Questions tagged [hyperplane-arrangements]

A hyperplane arrangement is a set of hyperplanes in a vector space or in a projective space. The complement of the union of these hyperplanes defines an algebraic variety, with interesting geometry and topology.

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Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that $\{L_1,\dots,L_m\}...
James Propp's user avatar
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Counting problems where unlabeled is easier than labeled

I was encouraged to post this question by Jim Propp during a meeting of the Cambridge Combinatorics and Coffee Club. It is a counterpoint to the MathOverflow question "Counting Problems where Labeled ...
Sam Hopkins's user avatar
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18 votes
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A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable?

The short version Here is an extremely natural hyperplane arrangement in $\mathbb{R}^n$, which I will call $R_n$ for resonance arrangement. Let $x_i$ be the standard coordinates on $\mathbb{R}^n$. For ...
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What are Sylvester-Gallai configurations in the complex projective plane?

A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. ...
Jérémy Blanc's user avatar
16 votes
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Coxeter Arrangements and an Identity

Let $\{A_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a central hyperplane arrangement). Such an arrangement is called Coxeter if reflecting across any ...
Greg Muller's user avatar
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15 votes
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Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space

I've got ten (projective) planes in projective 3-space: \begin{align} &x=0\\ &z=0\\ &t=0\\ &x+y=0\\ &x-y=0\\ &z+t=0\\ &x-y-z=0\\ &x+y+z=0\\ &x-y+t=0\\ &x+y-t=0 ...
მამუკა ჯიბლაძე's user avatar
12 votes
1 answer
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Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$. EDIT: I forgot to add that no ...
batconjurer's user avatar
12 votes
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Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree in $\mathbb C^{n+1})$. The fiber over $(\lambda:\mu) \in ...
Jorge Vitório Pereira's user avatar
11 votes
1 answer
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Chromatic number of a graph defined by $n$ lines on the plane

Given $n$ lines on the plane, consider all their intersection points. Find the minimal number $d=d(n)$ such that they may be always colored in $d$ colors so that on each line any two consecutive ...
Fedor Petrov's user avatar
11 votes
1 answer
344 views

Complexity of counting regions in hyperplane arrangements

Let $H_1,\ldots,H_n$ be hyperplanes in $\Bbb R^d$. Denote $\mathcal{H} :=\{H_1,\ldots,H_n\}$ and let $c(\mathcal{H})$ be the number of regions in the complement: $\Bbb R^d\setminus \bigcup H_i$. ...
Igor Pak's user avatar
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11 votes
2 answers
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Rigid line arrangements

What is already known about rigid line arrangements? By line arrangement, I mean a unions of lines in $\mathbb{P}^2_{\mathbb{C}}$ with fixed incidences. (Written in notation, I mean a collection of ...
DCT's user avatar
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10 votes
3 answers
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Number of regions formed by $n$ points in general position

Given $n$ points in $\mathbb{R}^d$ in general position, where $n\geq d+1$. For every $d$ points, form the hyperplane defined by these $d$ points. These hyperplanes cut $\mathbb{R}^d$ into several ...
Min Wu's user avatar
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9 votes
1 answer
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Hyperplane arrangements whose regions all have the same shape

Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal ...
Christian Gaetz's user avatar
9 votes
2 answers
763 views

Open problems in hyperplane/subspace arrangements?

What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and ...
9 votes
1 answer
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Action of Weyl group on regions of Shi arrangement

This is an elaboration of a question which was aked on MO several years ago, which was unanswered but deleted by the question-asker. I hope it is okay to elaborate on a deleted question like this; for ...
Sam Hopkins's user avatar
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9 votes
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Interactions between pseudoline arrangements and braid groups?

