Arithmetic non-very generic arrangements

TL;DR

This paper investigates the combinatorial structure of discriminantal hyperplane arrangements, providing arithmetic criteria to identify non-very generic intersections and correcting previous results on rank-two intersections.

Contribution

It introduces arithmetic criteria for non-very generic intersections and refines earlier findings on rank-two intersections in discriminantal arrangements.

Findings

Arithmetic criteria for non-very generic intersections

Complete and correct previous results on rank-two intersections

Enhanced understanding of the combinatorial structure of discriminantal arrangements

Abstract

A discriminantal hyperplane arrangement B(n,k,A) is constructed from a given (generic) hyperplane arrangement A, which is classified as either very generic or non-very generic depending on the combinatorial structure of B(n,k,A). In particular, A is considered non-very generic if the intersection lattice of B(n,k,A) contains at least one non-very generic intersection -- that is, an intersection that fails to satisfy a specific rank condition established by Athanasiadis in [1]. In this paper, we present arithmetic criteria characterizing non-very generic intersections in discriminantal arrangements and we complete and correct a previous result by Libgober and the third author concerning rank-two intersections in such arrangements.