Arithmetic non-very generic arrangements

Abstract A discriminantal hyperplane arrangement $$\mathcal {B}(n,k,\mathcal {A})$$ B ( n , k , A ) is constructed from a given generic hyperplane arrangement $$\mathcal {A}$$ A . The arrangement $$\mathcal {A}$$ A is classified as either very generic or non-very generic according to the combinatorial structure of $$\mathcal {B}(n,k,\mathcal {A})$$ B ( n , k , A ) . In particular, $$\mathcal {A}$$ A is regarded as non-very generic if the intersection lattice of $$\mathcal {B}(n,k,\mathcal {A})$$ B ( n , k , A ) contains at least one non-very generic intersection, namely, an intersection that does not satisfy a specific rank condition established by Athanasiadis. In this paper, we present arithmetic criteria characterizing non-very generic intersections in discriminantal arrangements, and we complete and correct a previous result of Libgober and the third author concerning rank 2 intersections in such arrangements.