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. [ABSTRACT FROM AUTHOR]