A geometric deletion-restriction formula

TL;DR

This paper introduces a geometric approach to compute the characteristic polynomial of hyperplane arrangements using the rational equivalence class of the logarithmic ideal, linking algebraic geometry with combinatorial invariants.

Contribution

It recovers the characteristic polynomial via geometric methods and proves the Solomon-Terao formula under the tame hypothesis, connecting Hilbert series and arrangement invariants.

Findings

Characteristic polynomial recovered through geometric methods

Proved Solomon-Terao formula under tame hypothesis

Linked Hilbert series of logarithmic ideal to arrangement invariants

Abstract

In this paper, we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement. The logarithmic ideal was introduced in [arXiv:0907.0896v2] in a study of the critical points of the master function. The above result is used to understand the asymptotic behavior the Hilbert series of the logarithmic ideal. As an application, we prove the Solomon-Terao formula under the tame hypothesis by identifying each side of the formula with a certain specialization of the Hilbert series of the logarithmic ideal.