Hyperplane Arrangements Research Articles

On Bernstein–Sato ideals for central line arrangements

The polynomial determines a plane central line arrangement α = 0. We compute explicitly multivariate Bernstein–Sato ideals of α by using the decomposition behavior of the D2-module given in earlier work by the authors. Our results are partially special cases of recent work in much greater generality, by Maisonobe on free hyperplane arrangements and Budur et al.) as well as Bath; however our proof is independent and gives some more information on different variants of Bernstein–Sato ideals in the simple plane case.

Bijections on r-Shi and r-Catalan arrangements

This paper aims to establish two bijections: from regions of the r-Shi arrangement to O-rooted labeled r-trees, and from regions of the r-Catalan arrangement to pairings of permutation and r-Dyck path. To this end, we introduce a cubic matrix for each region of the hyperplane arrangements. The first bijection is established by reading the positions of minimal positive entries in its column slices. The second one is obtained by reading the numbers of positive entries in its column slice, which turns out to be essentially the same as the bijection obtained by Duarte and Guedes de Oliveira [10] in 2019. Moreover, by reading the numbers of positive entries in its row slices we will recover the Pak-Stanley labeling, which is a celebrated bijection from regions of the r-Shi arrangement to r-parking functions.

The positive Dressian equals the positive tropical Grassmannian

The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces – the positive Dressian, and the positive tropical Grassmannian (which we introduced roughly fifteen years ago in [J. Algebraic Combin. 22 (2005), pp. 189–210]) – so it is natural to ask how these two positive spaces compare. In this paper we show that the positive Dressian equals the positive tropical Grassmannian. Using the connection between the positive Dressian and regular positroidal subdivisions of the hypersimplex, we use our result to give a new “tropical” proof of da Silva’s 1987 conjecture (first proved in 2017 by Ardila-Rincón-Williams) that all positively oriented matroids are realizable. We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer’s f f -vector theorem. Finally we give an example of a positroidal subdivision of the hypersimplex which is not regular, and make a connection to the theory of tropical hyperplane arrangements.

Combinatorially equivalent hyperplane arrangements

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.

Bordifications of hyperplane arrangements and their curve complexes

The complement of an arrangement of hyperplanes in $\mathbb C^n$ has a natural bordification to a manifold with corners formed by removing (or "blowing up") tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex $\mathcal{C}$, the vertices of which are the irreducible "parabolic subgroups" of the fundamental group of the arrangement complement. So, the complex $\mathcal{C}$ plays a similar role for an arrangement complement as the curve complex does for moduli space. Also, in analogy with curve complexes and with spherical buildings, we prove that $\mathcal{C}$ has the homotopy type of a wedge of spheres.

Positively hyperbolic varieties, tropicalization, and positroids

A variety of codimension c in complex affine space is positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension c. Positively hyperbolic hypersurfaces are defined by stable polynomials. We characterize these varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. Positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type A hyperplane arrangement defined by xi=xj, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of results of Choe–Oxley–Sokal–Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.

Singular loci of reflection arrangements and the containment problem

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals. Our work ties together and generalizes results in Bauer et al. (Int Math Res Not IMRN 24:7459–7514, 2019), Dumnicki et al. (J Algebra 393:24–29, 2013), Harbourne and Seceleanu (J Pure Appl Algebra 219(4):1062–1072, 2015) and Malara and Szpond (J Pure Appl Algebra 222(8):2323–2329, 2018) under a unified approach.

Twisted Alexander modules of hyperplane arrangement complements

We study torsion properties of the twisted Alexander modules of the affine complement M of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polynomials of the two spaces, compute the (first) twisted Alexander polynomial of a punctured stratified tubular neighborhood of an essential line arrangement, and study the possible roots of the twisted Alexander polynomials of both the complement and the punctured stratified tubular neighborhood of an essential hyperplane arrangement in higher dimensions. We apply our results to distinguish non-homeomorphic homotopy equivalent arrangement complements. We also relate the twisted Alexander polynomials of M with the corresponding twisted homology jump loci.

Parallel translates of represented matroids

Given an F-represented matroid (M,ρ) with the ground set [m], the representation ρ naturally defines a hyperplane arrangement Aρ. We will study its parallel translates Aρ,g of Aρ for all g∈Fm. Its intersection semi-lattices L(Aρ,g) and the characteristic polynomials χ(Aρ,g,t) will be classified by the intersection lattice of the derived arrangement Aδρ, which is a hyperplane arrangement associated with the derived matroid (δM,δρ) and also known as the discriminantal arrangement in the literature. As a byproduct, we obtain a comparison result and a decomposition formula on the characteristic polynomials χ(Aρ,g,t).

Helly meets Garside and Artin

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

Representations of automorphism groups on the homology of matroids

Given a group G of automorphisms of a matroid M, we describe the representations of G on the homology of the independence complex of the dual matroid M∗. These representations are related to the homology of the lattice of flats of M, and (when M is realizable) to the top cohomology of a hyperplane arrangement. Finally, we analyze in detail the case of the complete graph, which has applications to algebraic geometry.

Hyperplane Arrangements and Mixed Hodge Numbers of the Milnor Fiber

Abstract For a complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev’s theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking an arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of any simple normal crossing compactification of $F_{\mathcal{A}}$. As a direct consequence, we obtain a combinatorial formula for the top weight cohomology of $F_{\mathcal{A}}$, recovering a result of Dimca and Lehrer.

Whitney Numbers for Poset Cones

Hyperplane arrangements dissect \(\mathbb {R}^{n}\) into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes xi = xj inside \(\mathbb {R}^{n}\) for the symmetric group, thought of as the type An− 1 reflection group. Here, cones correspond to posets, chambers within the cone correspond to linear extensions of the poset, the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata’s theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.

Root Cones and the Resonance Arrangement

We study the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance arrangement, the number of resonance chambers has only been computed up to the $n=8$ dimensional case. We focus on data structures for labeling chambers, such as sign vectors and sets of alternating trees, with an aim at better understanding the structure of the resonance arrangement, and, in particular, enumerating its chambers. Along the way, we make connections with similar (and similarly difficult) enumeration questions. With the root polytope viewpoint, we relate resonance chambers to the chambers of polynomiality of the Kostant partition function. With the hyperplane viewpoint, we clarify the connections between resonance chambers and threshold functions. In particular, we show that the base-2 logarithm of the number of resonance chambers is asymptotically $n^2$.

On the Falk Invariant of Shi and Linial Arrangements

It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements.

Hyperplane arrangement complement with top degree Betti number being small

Hyperplane arrangement complement with top degree Betti number being small

Geometric recursion from polytope triangulations and twisted homology

A geometric approach to understanding recursion relations for scattering amplitudes is developed. We achieve this by studying intersection numbers of triangulated accordiohedra presented as hyperplane arrangements. The cancellation of spurious divergences is subsequently realized as a topological no-boundary condition.

Affine hyperplane arrangements and Jordan classes

We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.

Euclidean matchings and minimality of hyperplane arrangements

We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us to prove the isomorphism of Brieskorn’s Lemma by a simple bijection of the critical cells. Finally, in the case of line arrangements, we describe the algebraic Morse complex which computes the homology with coefficients in an abelian local system.

On polytopes and generalizations of the KLT relations

We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.