Singular loci of reflection arrangements and the containment problem

TL;DR

This paper investigates the algebraic structure of singular loci of hyperplane arrangements from complex reflection groups, providing minimal defining equations and analyzing the containment relations between their symbolic and ordinary powers.

Contribution

It introduces minimal equations for the radical ideals of these singular loci and explores the containment problem between symbolic and ordinary powers.

Findings

Derived minimal sets of equations for singular loci

Analyzed containment relations between symbolic and ordinary powers

Enhanced understanding of symmetry's role in polynomial vanishing

Abstract

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals.