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\}...
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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 ...
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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. ...
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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 ...
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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
...
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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 ...
12
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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 ...
11
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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 ...
11
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1
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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: $\Bbb R^d\setminus \bigcup H_i$.
...
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2
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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 ...
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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 ...
9
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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 ...
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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 ...
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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 ...
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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 ...
8
votes
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answers
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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 $\...
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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 ...
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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 \...
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answers
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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 $...
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3
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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-...
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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 ...
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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 $\...
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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 ...
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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? «...
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votes
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answer
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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 \...
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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 ...
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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 ...
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votes
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answer
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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 ...
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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 \...
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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 $R$ ...
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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 ...
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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}}...
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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 ...
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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}} + \...
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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}...
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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 ...
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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 $\...
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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....
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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):=...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
3
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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 ...
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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 ...
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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 ...
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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 (...
2
votes
1
answer
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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 $...