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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
28 views

Hyperplane arrangements and tropical linear spaces

I have been trying to understand Chapter 5.4 of this Brief Introduction to Tropical Geometry, but I am struggling because of my lack of mathematical background. I will ask a few questions after giving ...
7 votes
1 answer
385 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 ...
5 votes
1 answer
102 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 ...
0 votes
0 answers
29 views

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 ...
6 votes
0 answers
123 views

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 ...
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 $\...
2 votes
0 answers
96 views

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 ...
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 ...
45 votes
0 answers
909 views

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\}...
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: $\Bbb R^d\setminus \bigcup H_i$. ...
1 vote
0 answers
95 views

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+\...
2 votes
1 answer
216 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 &...
3 votes
0 answers
123 views

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 ...
9 votes
1 answer
294 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 ...
8 votes
2 answers
797 views

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 ...
5 votes
0 answers
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 $R$ ...
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\|_\...
2 votes
1 answer
128 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 ...
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 $...
1 vote
1 answer
135 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 ...
1 vote
0 answers
148 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 ...
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 ...
9 votes
1 answer
151 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 ...
3 votes
0 answers
95 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 ...
2 votes
0 answers
106 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 ...
4 votes
0 answers
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 $\...
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 $...
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 \...
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 ...
1 vote
2 answers
169 views

How does one map regression depth to undirected depth of a point?

The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray ...
1 vote
1 answer
103 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\...
1 vote
1 answer
103 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 $l^p_- = -l^p_+$....
2 votes
1 answer
265 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 (...
15 votes
2 answers
731 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 ...
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 ...
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 ...
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 ...
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 \...
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 $\...
0 votes
0 answers
189 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 ...
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 ...
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 ...
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? «...
2 votes
0 answers
80 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 ...
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....
11 votes
2 answers
340 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 ...
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 \...
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-...
4 votes
0 answers
209 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):=...
2 votes
0 answers
75 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 ...