Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the *valid order arrangement*, whose regions correspond to the different orders in which a line through p can cross the hyperplanes in A. If A is the set of affine spans of the facets of a convex polytope P and p lies in the interior of P, then the valid orderings with respect to p are just the line shellings of p where the shelling line contains p. When p is sufficiently generic, the intersection lattice of vo(A,p) is the *Dilworth truncation* of the semicone of A. Various applications and examples are given. For instance, we determine the maximum number of line shellings of a d-polytope with m facets when the shelling line contains a fixed point p. If P is the order polytope of a poset, then the sets of facets visible from a point involve a generalization of chromatic polynomials related to list colorings.
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it.
In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by "gluing" complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrang…
Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many non-free arrangements whose characteristic polynomial factors over the integer ring. On the other hand, the localization of a free arrangement is free, and its restriction is in many cases free, thus its characteristic polynomial factors. In this paper, we consider how their integer, or real roots behave.
Holm introduced $m$-free $\ell$-arrangements which is a generalization of free arrangements, while he asked whether all $\ell$-arrangements are $m$-free for $m$ large enough. Recently Abe and the author verified that this question is in the negative when $\ell\geq 4$. In this paper we verify that $3$-arrangements $\mathscr{A}$ are $m$-free and compute the $m$-exponents for all $m\geq |\mathscr{A}|+2$, where $|\mathscr{A}|$ is the cardinality of $\mathscr{A}$. Hence Holm's question is in the positive when $\ell=3$. Finally we prove that $3$-dimensional Weyl arrangements of types A and B are $m$…
The Coboundary Quasi-Polynomials of Hyperplane Arrangements over Residually Finite Dedekind Domains
Faculty of Engineering, Nippon Bunri University（日本布列大学工学部）；Department of Mathematics, Nagoya Institute of Technology（名古屋工业大学数学系）；Department of Mathematics, Hokkaido University of Education（北海道教育大学数学系）
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $σ$-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.
