Calogero-Moser spaces and the invariants of two matrices of degree 3

TL;DR

This paper explicitly describes the generators and relations of the coordinate ring of Calogero-Moser space for degree 3, providing new insights into invariants of two 3x3 matrices and their orbit structure.

Contribution

It offers a minimal generating set and explicit relations for the coordinate ring of Calogero-Moser space or degree 3, and presents a new description of the invariant matrix algebra.

Findings

Explicit generators and relations for or Calogero-Moser space or degree 3.

A new presentation of the algebra of 3x3 invariant matrices.

Description of the commuting variety of 3x3 matrices and its orbits.

Abstract

We find a minimal set of generators for the coordinate ring of Calogero-Moser space $\mathcal{C}_3$ and the algebraic relations among them explicitly. We give a new presentation for the algebra of $3\times3$ invariant matrices involving the defining relations of $\mathbb{C}[\mathcal{C}_3]$. We find an explicit description of the commuting variety of $3\times3$ matrices and its orbits under the action of the affine Cremona group.