An algebraic geometric classification of superintegrable systems in the Euclidean plane

TL;DR

This paper classifies non-degenerate second order superintegrable systems in the complex Euclidean plane using algebraic geometry, revealing their structure as a projective variety and confirming existing classifications through algebraic methods.

Contribution

It introduces an algebraic geometric framework for classifying superintegrable systems and explicitly solves the associated algebraic equations, providing a geometric interpretation.

Findings

The set of systems forms a projective variety.

Each system corresponds to a unique line triple arrangement.

The classification aligns with previous algebraic results.

Abstract

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving the corresponding system of homogeneous algebraic equations. We then solve these equations explicitly and give a detailed analysis of the algebraic geometric structure of the corresponding projective variety. This naturally associates a unique planar line triple arrangement to every superintegrable system, providing a geometric realisation of this variety and an intrinsic labelling scheme. In particular, our results confirm the known classification by independent, purely algebraic means.