If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement $\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship between the critical set of $\Phi_\lambda$, the variety defined by the vanishing of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$. For arrangements satisfying certain conditions, we show that if $\lambda$ is resonant in dimension $p$, then the critical set of $\Phi_\lambda$ has codimension at most $p$. These include all free arrangements and all rank $3$ arrangements.

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\lceil 2n/(n-d+2) \rceil-1$ flips for all $n \ge d+2 \ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension $d \ge 3$. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included.