Line-closed matroids, quadratic algebras, and formal arrangements

TL;DR

This paper explores the relationship between line-closed matroids, quadratic algebras, and formal arrangements, establishing conditions under which the Orlik-Solomon algebra is quadratic and linking these to geometric properties of arrangements.

Contribution

It introduces the concept of bb sets in matroids, proves their linear independence in quadratic closures, and clarifies the connections between line-closure, formality, and topological properties of arrangements.

Findings

Line-closed matroids imply quadratic Orlik-Solomon algebras.

Line-closure is not necessary or sufficient for arrangements to be $K(\pi,1)$ or free.

Examples show the limits of line-closure's implications for arrangement properties.

Abstract

Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the converse false. These results generalize to the degree $r$ closure of $\A(G)$. The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for \A to be free and for the complement $M$ of \A to be a $K(\pi,1)$ space. Formality of \A is also necessary for $A(G)$ to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of $G$ is not necessary or sufficient for $M$ to be a $K(\pi,1)$, or for \A to be free.