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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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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
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Recovery of hyperplane arrangements from homotopical data

Given a hyperplane arrangement, one can construct the homotopical data of the poset of faces and the assignment of each facet to the hyperplane in which it belongs. This should be enough to recover ...
Charles Wang's user avatar
6 votes
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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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8 votes
2 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 $\...
Denis Serre's user avatar
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2 votes
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Real-isability of a (relatively small) subconfiguration of the Klein configuration

The Klein configuration consists of $60$ points and $60$ planes in $\mathbb C\mathbf P^3$, each point lying on $15$ of the planes and each plane containing $15$ of the points. It appears, among many ...
მამუკა ჯიბლაძე's user avatar
3 votes
0 answers
219 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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11 votes
1 answer
366 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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cone structure of complement of hyperplanes

I want to know if in $\mathbb{R}^{m+3}$ we consider the following hyperplanes: \begin{cases} (1-g)y-\sum_{i\in I}x_i=0, & \text{if $I\subset\{1,\cdots,m+2\}$},|I|=g\\ gy-\sum_{i\in I}x_i+\...
tota's user avatar
  • 585
2 votes
1 answer
229 views

Possible "algebraic" direction in hyperplane arrangements

I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the &...
It'sMe's user avatar
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3 votes
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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
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152 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
2 votes
1 answer
130 views

Secant variety to a zero-dimensional projective variety

This is a reference request/nomenclature question. Let $A \subseteq \mathbb{P}^n$ be a finite set of points not contained in a hyperplane (over some field), and let $\sigma_r(A)$ be the $r$-th secant ...
Ben'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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2 votes
1 answer
178 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
2 votes
0 answers
76 views

Polyhedron coordinate bound

Given a polyhedron $$Ax\leq b$$ where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
Turbo's user avatar
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1 vote
1 answer
138 views

Extensions of combinatorially equivalent hyperplane arrangements

Let $A_1,A_2\subset \mathbb{C}^n$ be hyperplane arrangements with equivalent intersection lattices $L(A_1)\cong L(A_2)$. If $A_1\subset B_1$, where $B_1$ is third hyperplane arrangement, does there ...
user2520938's user avatar
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1 vote
0 answers
149 views

Reference for hyperplane arrangements

I am interested in the hyperplane arrangement in $\mathbb{C}^n$ defined by the polynomial $$ (x_i-x_j)(x_i+x_j) $$ for $1<i<j\leq n$. I vaguely recall that the completion of this arrangement ...
Igor Sikora's user avatar
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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 ...
Wille Liu's user avatar
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9 votes
1 answer
153 views

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
3 votes
0 answers
98 views

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
2 votes
0 answers
109 views

Sources for describing the characteristic polynomial of a nonintegral hyperplane arrangement in terms of point counting?

I have a family of hyperplane arrangements, and I'd like to describe their characteristic polynomials. When the hyperplanes are defined over the integers, the easiest way for me to do this is to use ...
Will Dana's user avatar
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0 answers
283 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
4 votes
0 answers
134 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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9 votes
1 answer
301 views

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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8 votes
1 answer
223 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
10 votes
3 answers
1k views

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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1 vote
1 answer
106 views

Regions of hyperplane arrangements and their faces

Consider a finite hyperplane arrangement $\mathcal{A}$ over $\mathbb{R}^n$. Let the regions given by $\mathcal{A}$ be $\mathcal{R}(\mathcal{A})=\{A_1,\dots A_m\}$ for some $m$. For any index set $I\...
Lemma1's user avatar
  • 157
1 vote
1 answer
107 views

Separation in $l^1$ (Kreps-YanTheorem)

I have a question about the hypotheses of the Kreps-Yan Separation Theorem. I use the notation $l^p_+$ for the subspace of vectors all of whose coordinates are non-negative and define $l^p_- = -l^p_+$....
MDR's user avatar
  • 188
2 votes
1 answer
269 views

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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3 votes
0 answers
107 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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1 vote
0 answers
96 views

What are the "ouverts convenables" used to prove Brieskorns lemma?

In the proof of Brieskorns lemma, see 3.3 here, Brieskorn mentions that we take "ouverts convenables" satisfying some properties, but, as far as I can tell, never specifies what these opens actually ...
user2520938's user avatar
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15 votes
2 answers
734 views

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
5 votes
0 answers
151 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
7 votes
0 answers
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
1 answer
166 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
  • 666
7 votes
1 answer
398 views

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
0 votes
0 answers
190 views

What breaks down in the theory of affine hyperplane arrangments?

It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...
MadcowD's user avatar
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9 votes
0 answers
184 views

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
6 votes
1 answer
284 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
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
2 votes
0 answers
82 views

Are the following hyperplane arrangements previously studied?

For a subset $I$ of $[n]$, a hyperplane $H_I \subset \mathbb{R}^n$ is defined by $$\sum_{i \in I} x_i= \sum_{j \not\in I} x_j.$$ Have you seen the following hyperplane arrangements? Is there anything ...
Hao Chen's user avatar
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4 votes
0 answers
101 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
  • 41
11 votes
2 answers
345 views

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
  • 1,537
2 votes
1 answer
131 views

Polyhedral structure of functions writable as a finite signed sum of max of linear functions

For any two positive integers $k,n$ consider the space of functions writable as, $\sum_i \sigma_i \max \{ L_{i1},L_{i2},..,L_{ik} \}$ (a finite sum) where each $L_{*} : \mathbb{R}^n \rightarrow \...
Student's user avatar
  • 555
7 votes
3 answers
391 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
  • 455
4 votes
0 answers
210 views

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
  • 243
2 votes
0 answers
78 views

Is there invariant for regions of central hyperplane arrangement?

Consider central hyperplane arrangement A with normal vectors with all combinations of -1 and 1 (is there name for it?). There is simple invariant for each chamber of A: sum of vectors, corresponding ...
DSblizzard's user avatar
16 votes
2 answers
792 views

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
11 votes
1 answer
340 views

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
1 vote
1 answer
152 views

Affine Hyperplane Arrangements in $\mathbb R^d$

Consider $\mathcal A=(u_i)_{i=1}^m $ to be a set of hyperplanes in $\mathbb R^d$, such that for every $1\leq i \leq m$: $u_i \in \mathbb R^d$. These hyperplanes are disconnecting $\mathbb R^d$ to ...
AvidLearner's user avatar