On the Orlik--Terao ideal and the relation space of a hyperplane arrangement

TL;DR

This paper explores the connection between the relation space and the Orlik--Terao ideal in hyperplane arrangements, providing new characterizations and insights into their algebraic properties.

Contribution

It generalizes the characterization of 2-formal arrangements and analyzes the prime ideals of subideals of the Orlik--Terao ideal.

Findings

Characterization of spanning sets of the relation space via the Orlik--Terao ideal

Extension of 2-formal arrangement characterization

Examples showing differences in codimension for subideals

Abstract

The relation space of a hyperplane arrangement is the vector space of all linear dependencies among the defining forms of the hyperplanes in the arrangement. In this paper, we study the relationship between the relation space and the Orlik--Terao ideal of an arrangement. In particular, we characterize spanning sets of the relation space in terms of the Orlik--Terao ideal. This result generalizes a characterization of 2-formal arrangements due to Schenck and Toh\v{a}neanu \cite[Theorem 2.3]{ST}. We also study the minimal prime ideals of subideals of the Orlik--Terao ideal associated to subsets of the relation space. Finally, we give examples to show that for a 2-formal arrangement, the codimension of the Orlik--Terao ideal is not necessarily equal to that of its subideal generated by the quadratic elements.