Geometric aspects of the Jacobian of a hyperplane arrangement

TL;DR

This paper explores the deep connections between hyperplane arrangements and rigidity theory, analyzing how geometric configurations influence algebraic invariants like the Jacobian ideal's saturation.

Contribution

It develops the theory of weak perspective representations to unify combinatorial and geometric aspects of hyperplane arrangements and rigidity.

Findings

Betti numbers vary when points lie on a conic

Framework flexibility depends on geometric placement

Connections between arrangements and rigidity are formalized

Abstract

An embedding of the complete bipartite graph $K_{3,3}$ in $\mathbb{P}^2$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $K_{3,3}$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. In this paper we develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that this example suggests. In particular, we seek to understand how the interplay of combinatorics and geometry influence algebraic structures associated to an arrangement, such as the saturation of the Jacobian ideal of the arrangement. We make connections between examples and constructions from rigidity theory and interesting phenomena in the study of hyperplane arrangements.