Singular loci of reflection arrangements and the containment problem

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. Our work ties together and generalizes results in Bauer et al. (Int Math Res Not IMRN 24:7459–7514, 2019), Dumnicki et al. (J Algebra 393:24–29, 2013), Harbourne and Seceleanu (J Pure Appl Algebra 219(4):1062–1072, 2015) and Malara and Szpond (J Pure Appl Algebra 222(8):2323–2329, 2018) under a unified approach.