A Geometric Condition for a Hyperplane Arrangement to be Free

Abstract Let G ( r , 1,  l ) be the complex arrangement { x i ,  x j − ξ h x k }, where ξ is a primitive r th root of unity. The matroids of these arrangements are the Dowling matroids Q l ( Z r ), where Z r is the group of r th roots of unity. We show that if E is a subset of G ( r , 1,  l ) which does not contain any matroid line, then deleting E gives an arrangement G ( r , 1,  l )\ E which is free. We also give necessary and sufficient conditions on a set E containing at least one matroid line but no matroid planes so that the deletion G ( r , 1,  l )\ E has a characteristic polynomial which factors completely over the integers. Two types of arrangements can be obtained in this way. We show that one type is always non-free. This yields examples of non-free complex arrangements whose characteristic polynomials factor completely over the integers. The same ideas also yield examples of non-free arrangements over any sufficiently large field (and hence, over the reals) with characteristic polynomials which factor completely over the integers. The matroids of these arrangements are non-supersolvable matroids whose characteristic polynomials factor completely over the integers.