On intersection lattices of hyperplane arrangements generated by generic points

TL;DR

This paper investigates the structure of intersection lattices of hyperplane arrangements generated by generic points, revealing a decomposition approach that enables computation of key combinatorial invariants up to dimension six.

Contribution

It introduces a novel decomposition of intersection lattices into smaller lattice products, facilitating the calculation of M"obius functions and characteristic polynomials for these arrangements.

Findings

Decomposition of intersection lattices into direct products of smaller lattices.

Computed M"obius functions for arrangements up to dimension six.

Derived characteristic polynomials for arrangements up to dimension six.

Abstract

We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the M\"obius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.