It is common to represent pseudoline arrangements as wiring diagrams:                     Fig. from: "Hamiltonicity and colorings of arrangement ...
Joseph O'Rourke's user avatar
8 votes
2 answers
506 views

Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices

My question is motivated by this one, but within real matrices instead of complex ones. ${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
Denis Serre's user avatar
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8 votes
2 answers
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a Littlewood–Offord-type problem concerning the "cubical lattice"

Fix even $n$ and consider the boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$, $f : (x_0, \ldots , x_{n - 1}) \mapsto (x_0 \vee x_1) \wedge (x_2 \vee x_3) \wedge \cdots \wedge (x_{n - 2} \vee ...
BD107's user avatar
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8 votes
1 answer
217 views

Counting polygons in arrangements

For an arrangement of lines $\cal{A}$ in the plane, an inducing polygon $P$ is a simple polygon satisfying: (a) every edge $e$ of $P$ lies on some line $\ell$ of $\cal{A}$, and (b) every line $\ell \...
Joseph O'Rourke's user avatar
8 votes
0 answers
276 views

Cohomology of the complement of the resonance hyperplane arrangement

Here was a question about resonance arrangement. It is defined as follows. Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
nikitamarkarian's user avatar
7 votes
3 answers
381 views

Bijection directly from (n,n+1)-core partitions to parking functions?

It is well-known that the increasing parking functions are counted by the Catalan numbers. The Catalan numbers also count the dominant alcoves in the Shi arrangement of type $A_{n}$. Athanasiadis-...
coolpapa's user avatar
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1 answer
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Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history

Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces. Counting the ...
Stefan Forcey's user avatar
7 votes
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93 views

Combinatorial region-halfplane incidence structures

I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate. Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\...
Mikhail Tikhomirov's user avatar
6 votes
2 answers
294 views

Functionals on oriented matroids

Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors. Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...
Benjamin Steinberg's user avatar
6 votes
1 answer
278 views

Singularities at worst like a hyperplane arrangement

Is there a standard name for the type of singularities a codimension-$1$ subvariety of a smooth algebraic variety has when it looks locally (possibly analytically) like an arrangement of hyperplanes? «...
Mariano Suárez-Álvarez's user avatar
6 votes
1 answer
165 views

The characteristic varieties of the complement of the braid arrangement

The characteristic varieties $V_d^i(X)$ of a (sufficiently nice) space $X$ are the cohomology jumping loci for 1-dimensional (complex) local systems on $X$. Assume that $H_1(X;\mathbb{Z}) \cong \...
K.K.'s user avatar
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Have the affine simplicial line arrangments been enumerated?

I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements. A line arrangment is a family of straight lines in $\Bbb R^2$. It is simplicial if all regions are ...
M. Winter's user avatar
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6 votes
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273 views

balls in arrangements of hyperplanes

The following theorem is from Aronov, Naiman, Pach and Sharir's An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem ...
Thomas Z's user avatar
5 votes
1 answer
101 views

The Salvetti complex of a non-realizable oriented matroid

Given a real hyperplane arrangement, the Salvetti complex of the associated oriented matroid is homotopy equivalent to the complement of the complexification of the arrangement. In particular, its ...
Nicholas Proudfoot's user avatar
5 votes
1 answer
302 views

Hyperplane arrangements and covering numbers

Let $H$ be a set of $(d-1)$-dimensional hyperplanes in $\mathbb{R}^d$. For each hyperplane $h \in H$ let $D(h)$ and $\bar{D}(h)$ be the corresponding half spaces of $\mathbb{R}^d$. For a point $x \...
Robby McKilliam's user avatar
5 votes
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149 views

Orlik-Solomon algebra and hyperplane complements in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$. Given a ring $R$ ...
Emiliano Ambrosi's user avatar
5 votes
0 answers
149 views

Random walks in arrangements of lines in the plane

Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$. (Simple: each pair of lines meet in a distinct point, i.e., no three lines pass through the same point.) Start a random walk at ...
Joseph O'Rourke's user avatar
5 votes
1 answer
592 views

Number of regions of a hyperplane arrangement avoiding a generic hyperplane

Let $\mathcal{A}$ be an essential arrangement of hyperplanes in $\mathbb{R}^n$. Zaslavsky's theorem says that the number of regions of $\mathcal{A}$ is given by $r(\mathcal{A})=(-1)^n\chi_{\mathcal{A}}...
Sam Hopkins's user avatar
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5 votes
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227 views

Tensor product of hyperplane arrangements

Let $(A_{1},V_{1})$ and $(A_{2},V_{2})$ be two central hyperplane arrangements, which 0 belongs to their intersections. Let $V=V_{1}\otimes V_{2}$. Define the tensor product arrangement $(A_{1}\otimes ...
Moh514's user avatar
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4 votes
1 answer
262 views

Criterion for being a simple vector

1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors $V\in\wedge^k(\mathbb R^n)$, $V=e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \...
Mircea's user avatar
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4 votes
3 answers
328 views

lines through A_n reflection arrangement and permutations

(updated; apologies for way too much room left for interpretation in the original post) Let $\mathcal{A} =A_{n-1}$ be the $A_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes $H_{ij}...
Dima Pasechnik's user avatar
4 votes
1 answer
141 views

Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?

Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$. The $n$-th type-A subdivision ...
darij grinberg's user avatar
4 votes
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132 views

Sheaves with specified singular support at infinity coming from hyperplane arrangements

Given a manifold $M$, we consider its cotangent bundle $T^*M$, and its cocircle bundle $T^\infty M$, quotienting out by the scaling action of the positive reals. Given a Legendrian submanifold $\...
Hugh Thomas's user avatar
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4 votes
0 answers
100 views

Bounds on k-tuple points for intersections of hyperplanes

Suppose that $H_1$,...,$H_d$ are hyperplanes in $\mathbb P^n$ (over some field -- you can pick). For $k \geq n$, let $t_k$ denote the number of points through which there pass exactly $k$ hyperplanes....
J L's user avatar
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4 votes
0 answers
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Counting the polytopes of the translates of the resonance hyperplane arrangement inside the unit hypercube

Let $n$ be a positive natural number. For all $\emptyset \subset S \subseteq \{1, \ldots, n\}$ and $k \in \mathbb{Z}$, define the hyperplane $H(S,k)$ in $\mathbb{R}^n$ given by the equations $$H(S,k):=...
calc's user avatar
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3 votes
3 answers
282 views

Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant $\...
fact's user avatar
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3 votes
2 answers
2k views

Is there a characterization of hyperplane arrangement intersection posets?

For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered ...
Sam Hopkins's user avatar
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3 votes
2 answers
599 views

Counting Regions in Hyperplane Arranglements

Consider the following: 1) How many connected regions can $n$ hyperplanes form in $\mathbb R^d$? 2) What if the set of hyperplanes are homogeneous? 3) Given a set of $n$ pairs of ...
AvidLearner's user avatar
3 votes
0 answers
210 views

When is a wonderful compactification a toric variety?

Given a (projective) hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^n$, in which we assume the intersection of all hyperplanes is $0$, and a building set $G$, De Concini and Procesi define the ...
Aidan's user avatar
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0 answers
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Salvetti complexes and cohomology of affine completion of Artin groups $E_6$ and $E_7$

After the solution of the Brieskorn-Arnold Pham conjecture on the asphericity of a space for affine Artin groups by Paolini and Salvetti MR4243019 (arXiv), I would like to know if there are ...
Nicolas Boerger's user avatar
3 votes
0 answers
112 views

Polynomiality of the inverse of equally weighted Varchenko matrices attached to hyperplane arrangements

Let $d\in \mathbb{Z}_{\ge 1}$, let $\sigma = (H_i)_{i\in \mathcal{I}}$ be a finite hyperplane arrangement in $\mathbb{R}^d$, where $H_i\subset \mathbb{R}^d$ is a hyperplane for $i\in \mathcal{I}$ (the ...
Wille Liu's user avatar
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3 votes
0 answers
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Cohomology of higher codimensional arrangements

Hyperplane arrangements are classical objects of study and there is a large literature on this subject, e.g. dealing with computing the cohomology of the complement. I am looking for similar results ...
Najib Idrissi's user avatar
3 votes
0 answers
104 views

Homotopy type of hyperplane arrangements intersected with real subspaces

The homotopy type, and especially the higher homotopy groups of complement of hyperplane arrangements in $\mathbb{C}^n$ has been extensively studied, for example Falk and Randell - On the homotopy ...
FKranhold's user avatar
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2 votes
1 answer
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Simplicial set represented by an (unordered) set

Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with $$ F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X) $$ where the right hand side denotes arbitrary maps of sets (...
cgodfrey's user avatar
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2 votes
1 answer
176 views

Seeking combinatorial interpretation of a quantity that comes up from central hyperplane arrangements

Let $\mathcal A$ be a central hyperplane arrangement in $\mathbb R^d$ and let's assume that it is essential, meaning the hyperplanes in $\mathcal A$ intersect in the origin. The intersection lattice $...
Benjamin Steinberg's user avatar