A class of Baker-Akhiezer arrangements

TL;DR

This paper classifies certain line arrangements with multiplicities that admit Baker-Akhiezer functions, explores their algebraic properties, and establishes conditions for Gorenstein algebras of quasi-invariants.

Contribution

It provides a complete description of Baker-Akhiezer arrangements with limited multiplicity complexity and analyzes their associated algebraic structures.

Findings

All such arrangements with at most one higher multiplicity line are described.

The associated algebras of quasi-invariants are Gorenstein.

No other arrangements with similar properties exist beyond the classified cases.

Abstract

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh-Veselov Baker-Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker-Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero-Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.