Extendability of the $$B_2$$-Arrangement

Source abstract

Abstract Let $$({\mathscr {A}},m)$$ ( A , m ) be a free multiarrangement, and let $${\mathscr {E}}$$ E be an extension of $$({\mathscr {A}},m)$$ ( A , m ) . It is well known that if $${\mathscr {A}}$$ A is the Coxeter arrangement of type $$A_2$$ A 2 , then a free extension of $$({\mathscr {A}},m)$$ ( A , m ) always exists. In this work, we demonstrate that if $${\mathscr {A}}$$ A is the Coxeter arrangement of type $$B_2$$ B 2 , there exist infinitely many multiplicities for which no free extension of $$({\mathscr {A}},m)$$ ( A , m ) exists. This result has immediate consequences for the existence of free extensions in higher rank.