  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.
