On a-fold products ideals of hyperplane arrangements

Abstract

Given Σ⊂R:=K[x1,…,xk], where K is a field of characteristic 0, any finite collection of linear forms defining a hyperplane arrangement in Pk−1 and any 1≤a≤|Σ| satisfying |Σ|−a≥k, we prove that the ideal generated by all a-fold products of Σ, denoted by Ia(Σ), is of fiber type. Moreover, we verify a conjecture of Mantero, Miranda-Neto and Nagel on symbolic powers for the class of ideals generated by (|Σ|−1)-fold products defined by a generic hyperplane arrangement.