces and the invariant commuting variety of a pair of matrices

TL;DR

This paper studies the algebraic and geometric structures of Calogero-Moser spaces and invariant matrix pairs, providing explicit descriptions, class computations, and new insights into related geometric objects.

Contribution

It offers a detailed description of coordinate rings and Poisson brackets for specific Calogero-Moser spaces and matrix invariants, including class calculations and geometric insights.

Findings

Explicit descriptions of coordinate rings and Poisson brackets for Calogero-Moser spaces.

Computed classes of these spaces in the Grothendieck ring.

Gained new geometric insights into the Hilbert scheme of points on the affine plane.

Abstract

This paper presents a comprehensive description of the coordinate rings and Poisson brackets associated with the fourth Calogero-Moser space and invariant commuting pairs of matrices of size four. As an application, we compute their respective classes in the Grothendieck ring of the category of complex varieties and we offer some novel insights about the geometry of the Hilbert scheme of points on the affine plane.