On the non-very generic intersections in discriminantal arrangements

In 1985 Crapo introduced in 3 a new mathematical object that he called geometry of circuits . Four years later, in 1989, Manin and Schechtman defined in 13 the same object and called it discriminantal arrangement , the name by which it is known now a days. Those discriminantal arrangements ℬ ( n , k , 𝒜 0 ) are builded from an arrangement 𝒜 0 of n hyperplanes in general position in a k -dimensional space and their combinatorics depends on the arrangement 𝒜 0 . On this basis, in 1997 Bayer and Brandt (see 2) distinguished two different type of arrangements 𝒜 0 calling very generic the ones for which the intersection lattice of ℬ ( n , k , 𝒜 0 ) has maximum cardinality and non-very generic the others. Results on the combinatorics of ℬ ( n , k , 𝒜 0 ) in the very generic case already appear in Crapo 3 and in 1997 in Athanasiadis 1 while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper 12 they provided a necessary and sufficient condition on 𝒜 0 for which the cardinality of rank 2 intersections in ℬ ( n , k , 𝒜 0 ) is not maximal anymore. In this paper we further develop their result providing a sufficient condition on 𝒜 0 for which the cardinality of rank r, r ≥ 2 , intersections in ℬ ( n , k , 𝒜 0 ) decreases.